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# Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness.

Pascal Auscher, Cyril Imbert, Lukas Niebel

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Pascal Auscher, Cyril Imbert, Lukas Niebel. Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness.. 2024. hal-04519638v2

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# Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness.

Pascal Auscher, Cyril Imbert, Lukas Niebel

► To cite this version:

Pascal Auscher, Cyril Imbert, Lukas Niebel. Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness.. 2025. hal-04519638v3

HAL Id: hal-04519638

https://hal.science/hal-04519638v3

Preprint submitted on 5 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# WEAK SOLUTIONS TO KOLMOGOROV-FOKKER-PLANCK EQUATIONS: REGULARITY, EXISTENCE AND UNIQUENESS

PASCAL AUSCHER, CYRIL IMBERT, AND LUKAS NIEBEL

ABSTRACT. In this article, we establish embeddings à la Lions and transfer of regularity à la Bouchut for a large scale of kinetic spaces. We use them to identify a notion of weak solutions to Kolmogorov-Fokker-Planck equations with (local or integral) diffusion and rough (measurable) coefficients under minimal requirements. We prove their existence and uniqueness for a large class of source terms, first in full space for the time, position and velocity variables and then for the kinetic Cauchy problem on infinite and finite time intervals.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Homogeneous kinetic spaces and Kolmogorov operators</td><td>9</td></tr><tr><td>3. Uniqueness and kinetic embeddings</td><td>20</td></tr><tr><td>4. Construction of weak solutions to rough equations</td><td>24</td></tr><tr><td>5. Kinetic Cauchy problems</td><td>27</td></tr><tr><td>6. The special case of local diffusion operators</td><td>32</td></tr><tr><td>7. Extensions and link with other kinetic embeddings</td><td>36</td></tr><tr><td>References</td><td>40</td></tr></table>

## 1. INTRODUCTION

1.1. **Kolmogorov-Fokker-Planck equations.** This work is concerned with the construction of weak solutions for kinetic equations of Kolmogorov-Fokker-Planck type. The prototypical example is given by the equation considered by A. Kolmogorov in [22],

$$ (1.1) \qquad (\partial_t + v \cdot \nabla_x)f - \Delta_v f = S, \quad (t, x, v) \in \Omega. $$

The unknown function $f = f(t, x, v)$ of the time $t$, position $x$ and velocity $v$ variables is defined in (a subset of) $\Omega = \mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$ and the source term $S$ is given. We are

<footnote>Date: March 25, 2024.</footnote>

<footnote n="2010">2010 Mathematics Subject Classification. Primary: 35K65, 35R05, 35D30, 35Q84, 35R09 Secondary: 35K70, 35B65.</footnote>

<footnote>Key words and phrases. Kolmogorov-Fokker-Planck equations, kinetic spaces, transfer of regularity, weak solutions, kinetic Cauchy problems.</footnote>

<footnote>The third author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics-Geometry-Structure.</footnote>
# WEAK SOLUTIONS TO KOLMOGOROV-FOKKER-PLANCK EQUATIONS: REGULARITY, EXISTENCE AND UNIQUENESS

PASCAL AUSCHER, CYRIL IMBERT, AND LUKAS NIEBEL

**ABSTRACT.** We prove existence, uniqueness and regularity of weak solutions of Kolmogorov-Fokker-Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the Cauchy problem and allow a broad class of source terms under minimal assumptions. The core of the analysis is a set of sharp kinetic embeddings à la Lions and transfer-of-regularity results à la Bouchut-Hörmander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Homogeneous kinetic spaces and Kolmogorov operators</td><td>13</td></tr><tr><td>3. Uniqueness and kinetic embeddings</td><td>25</td></tr><tr><td>4. Construction of weak solutions to rough equations when $\beta < d/2$</td><td>30</td></tr><tr><td>5. Kinetic Cauchy problems</td><td>33</td></tr><tr><td>6. Statements on inhomogeneous spaces</td><td>36</td></tr><tr><td>7. Proof of Theorem 1.5 and Theorem 5.2 when $\beta \ge d/2$.</td><td>38</td></tr><tr><td>8. The special case of local diffusion operators</td><td>40</td></tr><tr><td>9. Link with other kinetic embeddings</td><td>44</td></tr><tr><td>References</td><td>47</td></tr></table>

## 1. INTRODUCTION

Motivated by the study of non-linear kinetic equations, this work is concerned with proving the existence and uniqueness of weak solutions for linear kinetic Kolmogorov-Fokker-Planck equations of both local and non-local (fractional) type, with rough diffusion coefficients or kernels. To the best of our knowledge, this work establishes for

Date: December 5, 2025.

2010 Mathematics Subject Classification. Primary: 35K65, 35R05, 35D30, 35Q84, 35R09 Secondary: 35K70, 35B65.

Key words and phrases. Kolmogorov-Fokker-Planck equations, kinetic spaces, transfer of regularity, weak solutions, kinetic Cauchy problems.

The third author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 - 390685587, Mathematics Münster: Dynamics-Geometry-Structure. A CC-BY 4.0 https://creativecommons.org/licenses/by/4.0/ public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.
also interested in the fractional counterpart of the previous equation [19, 27, 28],

$$ (1.2) \qquad (\partial_t + v \cdot \nabla_x) f + (-\Delta_v)^\beta f = S $$

where $\beta \in (0, 1)$ – see below for a definition of the fractional Laplacian $(-\Delta_v)^\beta$.

Motivated by the study of non-linear equations, we choose a framework that encompasses equations with coefficients. For this reason, we consider equations under the following general form,

$$ (1.3) \qquad (\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S $$

where $\mathcal{A}$ is a linear operator acting on the velocity variable $v$ (in a uniform fashion with respect to the other variables $t, x$). Kinetic Fokker-Planck equations with rough coefficients correspond to

$$ (1.4) \qquad \mathcal{A} f(t, x, v) = -\nabla_v \cdot (\mathbf{A} \nabla_v f)(t, x, v) $$

where $\mathbf{A} = \mathbf{A}(t, x, v)$ is a real symmetric matrix satisfying for a.e. $(t, x, v) \in \Omega$, eigenvalues of $\mathbf{A}(t, x, v)$ lie in $[\lambda, \Lambda]$

for some $0 < \lambda \le \Lambda < \infty$. Another important class of examples are kinetic equations with integral diffusion. They correspond to

$$ (1.5) \qquad \mathcal{A} f(t, x, v) = \text{PV} \int_{\mathbb{R}^d} (f(t, x, v) - f(t, x, v')) k(t, x, v, v') \, dv' $$

where the function $k: \Omega \times \mathbb{R}^d \to \mathbb{R}$ is non-negative and satisfies appropriate ellipticity conditions. The main example is $\mathcal{A}f(t, x, v) = (-\Delta_v)^\beta f(t, x, v)$ for which $k(t, x, v, v') = c_{d,\beta}|v' - v|^{-d-2\beta}$ for some explicit constant $c_{d,\beta} > 0$. In the general integro-differential case, since the kernel $k$ can be singular, the meaning of $\mathcal{A}f$ has to be understood in the principal value (PV) sense. The simplest form of the ellipticity conditions for the kernel $k$ are the following ones,

$$ \left\{ \begin{array}{l} k(t, x, v, v') = k(t, x, v', v) \ge 0, \\ \frac{\lambda}{|v'-v|^{d+2\beta}} \le k(t, x, v, v') \le \frac{\Lambda}{|v'-v|^{d+2\beta}}. \end{array} \right. $$

These conditions are considered in many articles concerned with the regularity of parabolic equations with integral diffusion, see for instance [9, 10, 21]. Our approach allows us to deal with a larger class of kernels, satisfying weaker ellipticity conditions. More precisely, our results apply to the class of kinetic equations with integral diffusion introduced in [19]. They naturally appear in the context of the study of the Boltzmann equation without Grad's cut-off assumption. See Remark 1.8 below.

## 1.2. Kinetic embedding and transfer of regularity.
A question one may ask is: what is the correct definition of a weak solution? By correct, we mean a definition that allows one to prove existence and uniqueness even for rough coefficients or kernels. Recall that the linear operator $\mathcal{A}$ is diffusive in the velocity variable so one necessarily requires regularity of order $\beta$ in the $v$-variable. Do we need more a priori regularity conditions in the definition? The answer is no. Indeed, it is known since the discovery of hypoellipticity and averaging lemmas that the transport operator transfers $v$-regularity to the other variables. The expected amount of transfered regularity in the $x$ variable with respect to the diffusion parameter $\beta > 0$ can be computed by the underlying scaling invariance $(t, x, v) \mapsto (r^{2\beta}t, r^{2\beta+1}x, rv)$ of the
the first time the well-posedness of the non-local Kolmogorov–Fokker–Planck equation with a rough diffusion kernel. An essential ingredient in our proof is a kinetic transfer-of-regularity result for weak solutions, which appears here for the first time in the non-local case and improves on the results of Hörmander [21] and Bouchut [10] in the local case. In all cases, we obtain the regularity exponents implied by the kinetic scaling. Kinetic regularity also allows us to deal with large classes of source terms. In particular, this precise solution theory allows us to define a notion of fundamental solution for such equations (presented in a subsequent work [4]).

In this introduction, we first define the class of equations and their weak solutions, and we state our main existence and uniqueness result together with some consequences. Then we present our main instrumental result on kinetic regularity. We next highlight a few aspects of our proofs and present tools that we use. We also explain the recourse to homogeneous Sobolev spaces to obtain estimates that are invariant under kinetic scaling. We continue with a short review of the literature and the state of the art, and we describe the organisation of the article. This work is the first of a series, and we also describe subsequent papers.

1.1. **Kolmogorov–Fokker–Planck equations.** The equations take the following general form:

$$ (1.1) \qquad (\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S, $$

where the unknown function $f = f(t, x, v)$, depending on time $t$, position $x$, and velocity $v$, is defined on $\Omega = \mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$, with $d \ge 1$, and the source term $S$ is given. The operator $\mathcal{A}$ is a diffusive linear operator acting on the velocity variable $v$. Later, we shall also consider restrictions of time to finite intervals, still with $x$ and $v$ in $\mathbb{R}^d$.

A first example is the kinetic Fokker–Planck equation corresponding to

$$ (1.2) \qquad \mathcal{A} f(t, x, v) = -\nabla_v \cdot (\mathbf{A} \nabla_v f)(t, x, v) $$

where $\mathbf{A} = \mathbf{A}(t, x, v)$ is a real symmetric matrix satisfying for a.e. $(t, x, v) \in \Omega$,

eigenvalues of $\mathbf{A}(t, x, v)$ lie in $[\lambda, \Lambda]$

for some $0 < \lambda \le \Lambda < \infty$. We shall see that coercivity and boundedness of the matrix $\mathbf{A}$ are sufficient (real symmetry plays no role), and no regularity other than measurability of the coefficients is required. The case $-\mathcal{A} = \Delta_v$, the Laplace operator in the variable $v$, yields the well-known Kolmogorov equation.

Another important class of examples consists of kinetic equations with integral diffusion. They correspond to

$$ (1.3) \qquad \mathcal{A} f(t, x, v) = \mathrm{PV} \int_{\mathbb{R}^d} (f(t, x, v) - f(t, x, v')) k(t, x, v, v') \, dv' $$

where the function $k: \Omega \times \mathbb{R}^d \to \mathbb{R}$ is non-negative and satisfies appropriate ellipticity conditions. The main example is $\mathcal{A}f(t, x, v) = (-\Delta_v)^\beta f(t, x, v)$ for which $k(t, x, v, v') = c_{d,\beta}|v' - v|^{-d-2\beta}$, $0 < \beta < 1$, for some explicit constant $c_{d,\beta} > 0$ and its regularity theory will be instrumental. In the general integro-differential case, since the kernel $k$ can be singular, the meaning of $\mathcal{A}f$ has to be understood in the principal value (PV) sense. The simplest form of the ellipticity condition on the kernel $k$ is the
fractional Kolmogorov equation. Our first main result quantifies this in a way that is both reminiscent of Bouchut’s transfer of regularity result in [8] and Lions’ embedding theorem in [24].

**Theorem 1.1** (Kinetic embedding and transfer of regularity). Let $\beta \in (0, d/2)$. For any $f \in D'(\Omega)$ such that $f \in L_{t,x}^2 \dot{H}_v^\beta$ and $(\partial_t + v \cdot \nabla_x)f = S_1 + S_2 + S_3$, with $S_1 \in L_{t,x}^2 \dot{H}_v^{-\beta}$, $S_2 \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}}$ and $S_3 \in L_t^1 L_{x,v}^2$, we have,

(i) $f \in L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2)$ with

$$ \|f\|_{L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}}} + \sup_{t \in \mathbb{R}} \|f(t)\|_{L_{x,v}^2} \lesssim_{d,\beta} \|f\|_{L_{t,x}^2 \dot{H}_v^\beta} + \|S_1\|_{L_{t,x}^2 \dot{H}_v^{-\beta}} + \|S_2\|_{L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}}} + \|S_3\|_{L_t^1 L_{x,v}^2}. $$

(ii) The map $t \mapsto \|f(t)\|_{L_{x,v}^2}$ is absolutely continuous on $\mathbb{R}$ with

$$ (1.6) \quad \frac{d}{dt} \|f(t)\|_{L_{x,v}^2}^2 = 2 \operatorname{Re} \left( \int_{\mathbb{R}^d} \langle S_1, f \rangle \, dx + \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv + \iint_{\mathbb{R}^{2d}} S_3 \bar{f} \, dx \, dv \right). $$

In (1.6), the first bracket denotes the sesquilinear duality between $\dot{H}_v^{-\beta}$ and $\dot{H}_v^{\beta}$, giving an integrable function of $(t, x)$, and the second one refers to the duality between $\dot{H}_x^{-\frac{\beta}{2\beta+1}}$ and $\dot{H}_x^{\frac{\beta}{2\beta+1}}$, giving an integrable function of $(t, v)$. Integrals are in the sense of Lebesgue.

**Remark 1.2.** We use homogeneous spaces and obtain scale invariant estimates. The homogeneous Sobolev spaces are defined in Section 2 where the limitation on $\beta$ will be explained.

**Remark 1.3.** In Section 6 we also consider a different definition in the local case $\beta = 1$ that eliminates the constraint on $\beta$.

**Remark 1.4.** We shed light on the fact that we do not assume a priori control on $f(t) \in L_{x,v}^2$ at any time, which could look surprising. In Section 7, we will see that assuming it, but only qualitatively, allows to deal with $\beta \ge d/2$.

**Remark 1.5.** We allow the free transport of $f$ to belong to a larger space than just the dual of $L_{t,x}^2 \dot{H}_v^\beta$.

**Remark 1.6.** Not only do we obtain continuity of $t \mapsto f(t) \in L_{x,v}^2$ but we also get absolute continuity of $t \mapsto \|f(t)\|_{L_{x,v}^2}$ and zero limit at infinity, both being crucial when coming to weak solutions.

**Remark 1.7.** Theorem 1.1 is a special case of a more general result, see Theorem 3.6, which itself is a consequence of embeddings obtained for a scale of kinetic spaces, see Theorem 2.2. Such generalizations could be useful for further developments in the field for weak solutions and we shall prove them in this article.
following:

$$ \left\{ \begin{array}{l} k(t, x, v, v') = k(t, x, v', v), \\ \frac{\lambda}{|v'-v|^d} \le k(t, x, v, v') \le \frac{\Lambda}{|v'-v|^{d+2\beta}}. \end{array} \right. $$

These conditions are considered in many articles concerned with the regularity of parabolic equations with integral diffusion, see for instance [11, 12, 25]. Our approach allows us to deal with a larger class of kernels, satisfying weaker ellipticity conditions. More precisely, our results apply to the class of kinetic equations with integral diffusion introduced in [23]. They naturally appear in the context of the study of the Boltzmann equation without Grad's cut-off assumption. See Remark 1.1 below.

## 1.2. Weak solutions: definition, main result and consequences.
Not only do we want to solve Cauchy problems on finite time intervals, but also to obtain estimates that are scale invariant. Thus, we start with homogeneous assumptions to state our main result.

For the Kolmogorov-Fokker-Planck equation (1.1) with $A = (-\Delta_v)^\beta$, distributional solutions can be built via Fourier transform. For rough coefficients, we shall use a form method.

The inhomogeneous Sobolev spaces $H_v^\beta$ on $\mathbb{R}^d$ will be defined in a standard way. By using the Fourier transform, the homogeneous semi-norms $\|f\|_{\dot{H}_v^\beta} = \|D_v^\beta f\|_{L_v^2}$, with $D_v = (-\Delta_v)^{1/2}$, make sense for all $\beta \in \mathbb{R}$ when, say, $f \in L_v^2$ to start with. A completion of the space $f \in L_v^2$ for the semi-norm $\|f\|_{\dot{H}_v^\beta}$ can always be obtained within the space of tempered distributions modulo polynomials. But this is not convenient when dealing with partial differential equations. Instead, one looks for a completion contained in the space of tempered distributions, which we denote by $\dot{H}_v^\beta$ and refer to as a homogeneous Sobolev space. When $\beta < d/2$, this completion exists as a Hilbert space (see Section 2 for a definition). Otherwise, if $\beta \ge d/2$, such a completion exists but only as a semi-normed space, which poses technical difficulties in proofs and statements, so we decide not to consider them in this article,<fnref n="1" /> although the norms above are still defined when, in addition, $f \in L_v^2$.

In any case for $\beta > 0$, $\dot{H}_v^\beta \cap L_v^2 = H_v^\beta$. Hence, the constraint $\beta < d/2$ will arise in some proofs for technical reasons, and the case $\beta \ge d/2$ needs to be treated differently. We thought it interesting to present things like this as it directly covers the new cases of interest when $d = 2, 3$ and $0 < \beta < 1$.

As suggested above, we want that for all $(t, x) \in \mathbb{R} \times \mathbb{R}^d$, $A$ maps $H_v^\beta$ to $H_v^{-\beta}$ in a uniform fashion and in a scale-invariant way. To make it rigorous and valid for all $\beta > 0$, we assume that $\mathbb{R} \times \mathbb{R}^d \ni (t, x) \mapsto a_{t,x}$ is a family of continuous sesquilinear forms $a_{t,x}: H_v^\beta \times H_v^\beta \to \mathbb{C}$ with the homogeneous estimate

$$ (1.4) \qquad |a_{t,x}(f,g)| \le \Lambda \|f\|_{\dot{H}_v^\beta} \|g\|_{\dot{H}_v^\beta} $$

for some $\Lambda < \infty$, uniformly for $(f,g) \in H_v^\beta \times H_v^\beta$, $(t,x) \in \mathbb{R} \times \mathbb{R}^d$, and that $(t,x) \mapsto a_{t,x}(f,g)$ is measurable (here, and throughout, ambient measure in $(t,x,v)$ is Lebesgue

<footnote n="1">¹With this adequate definition when $\beta \ge d/2$, one can simplify the definition of weak solutions below. Actually, most of our results here in which the threshold $d/2$ appears can be improved to results with a threshold $K/2$ where $K = 2\beta + (2\beta + 2)d$ is the homogeneous dimension for the scaling. We refer to the forthcoming article [5] for details.</footnote>
**1.3. Weak solutions to the Kolmogorov-Fokker-Planck equation.** We explained above that we are interested in constructing solutions of the Kolmogorov-Fokker-Planck equation (1.3) for some general operator $\mathcal{A}$. Keeping in mind examples (1.4) and (1.5), we now make precise the definition of this operator and the assumptions that we need. As mentioned above, we want that for all $(t, x) \in \mathbb{R} \times \mathbb{R}^d$, $\mathcal{A}$ maps $\dot{\mathbf{H}}_v^\beta$ to $\dot{\mathbf{H}}_v^{-\beta}$ in a uniform fashion. To make it rigorous, we assume that $\mathbb{R} \times \mathbb{R}^d \ni (t, x) \mapsto a_{t,x}$ is a family of continuous sesquilinear forms $a_{t,x}: \dot{\mathbf{H}}_v^\beta \times \dot{\mathbf{H}}_v^\beta \to \mathbb{C}$ with

$$ (1.7) \qquad |a_{t,x}(f,g)| \le \Lambda \|f\|_{\dot{\mathbf{H}}_v^\beta} \|g\|_{\dot{\mathbf{H}}_v^\beta} $$

for some $\Lambda < \infty$, uniformly for $(f,g) \in \dot{\mathbf{H}}_v^\beta \times \dot{\mathbf{H}}_v^\beta$, $(t,x) \in \mathbb{R} \times \mathbb{R}^d$, and $(t,x) \mapsto a_{t,x}(f,g)$ measurable. We also assume that there exists $\lambda > 0$ such that for all $(t,x) \in \mathbb{R} \times \mathbb{R}^d$ and all $f \in \dot{\mathbf{H}}_v^\beta$,

$$ (1.8) \qquad \mathrm{Re} \, a_{t,x}(f, f) \ge \lambda \|f\|_{\dot{\mathbf{H}}_v^\beta}^2. $$

The space

$$ \dot{\mathbf{F}}^\beta = L_{t,x}^2 \dot{\mathbf{H}}_v^\beta $$

plays a central role and we set

$$ (1.9) \qquad \forall f, g \in \dot{\mathbf{F}}^\beta, \qquad a(f,g) = \iint_{\mathbb{R} \times \mathbb{R}^d} a_{t,x}(f(t,x), g(t,x)) \, dt \, dx. $$

For short, we write $a_{t,x}(f(t,x), g(t,x))$ as $a_{t,x}(f,g)$. Then, $a$ is a continuous and coercive sesquilinear form on $\dot{\mathbf{F}}^\beta \times \dot{\mathbf{F}}^\beta$. Denote by $(\dot{\mathbf{F}}^\beta)'$ the dual of $\dot{\mathbf{F}}^\beta$ realized in the duality extending the $L_{t,x,v}^2$ inner product and $\mathcal{A}: \dot{\mathbf{F}}^\beta \to (\dot{\mathbf{F}}^\beta)'$ the isomorphism associated to $a$ by

$$ \langle \mathcal{A} f, g \rangle = a(f, g), \quad f, g \in \dot{\mathbf{F}}^\beta. $$

**Remark 1.8.** We can work with weaker forms of boundedness and ellipticity conditions using inhomogeneous spaces, see (5.3) and (5.4). The situation is well-understood when $\mathcal{A}$ is given by (1.4). When $\mathcal{A}$ is given by a diffusion kernel (1.5), [19] gives sufficient conditions on the kernel for obtaining the weaker forms: see Condition (1.3) with $\bar{R} = +\infty$ for ellipticity and Conditions (1.4), (1.5), (1.6), (1.7) with $\bar{R} = +\infty$ for boundedness. The interested reader is referred to [19, Theorem 4.1] and [26, Theorem 2.1].

With the kinetic embedding and the transfer of regularity at hand, and the general form of the elliptic operator $\mathcal{A}: L_{t,x}^2 \dot{\mathbf{H}}_v^\beta \to L_{t,x}^2 \dot{\mathbf{H}}_v^{-\beta}$ with rough coefficients, we define weak solutions of $(\partial_t + v \cdot \nabla_x) + \mathcal{A}$ and state our second main result concerned with their construction for fairly general source terms $S$ in appropriate spaces.

**Definition 1.9 (Weak solutions).** Let $\beta \in (0, d/2)$ and $S \in \mathcal{D}'(\Omega)$. A distribution $f \in \mathcal{D}'(\Omega)$ is said to be a weak solution to (1.3) if $f \in L_{t,x}^2 \dot{\mathbf{H}}_v^\beta$ and for all $h \in \mathcal{D}(\Omega)$,

$$ - \iiint_{\Omega} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + a(f, h) = \langle S, h \rangle. $$

**Remark 1.10.** We may also consider the adjoint equation

$$ (1.10) \qquad -(\partial_t + v \cdot \nabla_x) f + \mathcal{A}^* f = S, $$
measure). We also assume homogeneous coercivity, namely that there exists $\lambda > 0$ such that for all $(t, x) \in \mathbb{R} \times \mathbb{R}^d$ and all $f \in H_v^\beta$,

$$ (1.5) \qquad \operatorname{Re} a_{t,x}(f, f) \geq \lambda \|f\|^2_{H_v^\beta}. $$

The mixed space

$$ F^\beta = L_{t,x}^2 H_v^\beta $$

(and its homogeneous version $\dot{F}^\beta = L_{t,x}^2 \dot{H}_v^\beta$ when $\beta < d/2$) plays a central role and we set

$$ (1.6) \qquad \forall f, g \in F^\beta, \qquad a(f, g) = \iint_{\mathbb{R} \times \mathbb{R}^d} a_{t,x}(f(t, x), g(t, x)) \, dt \, dx. $$

For short, we write $a_{t,x}(f(t, x), g(t, x))$ as $a_{t,x}(f, g)$. We define by $\mathcal{A}: F^\beta \to (F^\beta)'$, the complex dual of $F^\beta$ realized in the extended $L_{t,x,v}^2$ inner product, the operator associated to $a$ by

$$ \langle \mathcal{A} f, g \rangle = a(f, g), \quad f, g \in F^\beta. $$

When $\beta < d/2$, this can be extended to $\dot{F}^\beta$: In this case, $a$ (identified with its extension) becomes a continuous and coercive sesquilinear form on $\dot{F}^\beta \times \dot{F}^\beta$ and $\mathcal{A}: \dot{F}^\beta \to (\dot{F}^\beta)'$ is an isomorphism. We shall not distinguish notation.

**Remark 1.1.** We can work with weaker inhomogeneous formulations of boundedness and ellipticity conditions, see (6.1) and (6.2). The situation is well-understood when $\mathcal{A}$ is given by (1.2) plus lower order terms. When $\mathcal{A}$ is given by a diffusion kernel (1.3), the second author and L. Silvestre give sufficient conditions in [23] on the kernel for obtaining the weaker forms: see Condition (1.3) with $\bar{R} = +\infty$ for ellipticity and Conditions (1.4), (1.5), (1.6), (1.7) with $\bar{R} = +\infty$ for boundedness. The interested reader is referred to [23, Theorem 4.1] and [30, Theorem 2.1].

We now define our classes of weak solutions to the equation $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ and state our main result concerned with their construction for fairly general source terms $S$ in appropriate spaces. Define

$$ \dot{U}^\beta = \begin{cases} L_{t,x}^2 \dot{H}_v^\beta & , \text{ if } \beta < d/2, \\ L_{\text{loc},t}^2 L_{x,v}^2 \cap L_{t,x}^2 \dot{H}_v^\beta & , \text{ if } \beta \ge d/2, \end{cases} $$

and set

$$ \|f\|_{\dot{U}^\beta} = \|f\|_{L_{t,x}^2 \dot{H}_v^\beta} = \|D_v^\beta f\|_{L_{t,x,v}^2} $$

which makes sense for all $\beta > 0$ by the above remarks. The threshold $d/2$ is only there because our definition here of the homogeneous Sobolev spaces has this restriction.

**Definition 1.2** (Weak solutions). Let $S \in \mathcal{D}'(\Omega)$. A distribution $f \in \mathcal{D}'(\Omega)$ is said to be a weak solution to (1.1) if $f \in \dot{U}^\beta$ and for all $h \in \mathcal{D}(\Omega)$,

$$ - \iiint_{\Omega} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + a(f, h) = \langle S, h \rangle. $$

**Remark 1.3.** We may also consider the adjoint equation

$$ (1.7) \qquad -(\partial_t + v \cdot \nabla_x) f + \mathcal{A}^* f = S, $$
where $\mathcal{A}^*$ is the adjoint of $\mathcal{A}$. We refer to it as the backward kinetic equation and we define weak solutions similarly. The theory we develop will need to consider both equations (1.3) and (1.10).

**Remark 1.11.** We do not assume more defining conditions for weak solutions than being in $L_{t,x}^2 \dot{H}_v^\beta$ and in particular no a priori $L_{x,v}^2$ condition like the common assumption $L_{t,loc}^\infty L_{x,v}^2$ (see for example [12, 15, 16, 19]). Our definition of $\dot{H}_v^\beta$ and the restriction $0 < \beta < d/2$ will guarantee that a weak solution belongs to $L_{\text{loc}}^2$ in all variables and is a tempered distribution, that is $f \in S'(\Omega)$. Thus the source $S$ must also be in $S'(\Omega)$. The sesquilinear duality $\langle S, h \rangle$ between the distribution $S$ and test functions will be specified depending on the assumptions on $S$. In Section 7, we explain how to treat the case $\beta \ge d/2$.

**Theorem 1.12 (Existence and uniqueness of weak solutions).** Let $\beta \in (0, d/2)$. Assume that the family of sesquilinear forms satisfies (1.7) and (1.8). Let $S = S_1 + S_2 + S_3$ with $S_1 \in L_{t,x}^2 \dot{H}_v^{-\beta}$, $S_2 \in L_{t,x}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}}$ and $S_3 \in L_t^1 L_{x,v}^2$. There exists a unique weak solution $f \in L_{t,x}^2 \dot{H}_v^\beta$ to (1.3) in the sense of Definition 1.9. Moreover, this solution belongs to $L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2)$ and

$$ \sup_{t \in \mathbb{R}} \|f(t)\|_{L_{x,v}^2} + \|D_x^{\frac{\beta}{2\beta+1}} f\|_{L_{t,x,v}^2} + \|D_v^\beta f\|_{L_{t,x,v}^2} \\ \lesssim \|D_x^{-\frac{\beta}{2\beta+1}} S_1\|_{L_{t,x,v}^2} + \|D_v^{-\beta} S_2\|_{L_{t,x,v}^2} + \|S_3\|_{L_t^1 L_{x,v}^2} $$

for an implicit constant depending only on $d, \beta, \lambda, \Lambda$, where $D_x$ and $D_v$ denote $(-\Delta_x)^{\frac{1}{2}}$ and $(-\Delta_v)^{\frac{1}{2}}$, respectively. In addition, the solution satisfies the energy equality: for any $s, t \in \mathbb{R}$ with $s < t$,

$$ \|f(t)\|_{L_{x,v}^2}^2 + 2 \operatorname{Re} \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau = \\ \qquad (1.11) \quad \|f(s)\|_{L_{x,v}^2}^2 + 2 \operatorname{Re} \int_s^t \left( \int_{\mathbb{R}^d} \langle S_1, f \rangle \, dx + \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv + \iint_{\mathbb{R}^{2d}} S_3 \bar{f} \, dx \, dv \right) \, d\tau. $$

**Remark 1.13.** As the problem is linear, all sources that belong to any interpolation space that can be obtained from the above three ones give rise to a weak solution. We shall not attempt to write such statements.

**Remark 1.14.** The time variable describes $\mathbb{R}$ and we obtain homogeneous estimates. We also construct solutions for the kinetic Cauchy problem on half-infinite time intervals with homogeneous estimates, see Theorem 5.2.

**Remark 1.15.** We extend our approach to inhomogeneous spaces and construct solutions for the kinetic Cauchy problem in the corresponding inhomogeneous framework on finite time intervals, under weaker ellipticity conditions (5.3) and (5.4), see Theorem 5.7.

**Remark 1.16.** The same statements and remarks hold for the backward equations corresponding to the operator in (1.10).
where $\mathcal{A}^*$ is the adjoint of $\mathcal{A}$. We refer to it as the backward kinetic equation, and we define its weak solutions similarly. The theory we develop will need to consider both equations (1.1) and (1.7).

**Remark 1.4.** We do not assume $L_{\text{loc},t}^{\infty} L_{x,v}^{2}$ in this definition (compare, for example, [16, 19, 20, 23]). The choice of $L_{\text{loc},t}^{2} L_{x,v}^{2}$ when $\beta \ge d/2$ is somewhat arbitrary. It would suffice to have a subspace of $L_{\text{loc}}^{1}(\Omega)$ that embeds in $L_{\text{loc},t}^{1} S'_{x,v}$ (the space of locally integrable functions on $\mathbb{R}_t$ valued in the Schwartz space $S'_{x,v} = S(\mathbb{R}^{2d})$) and on which we can compute $\|D_{v}^{\beta} f\|_{L_{t,x,v}^{2}}$. This is because we use the partial Fourier transform in $(x, v)$ in our arguments. In the end, the weak solutions we construct belong to $C_{0}(\mathbb{R}_{t}; L_{x,v}^{2})$ anyway.

**Theorem 1.5 (Existence and uniqueness of weak solutions).** Let $\beta > 0$. Assume that the family $\mathcal{A}$ of sesquilinear forms satisfies (1.4) and (1.5). Let $S = S_1 + S_2 + S_3$ with $S_1 \in L_{t,x}^2 \dot{H}_v^{-\beta}$, $S_2 \in L_{t,x}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}}$ and $S_3 \in L_t^1 L_{x,v}^2$. There exists a unique weak solution $f \in \dot{U}^{\beta}$ to (1.1) in the sense of Definition 1.2. Moreover, $f \in C_0(\mathbb{R}_t; L_{x,v}^2)$ and $f \in L_{t,x}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}}$ with

$$ \sup_{t \in \mathbb{R}} \|f(t)\|_{L_{x,v}^2} + \|D_x^{\frac{\beta}{2\beta+1}} f\|_{L_{t,x,v}^2} + \|D_v^{\beta} f\|_{L_{t,x,v}^2} \\ \lesssim \|D_v^{-\beta} S_1\|_{L_{t,x,v}^2} + \|D_x^{-\frac{\beta}{2\beta+1}} S_2\|_{L_{t,x,v}^2} + \|S_3\|_{L_t^1 L_{x,v}^2} $$

for an implicit constant depending only on $d, \beta, \lambda, \Lambda$, and where $D_x$ and $D_v$ denote $(-\Delta_x)^{\frac{1}{2}}$ and $(-\Delta_v)^{\frac{1}{2}}$, respectively. In addition, the solution satisfies the energy equality: for any $s, t \in \mathbb{R}$ with $s < t$,

$$ \|f(t)\|_{L_{x,v}^2}^2 + 2 \operatorname{Re} \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau = \\ \qquad (1.8) \quad \|f(s)\|_{L_{x,v}^2}^2 + 2 \operatorname{Re} \int_s^t \left( \int_{\mathbb{R}^d} \langle S_1, f \rangle \, dx + \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv + \iint_{\mathbb{R}^{2d}} S_3 \bar{f} \, dx \, dv \right) \, d\tau. $$

**Remark 1.6.** Let us explain the statement. The notation $C_0(\mathbb{R}; X)$ is the space of continuous functions on $\mathbb{R}$ taking values in $X$ with zero limits at $\pm\infty$. Note that since the solution $f(t) \in L_{x,v}^2$ at all times, the expressions $D_v^\beta f$ and $D_x^{\frac{\beta}{2\beta+1}} f$ make sense, so that $f \in H_v^\beta$ and $f \in H_x^{\frac{\beta}{2\beta+1}}$, at least almost everywhere in the other variables. But the point of this formulation is that estimates are homogeneous. In (1.8), the bracket $\langle S_1, f \rangle$ denotes the extension of the sesquilinear duality on $L_v^2$ to $\dot{H}_v^{-\beta} \times H_v^\beta$, giving an integrable function of $(t, x)$, and the second one $\langle S_2, f \rangle$ refers to the extension of the sesquilinear duality on $L_x^2$ to $\dot{H}_x^{-\frac{\beta}{2\beta+1}} \times H_x^{\frac{\beta}{2\beta+1}}$, giving an integrable function of $(t, v)$. Integrals are in the sense of Lebesgue.

**Remark 1.7.** As the problem is linear, all sources that belong to any interpolation space that can be obtained from the above three ones give rise to a weak solution. We shall not attempt to write such statements.

Note that the source terms $S_1$ have been considered in the literature (see below), but having the source terms $S_2$ and $S_3$ is new.
**1.4. Highlights and observations.** The theory of kinetic equations related to this work is much developed and we shall provide a review. We wish to highlight some points in our contribution and make a few observations on tools that we use or not.

The Hilbertian framework proposed by J.-L. Lions to construct weak solutions had already been used for some particular kinetic equations. We show it applies in full generality, thanks to the complete description of the kinetic embedding and transfer of regularity for scales of adapted kinetic spaces: it allows us to obtain uniqueness and also to obtain existence with more source terms.

A second highlight concerns the strategy of proof of the kinetic embedding and transfer of regularity: it boils down to estimates and appropriate uniqueness of distributional solutions of the constant coefficients Kolmogorov equation $(\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^\beta f = S$ and/or its adjoint. This also shows isomorphism properties of the integral Kolmogorov operators which is interesting in its own right.

As for the techniques of proofs, they only use partial Fourier transformation for the position and velocity variables and the Galilean change of variables. Using the homogeneous norms also provides a very convenient setup in which estimates are quite elementary and do not require subtle decompositions in Fourier space.

We also mention that there are common tools in the field that we do not need. We neither use the Galilean group law associated with the fields $(\partial_t + v \cdot \nabla_x)$, $\nabla_v$ nor rely on commutator techniques à la Hörmander in the context of hypoelliptic equations.

Our final observation is that we do not use the local regularity theory of weak solutions, such as the De Giorgi-Nash-Moser estimates. In other words, the construction of weak solutions does not require knowing that weak solutions have local regularity properties.

**1.5. From Lions embedding to kinetic embeddings.** We think it would be useful for the reader to focus on the genesis of our kinetic embedding and the subtlety of the homogeneous situation by describing the parabolic case where there is no transport and no position variable $x$.

The starting point of J.-L. Lions in his 1957 article [24] is his famous and well-known abstract embedding. Three Hilbert spaces $(V, H, V')$ form a *Gelfand triple* if there are continuous and dense inclusions $V \hookrightarrow H \hookrightarrow V'$. With such a triple, Lions established that if $f \in L^2(0, T; V)$ and $f \in H^1(0, T; V')$ then $f \in C([0, T]; H)$ and $t \mapsto \|f(t)\|_H^2$ is absolutely continuous. His proof uses the *a priori* knowledge that $u(t)$ exists in $H$ almost everywhere, approximation procedures that are compatible in the three spaces $V, H, V'$ and integration theory for vector-valued functions.

For example, with $V = H^1(\mathbb{R}^d)$ the classical Sobolev space, this shows that if $f \in L^2(0, T; H^1(\mathbb{R}^d))$ and $f \in H^1(0, T; H^{-1}(\mathbb{R}^d))$ then $f \in C([0, T]; L^2(\mathbb{R}^d))$. Given the natural scaling, the embedding could be envisioned with $H^1, H^{-1}$ replaced by their homogeneous versions $\dot{H}^1, \dot{H}^{-1}$: however, it is not true when $T < \infty$, for the simple reason that both embeddings $\dot{H}^1(\mathbb{R}^d) \hookrightarrow L^2(\mathbb{R}^d) \hookrightarrow \dot{H}^{-1}(\mathbb{R}^d)$ fail. Indeed, take any $g \in \dot{H}^1(\mathbb{R}^d)$ which is neither an element of $L^2(\mathbb{R}^d)$ nor $\dot{H}^{-1}(\mathbb{R}^d)$ and consider $f(t) \equiv g \in L^2(0, 1; \dot{H}^1(\mathbb{R}^d)) \cap \dot{H}^1(0, 1; \dot{H}^{-1}(\mathbb{R}^d))$ which cannot be an element of $C([0, 1]; L^2(\mathbb{R}^d))$. In other words, there is no possible temporal trace theory with homogeneous spaces on finite time intervals.
**Remark 1.8.** We also consider the kinetic Cauchy problem on half-infinite time intervals and construct solutions with homogeneous estimates; see Theorem 5.2.

**Remark 1.9.** We extend our approach to inhomogeneous spaces and construct solutions for the kinetic Cauchy problem in the corresponding inhomogeneous framework on finite time intervals, under weaker ellipticity conditions (6.1) and (6.2), see Theorem 6.3.

**Remark 1.10.** The same statements and remarks hold for the backward equations corresponding to the operator in (1.7).

Keeping in mind Remark 1.10, let us mention a corollary of Theorem 1.5, that concerns the boundedness of the following two operators

$$ (1.9) \qquad \mathcal{K}_{\mathcal{A}}^{\pm} = (\pm(\partial_t + v \cdot \nabla_x) + \mathcal{A})^{-1} $$

defined as follows: For $S \in \mathcal{D}'(\Omega)$, $\mathcal{K}_{\mathcal{A}}^{\pm}S$ is the unique weak solution in $\dot{U}^{\beta}$, whenever it exists, to $\pm(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ in the sense of Definition 1.2, respectively. Remark that as $\mathcal{K}_{\mathcal{A}}^{\pm}S$ satisfy an equation and $\mathcal{A}\mathcal{K}_{\mathcal{A}}^{\pm}S \in L_{t,x}^2 \dot{H}_v^{-\beta}$, the transport term $(\partial_t + v \cdot \nabla_x)\mathcal{K}_{\mathcal{A}}^{\pm}S$ is a distribution (tempered if $S \in \mathcal{S}'(\Omega)$). This motivates the introduction of a kinetic space with only conditions on the transport and regularity with respect to the velocity.

**Corollary 1.11 (Boundedness).** Let $\beta > 0$. Let

$$ \begin{align*} \dot{Z}^{\beta} &:= L_{t,x}^{2} \dot{H}_{v}^{-\beta} + L_{t,v}^{2} \dot{H}_{x}^{-\frac{\beta}{2\beta+1}} + L_{t}^{1} L_{x,v}^{2}, \\ \dot{Y}^{\beta} &:= L_{t,x}^{2} \dot{H}_{v}^{\beta} \cap L_{t,v}^{2} \dot{H}_{x}^{\frac{\beta}{2\beta+1}} \cap C_{0}(\mathbb{R}_{t}; L_{x,v}^{2}), \\ \dot{\mathcal{L}}_{\beta}^{\beta} &:= \{f \in \dot{U}^{\beta} ; (\partial_{t} + v \cdot \nabla_{x})f \in \dot{Z}^{\beta}\}, \end{align*} $$

and equip them with the natural norms. Under the assumptions of Theorem 1.5 on $\mathcal{A}$, the maps

$$ (1.10) \qquad \begin{array}{l} \mathcal{K}_{\mathcal{A}}^{\pm} : \dot{Z}^{\beta} \to \dot{Y}^{\beta} \\ \text{are continuous, and} \\ \qquad \pm(\partial_t + v \cdot \nabla_x) + \mathcal{A} : \dot{\mathcal{L}}_{\beta}^{\beta} \to \dot{Z}^{\beta} \end{array} $$

are isomorphisms. In particular,

$$ (1.11) \qquad \| \pm (\partial_t + v \cdot \nabla_x) f + \mathcal{A} f \|_{\dot{Z}^\beta} \sim \|f\|_{\dot{U}^\beta} + \|(\partial_t + v \cdot \nabla_x) f\|_{\dot{Z}^\beta}, \quad f \in \dot{\mathcal{L}}_{\beta}^{\beta} $$

and,

$$ (1.12) \qquad \|\mathcal{K}_{\mathcal{A}}^{\pm} S\|_{\dot{Y}^{\beta}} \lesssim \|S\|_{\dot{Z}^{\beta}} \sim \|\mathcal{K}_{\mathcal{A}}^{\pm} S\|_{\dot{U}^{\beta}} + \|(\partial_t + v \cdot \nabla_x) \mathcal{K}_{\mathcal{A}}^{\pm} S\|_{\dot{Z}^{\beta}}, \quad S \in \dot{Z}^{\beta}. $$

The implicit constants depend only on $\beta, d, \lambda, \Lambda$.

**Remark 1.12.** Note that as we have not defined homogeneous Sobolev spaces with large exponents, it would be more rigorous to write $\dot{Y}^{\beta} = \dot{U}^{\beta} \cap L_{t,v}^{2} \dot{H}_{x}^{\frac{\beta}{2\beta+1}} \cap C_{0}(\mathbb{R}_{t}; L_{x,v}^{2})$ but this amounts to the same in terms of norms (as $\beta/(2\beta + 1) < 1/2 \le d/2$, $\dot{H}_{x}^{\frac{\beta}{2\beta+1}}$ is well-defined). We make the abuse of notation for clarity. This space is a closed subspace of $L_{t,x}^{2} \dot{H}_{v}^{\beta} \cap L_{t,v}^{2} \dot{H}_{x}^{\frac{\beta}{2\beta+1}} \cap L_{t}^{\infty} L_{x,v}^{2}$, the dual space to $\dot{Z}^{\beta}$ for the duality extending the $L_{t,x,v}^{2}$ duality.
This failure is not an obstacle when $T = \infty$. Indeed, it is proved by the first author with S. Monniaux and P. Portal in [5] that the embedding is true provided one works on an infinite interval. The strategy is to prove first a uniqueness result for the backward heat equation so that any distribution $f$ that belongs to $L^2(0, \infty; \dot{H}^1(\mathbb{R}^d)) \cap \dot{H}^1(0, \infty; \dot{H}^{-1}(\mathbb{R}^d))$ is represented as a Duhamel integral for the backward heat equation (up to a constant, depending on how homogeneous Sobolev spaces are defined) with source term $\partial_t f + \Delta f$. As this integral belongs to $C_0([0, \infty); L^2(\mathbb{R}^d))$ by construction, this yields the desired conclusion for the embedding. This can also be done on $\mathbb{R}$.

With another strategy using half-time derivatives, it was shown by the first author and M. Egert in [3] that the condition $f \in \dot{H}^1(\mathbb{R}; \dot{H}^{-1}(\mathbb{R}^d))$, that is, $\partial_t f \in L^2(\mathbb{R}; \dot{H}^{-1}(\mathbb{R}^d))$, can even be relaxed for the embedding, so as to allow more source terms in the equation.

Our proof of continuity in time in Theorem 1.1 will rely on using the strategy of [5] in a kinetic context: working on infinite intervals is required to obtain the scale-invariant estimates; and we allow as general source terms as possible, in the spirit of [3]. We also make it work for fractional diffusion.

**1.6. Review of literature.** This work is concerned with the study of kinetic equations with both local (typically $\beta = 1$) and integral diffusion. It is related to the existing literature steaming from various trends of research.

**1.6.1. Kolmogorov equations.** The Kolmogorov equation was first exhibited by A. Kolmogorov in 1934 in [22]. He derived explicitly the fundamental solution of (1.1), which allows us to observe the regularizing effect of the, at first glance, degenerate diffusion equation. This was the starting point of Hörmander's hypoellipticity theory [17]. One of the examples in his paper gave birth to a trend of research devoted to a class of equations now referred to as ultra-parabolic equations or equations of Kolmogorov type. It was launched by E. Lanconelli and S. Polidoro [23]. In this article, they first considered such equations with constant coefficients, exhibited a Lie group structure associated to the equation (extending the Galilean invariance of the original Kolmogorov equation) and established a Harnack inequality.

**1.6.2. Construction of weak solutions.** Weak solutions to the kinetic Fokker-Planck equation ($\beta = 1$) for constant diffusion coefficients in bounded domains were constructed in inhomogeneous function spaces using Lions existence theorem by J. A. Carrillo in [11]. More recently, D. Albritton, S. Armstrong, J. C. Mourrat and M. Novack [1] defined a kinetic space associated to a class of Fokker-Planck equations ($\beta = 1$) and apply again Lions's variational approach for well-posedness. This space is made of functions $f \in L_{t,x}^2 H_v^1(\gamma dv)$ such that $(\partial_t + v \cdot \nabla_x)f \in L_{t,x}^2 H_v^{-1}(\gamma dv)$. Because a term associated with confinement appears in the equation they are interested in, they work with a Gaussian weight $\gamma$ in the velocity variable. Solutions are proven to be continuous in time with values in $L_{x,v}^2(\gamma dv)$.

K. Nyström and M. Litsgård considered the Cauchy-Dirichlet Problem for the Fokker-Planck equation ($\beta = 1$) in possibly unbounded domains in [25]. Weak solutions are constructed on the full space in the case where coefficients are allowed to be rough using a minimization procedure of a convex functional. For the uniqueness result, coefficients are allowed to be rough only in a compact set; they are assumed to
**Remark 1.13.** The formal adjoint of $K_A^+$ is $(K_A^+)^* = - (\partial_t + v \cdot \nabla_x) + A^*)^{-1} = K_{A^*}^-$.

There is a particularly important special case which is, in fact, proved along the way.

**Corollary 1.14 (The fractional Kolmogorov equation).** Let $\beta > 0$ and assume $A = (-\Delta_v)^\beta$. For $S \in \dot{Z}^\beta$, the weak solutions $K_{(-\Delta_v)^\beta}^\pm S$ are the Kolmogorov solutions (noted $K_\beta^\pm S$) given by the fundamental solution for the forward and backward operators $\pm(\partial_t + v \cdot \nabla_x) + (-\Delta_v)^\beta$.

Of course, the action of the fundamental solution is meant through an extension from a dense class of sources for which the usual formula via the Fourier transform can be used.

**Remark 1.15.** Let us make the following comment, which illustrates our strategy. The story begins with the model operators studied by Kolmogorov, and we only have access to Kolmogorov solutions $K_\beta^\pm S$. We first establish (new) bounds for them, that is, the bounds (1.10) for $K_\beta^\pm$. Next, we prove a novel uniqueness statement for each equation $\pm(\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^\beta f = 0$ when $f \in \dot{U}^\beta$ that implies Lemma 3.4. These two facts already show Corollary 1.14. But we can say more: they imply that $K_\beta^\pm$ is an isomorphism between the source space and the kinetic space, materialised by the equivalences in (1.11) and (1.12). This isomorphism property is at the heart of kinetic embeddings, yielding in particular a posteriori continuity valued in $L^2_{x,v}$ of weak solutions for rough $A$ and that are eventually injected into the proof of Theorem 1.5.

**1.3. Kinetic embedding and transfer of regularity.** As just pointed out, our main tool in establishing the above existence and uniqueness statement is the following result that is independent of any equation. This result is both reminiscent of Bouchut's and Hörmander's transfer of regularity results in [10, 21] (only proved for weak solutions in the local case there) and Lions' embedding theorem for time continuity [28] (used essentially for parabolic equations, which is sometimes called the homogeneous case in kinetic theory as it means no dependence in the position variable). The fractional setting is new, and we also treat a broader class of source terms than has previously been considered in the literature.

**Theorem 1.16 (Kinetic embedding and transfer of regularity).** Let $\beta > 0$. Let $f \in D'(\Omega)$ be such that $f \in \dot{U}^\beta$ and $(\partial_t + v \cdot \nabla_x)f = S_1 + S_2 + S_3$, with $S_1 \in L^2_{t,x} \dot{H}_v^{-\beta}$, $S_2 \in L^2_{t,v} \dot{H}_x^{-\frac{\beta}{2\beta+1}}$ and $S_3 \in L^1_t L^2_{x,v}$, that is $f \in \dot{L}_\beta^\beta$. Then, we have

(i) $f \in C_0(\mathbb{R}_t; L^2_{x,v})$ and $f \in L^2_{t,v} \dot{H}_x^{\frac{\beta}{2\beta+1}}$ with

$$ \| D_x^{\frac{\beta}{2\beta+1}} f \|_{L^2_{t,x,v}} + \sup_{t \in \mathbb{R}} \| f(t) \|_{L^2_{x,v}} $$

$$ \lesssim_{d,\beta} \| D_v^\beta f \|_{L^2_{t,x,v}} + \| D_v^{-\beta} S_1 \|_{L^2_{t,x,v}} + \| D_x^{-\frac{\beta}{2\beta+1}} S_2 \|_{L^2_{t,x,v}} + \| S_3 \|_{L^1_t L^2_{x,v}} $$

<footnote n="2">²Strictly speaking we should write $I_{t,x} \otimes (-\Delta_v)^\beta$ with $I_{t,x}$ being the identity on $L^2_{t,x}$.</footnote>
be constant in the rest of the domain. See also the work of F. Anceschi and A. Rebucci in [2] for a follow-up on an obstacle problem.

In [28], the third author together with R. Zacher introduced and developed a notion called kinetic maximal $L^2$-regularity. This concept is applied to distributional solutions of the fractional Kolmogorov ($0 < \beta \le 1$) with constant diffusion coefficients. The Cauchy problem is studied on $[0, T]$ and in inhomogeneous spaces in $(x, v)$-variables on the full space. Continuity in time is proven for the space of solutions and the temporal trace space is characterized in terms of an anisotropic Sobolev space. There are similarities and differences with our approach of kinetic embedding that we shall explain in the last section.

### 1.6.3. Transfer of regularity.
The regularizing effect of equations combining free transport $(\partial_t + v \cdot \nabla_x)$ and diffusion in the velocity variable $v$ was first exhibited by A. Kolmogorov in [22]. It can be seen from his explicit computation of the fundamental solution of (1.1). This transfer of regularity has been quantified in Hörmander's work, although only in a crude way for the regularity exponents.

An important step in the study of the regularity of kinetic equations is the observation that the means of solutions in the velocity variable $v$ are more regular than the solution itself. This was observed in particular by F. Golse, B. Perthame and R. Senbis [14] (see also [13]). Results of this type are nowadays called *averaging lemmas*. The theory was further developed by the study of the regularity of the solution itself, exhibiting the *transfer of regularity* property of the free transport. In this perspective, F. Bouchut's article [8] was influential as it was among the first ones to provide optimal and global estimates for this phenomenon, which is sometimes also called *kinetic regularisation*. We emphasize that the results of [8] apply to weak solutions only when $\beta = 1$ and in inhomogeneous spaces.

This regularising effect can also be seen for the weak solutions constructed in [1] by working with a commutator method inspired by the work of Hörmander. Sharp estimates are proven in adapted Besov spaces.

### 1.6.4. Regularity of weak solutions.
Although we do not use the local regularity of weak solutions at all, recent developments in such topics have been crucial to our understanding. In the case of rough and real coefficients satisfying an ellipticity condition but without any smoothness assumptions on coefficients, a local $L^\infty$ bound for weak solutions was obtained through Moser's iterative method by A. Pascucci and S. Polidoro [31]. Then W. Wang and L. Zhang [33] proved a local Hölder regularity thanks to a weak Poincaré inequality satisfied by weak subsolutions. In both works, weak solutions $f$ together with $\nabla_v f$ and $(\partial_t + v \cdot \nabla_x)f$ are assumed to be locally square integrable, which has no reason to hold for rough coefficients. Another proof of the local Hölder estimate was proposed in [12] by F. Golse, the second author, C. Mouhot and A. Vasseur, where a Harnack inequality is established. This latter work focuses on (1.3) with $Af = -\nabla_v \cdot (A\nabla_v f)$ and weak solutions are assumed to be locally in $L^\infty \cap L^2_{x,v} \cap L^2_{t,x} H_v^1$ and such that $(\partial_t + v \cdot \nabla_x)f \in L^2_{t,x} H_v^{-1}$. See also the works of J. Guerand with the second author and with C. Mouhot, and also the work of the third author with R. Zacher [15, 16, 30] for some further developments.

We conclude this review of literature by mentioning that we focused mainly on literature concerning weak solutions. For an overview of the literature on strong $L^p$
(ii) The map $t \mapsto \|f(t)\|_{L_{x,v}^2}$ is absolutely continuous on $\mathbb{R}$, and for a.e. $t \in \mathbb{R}$,

$$ (1.13) \quad \frac{d}{dt} \|f(t)\|_{L_{x,v}^2}^2 = 2 \operatorname{Re} \left( \int_{\mathbb{R}^d} \langle S_1, f \rangle dx + \int_{\mathbb{R}^d} \langle S_2, f \rangle dv + \iint_{\mathbb{R}^{2d}} S_3 \bar{f} \, dx \, dv \right). $$

**Remark 1.17.** The same comments as in Remark 1.6 apply.

**Remark 1.18.** Note that estimates are invariant under the kinetic scaling $(t, x, v) \mapsto (\delta^{2\beta}t, \delta^{2\beta+1}x, \delta v)$.

**Remark 1.19.** In the local case $\beta = 1$, it is possible to replace the assumption $f \in \dot{\mathbb{U}}^1$ by $f \in L_{t,x}^2 \dot{W}_v^{1,2}$ with $\|\nabla_v f\|_{L_{t,x,v}^2} < \infty$. See Section 8.

**Remark 1.20.** We shed light on the fact that we do not assume a priori control on $f(t) \in L_{x,v}^2$ at any time when $\beta < d/2$, which may seem surprising. When $\beta \ge d/2$, we make this assumption only qualitatively (one could even take a much weaker one), and it does not appear in the estimate.

**Remark 1.21.** We allow the free transport of $f$ to belong to a larger space than just $L_{t,x}^2 \dot{H}_v^{-\beta}$.

**Remark 1.22.** Not only do we obtain continuity of $t \mapsto f(t)$ in $L_{x,v}^2$ but we also get absolute continuity of $t \mapsto \|f(t)\|_{L_{x,v}^2}^2$ and zero limit at infinity. Both facts are crucial when dealing with weak solutions.

**Remark 1.23.** Theorem 1.16 is a special case of a more general result, see Theorem 3.9, which itself is a consequence of embeddings obtained for a scale of kinetic spaces, see Theorem 2.2. Such generalisations could be useful for further developments in the study weak solutions. As their proofs follow a similar pattern, we shall prove them all in this article.

**Remark 1.24.** Theorem 1.16 implies the sharp kinetic Sobolev embedding as

$$ \|f\|_{L_{t,x,v}^{2\kappa}} \lesssim_{d,\beta} \|D_v^{\beta} f\|_{L_{t,x,v}^2} + \|D_v^{-\beta} S_1\|_{L_{t,x,v}^2} + \|D_x^{-\frac{\beta}{2\beta+1}} S_2\|_{L_{t,x,v}^2} + \|S_3\|_{L_t^1 L_{x,v}^2} $$

with the optimal gain of integrability $\kappa = 1 + \frac{\beta}{(\beta+1)d}$. This is a direct consequence of the Sobolev embedding for the anisotropic space $L_x^2 H_v^\beta \cap L_v^2 H_x^{\frac{\beta}{2\beta+1}}$ at every fixed time. Integrating this estimate and combining it with the $L_t^\infty L_{x,v}^2$ estimate yields the claim. This strategy was also used in [16, 23] in order to prove a gain of integrability for weak subsolutions. We refer to [15] for an alternative proof in the case $\beta = 1$ based on kinetic trajectories.

## 1.4. Highlights and observations.
The theory of kinetic equations related to this work has been much developed, and we shall provide a short review. We wish to highlight some points in our contribution and make a few observations on tools that we do or do not use.

The Hilbertian framework proposed by J.-L. Lions to construct weak solutions had already been used for some particular kinetic equations. Our approach shows that it applies in a general setting, thanks to the complete description of the kinetic embedding and transfer of regularity for scales of adapted kinetic spaces: it allows us
solutions or strong Hölder solutions we refer to [27, 29] and [18, 20] and the references therein. Here, the pioneering work of L. Rothschild and E. Stein [32] was influential.

1.7. **Future research.** We announce some subsequent works in upcoming articles. We shall use the material of this article to provide a construction of the fundamental solution. In the local case, we also obtain Gaussian estimates, see [4]. We shall also establish self-improvement of the regularity properties of local weak solutions with improved source terms, both in the local and non-local cases. The authors also plan to treat $L^p$ estimates in the spirit of kinetic maximal $L^p$-regularity as in [29] for weak solutions of the (fractional) Kolmogorov equation with constant (and even possibly rough) coefficients. We will also develop further regularity theory along the characteristics with some new variational methods, with consequences for weak solutions.

Among possible other developments is the construction of weak solutions on bounded domains.

It is an interesting question how far one can stretch the methods of the article. While it is natural to apply them to operators of Kolmogorov type as considered in [23], another problem is to tackle Hörmander sum of squares with a drift term.

1.8. **Organisation of the article.** In Section 2, homogeneous kinetic spaces are introduced and estimates for Kolmogorov operators are obtained. Section 3 concerns kinetic embeddings. They rely on an appropriate uniqueness result for the equation associated with Kolmogorov operators. Weak solutions for equations with rough coefficients are constructed in Section 4. The kinetic Cauchy problem is treated in Section 5. In particular, inhomogeneous kinetic spaces are introduced and they are used to solve the kinetic Cauchy problem on finite time intervals. In Section 6, we make a special focus on the case of local diffusion operators ($\beta = 1$). Section 7 contains an alternative definition of homogeneous kinetic spaces that permits us to relax the constraint $\gamma \le 2\beta$ and allows us to compare our findings with previously known kinetic embeddings. We also explain in this final section how to lift the limitation $\beta < d/2$.

1.9. **Notation.** The full space $\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$ is denoted by $\Omega$ while $\Omega_+$ denotes $(0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d$. The set of distributions in $\Omega$ (resp. tempered distributions in $\Omega$) is denoted by $\mathcal{D}'(\Omega)$ (resp. $\mathcal{S}'(\Omega)$). The operators $(-\Delta_v)^{1/2}$ and $(-\Delta_x)^{1/2}$ are denoted by $D_v$ and $D_x$ respectively. We work with functions of $(t, x, v)$. As we use mixed spaces, we shall put variables in the index, e.g. $L_{t,x}^2 \hat{H}_v^\beta$. The Galilean change of variables is denoted by $\Gamma f(t, x, v) = f(t, x + tv, v)$. Given a Banach space B, $C_0(\mathbb{R}; B)$ denotes the space of continuous functions valued into B, with limit zero at $\pm\infty$. Given two Banach spaces X and Y, $X \hookrightarrow Y$ means that X is continuously embedded in Y. We use the same bracket notation for the duality pairing in different contexts with subsequent explanations, hoping this does not create any confusion.

## 2. HOMOGENEOUS KINETIC SPACES AND KOLMOGOROV OPERATORS

2.1. **Main functional spaces.** Before defining rigorously the homogeneous kinetic spaces that we will work with, we give here a short list and state the main result for the reader's convenience. The ranges of $\gamma, \beta$ will be specified below.
to obtain uniqueness under minimal assumptions and also to obtain existence with more source terms.

A second highlight concerns the strategy of proof of the kinetic embedding and transfer of regularity: it boils down to estimates and appropriate uniqueness of distributional solutions of the constant coefficients Kolmogorov equation $(\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^\beta f = S$ and its adjoint. This also shows the isomorphism properties of the integral Kolmogorov operators. This is interesting in its own right and new at this level of generality.

As for the techniques of proofs, they only use partial Fourier transformation for the position and velocity variables and the Galilean change of variables. Using homogeneous norms also provides us with a very convenient setup in which estimates are quite elementary and do not require subtle decompositions in Fourier space.

We also mention that there are common tools in the field that we do not need. We neither use the Galilean group law associated with the fields $(\partial_t + v \cdot \nabla_x), \nabla_v$ nor rely on commutator techniques à la Hörmander in the context of hypoelliptic equations. Our final observation is that we do not use the recently developed local regularity theory of weak solutions, such as the De Giorgi-Nash-Moser estimates. In other words, the construction of weak solutions does not require knowing that weak solutions have local regularity properties.

**1.5. From Lions embedding to kinetic embeddings.** We think it could be useful for the reader to focus on the genesis of our kinetic embedding and the subtlety of using homogeneous spaces by describing the parabolic case, where there is no transport and no position variable $x$.

The starting point of J.-L. Lions in his 1957 article [28] is his famous abstract embedding. Three Hilbert spaces $(V, H, V')$ form a *Gelfand triple* if there are continuous and dense inclusions $V \hookrightarrow H \hookrightarrow V'$. With such a triple, Lions established that if $f \in L^2(0, T; V)$ and $f \in H^1(0, T; V')$ then $f \in C([0, T]; H)$ and $t \mapsto \|f(t)\|_H^2$ is absolutely continuous. His proof uses the *a priori* knowledge that $f(t)$ exists in $H$ almost everywhere, approximation procedures that are compatible with the topologies of the three spaces $V, H, V'$, and integration theory for vector-valued functions.

For example, with $V = H^1(\mathbb{R}^d)$ the classical Sobolev space, this shows that if $f \in L^2(0, T; H^1(\mathbb{R}^d))$ and $f \in H^1(0, T; H^{-1}(\mathbb{R}^d))$ then $f \in C([0, T]; L^2(\mathbb{R}^d))$. Given the natural scaling, the embedding could be envisioned with $H^1, H^{-1}$ replaced by their homogeneous versions $\dot{H}^1, \dot{H}^{-1}$: however, it is not true when $T < \infty$, for the simple reason that both embeddings $\dot{H}^1(\mathbb{R}^d) \hookrightarrow L^2(\mathbb{R}^d) \hookrightarrow \dot{H}^{-1}(\mathbb{R}^d)$ fail. Indeed, take any $g \in \dot{H}^1(\mathbb{R}^d)$ which is neither an element of $L^2(\mathbb{R}^d)$ nor an element of $\dot{H}^{-1}(\mathbb{R}^d)$ and consider $f(t) = g \in L^2(0, 1; \dot{H}^1(\mathbb{R}^d)) \cap \dot{H}^1(0, 1; \dot{H}^{-1}(\mathbb{R}^d))$ which cannot be an element of $C([0, 1]; L^2(\mathbb{R}^d))$. In other words, there is no possible temporal trace theory with homogeneous spaces on finite time intervals.

This failure is not an obstacle when $T = \infty$. Indeed, it is proved by the first author with S. Monniaux and P. Portal in [6] that the embedding for homogeneous spaces is true provided one works on an infinite interval. The strategy is to prove first a uniqueness result for the backward heat equation so that any distribution $f$ that belongs to $L^2(0, \infty; \dot{H}^1(\mathbb{R}^d)) \cap \dot{H}^1(0, \infty; \dot{H}^{-1}(\mathbb{R}^d))$ is represented as a Duhamel integral for the backward heat equation (up to a constant, depending on how homogeneous
• The space $\dot{H}^{\alpha}$ is the homogeneous Sobolev space on $\mathbb{R}^d$.

• The space $\dot{F}^{\beta}$ is $L_{t,x}^2 \dot{H}_v^{\beta}$.

• The space $\dot{X}_{\beta}^{\gamma}$ equals $L_x^2 \dot{H}_v^{\gamma} \cap L_v^2 \dot{H}_x^{2\beta+1}^\gamma$ when $\gamma \ge 0$ and $\dot{X}_{\beta}^{\gamma} = L_x^2 \dot{H}_v^{\gamma} + L_v^2 \dot{H}_x^{2\beta+1}^\gamma$ if $\gamma \le 0$.

• The space $\dot{\mathcal{F}}_{\beta}^{\gamma}$ is made of $f \in L_{t,x}^2 \dot{H}_v^{\gamma}$ such that $(\partial_t + v \cdot \nabla_x)f \in L_{t,x}^2 \dot{H}_v^{\gamma-2\beta}$.

• The space $\dot{\mathcal{G}}_{\beta}^{\gamma}$ is made of $f \in L_{t,x}^2 \dot{H}_v^{\gamma}$ such that $(\partial_t + v \cdot \nabla_x)f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta}$.

• The space $\dot{\mathcal{L}}_{\beta}^{\gamma}$ is made of $f \in L_{t,x}^2 \dot{H}_v^{\gamma}$ such that $(\partial_t + v \cdot \nabla_x)f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}$.

We use calligraphic letters whenever we are concerned with kinetic spaces, that is spaces which describe also the regularity in terms of the kinetic operator $\partial_t + v \cdot \nabla_x$.

**Remark 2.1.** The calligraphic scales are adapted to the fractional Laplacian of order $\beta$: formally, $(-\Delta_v)^\beta$ maps $\dot{\mathcal{L}}_\beta^\gamma$ into $L_{t,x}^2 \dot{H}_v^{\gamma-2\beta} \subset L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ when $\gamma-2\beta \le 0$. The spaces $\dot{\mathcal{F}}_\beta^\beta$ and $\dot{\mathcal{L}}_\beta^\gamma$ are the most important ones here. Having full scales is important for the potential extension of the theory (and proofs are not much different anyway). The spaces $\dot{\mathcal{G}}_\beta^\gamma$ are also related to other kinetic spaces defined in Section 7.

**Theorem 2.2** (Kinetic embeddings and transfers of regularity: homogeneous case). Assume $\gamma < d/2$ and $0 \le \gamma \le 2\beta$.

(i) $\dot{\mathcal{L}}_{\beta}^{\gamma} \hookrightarrow L_{t,v}^2 \dot{H}_x^{2\beta+1}$ with $\|D_x^{2\beta+1} f\|_{L_{t,x,v}^2} \lesssim_{d,\beta,\gamma} \|f\|_{\dot{\mathcal{L}}_{\beta}^{\gamma}}$.

(ii) $\dot{\mathcal{L}}_{\beta}^{\gamma} \hookrightarrow C_0(\mathbb{R}_t; \dot{X}_{\beta}^{\gamma-\beta})$ with $\sup_{t \in \mathbb{R}} \|f(t)\|_{\dot{X}_{\beta}^{\gamma-\beta}} \lesssim_{d,\beta,\gamma} \|f\|_{\dot{\mathcal{L}}_{\beta}^{\gamma}}$.

(iii) There is a subspace of $\mathcal{S}(\Omega)$ which for all $\gamma$ is dense in $\dot{\mathcal{F}}_{\beta}^{\gamma}, \dot{\mathcal{G}}_{\beta}^{\gamma}$ and $\dot{\mathcal{L}}_{\beta}^{\gamma}$.

(iv) The families of spaces $(\dot{\mathcal{F}}_{\beta}^{\gamma})_{\gamma}$ and $(\dot{\mathcal{G}}_{\beta}^{\gamma})_{\gamma}$ have the complex interpolation property.

**Remark 2.3.** It is not clear whether the complex interpolation property is satisfied by $(\dot{\mathcal{L}}_{\beta}^{\gamma})_{\gamma}$ or not.

## 2.2. Homogeneous kinetic spaces. We come to precise definitions.

### 2.2.1. Homogeneous Sobolev spaces $\dot{H}^{\alpha}(\mathbb{R}^d)$.
To define homogeneous Sobolev spaces, we use the Fourier transform on tempered distributions in $\mathbb{R}^d$, equipped with Euclidean norm $|x|$ and inner product $x \cdot y$.

For $0 \le \alpha < d/2$, we define the homogeneous Sobolev space $\dot{H}^{\alpha}(\mathbb{R}^d)$ as

$$ (2.1) \qquad \dot{H}^{\alpha}(\mathbb{R}^d) = \{f \in \mathcal{S}'(\mathbb{R}^d); \exists g \in L^2(\mathbb{R}^d), \hat{f} = |\xi|^{-\alpha} \hat{g}\} $$

and we set $(-\Delta)^{\alpha/2} f = g$ and $\|f\|_{\dot{H}^{\alpha}} = \|g\|_2$. Note that when $g \in L^2(\mathbb{R}^d)$ and $0 \le \alpha < d/2$, $|\xi|^{-\alpha} \hat{g} \in \mathcal{S}'(\mathbb{R}^d)$ and also $|\xi|^{-\alpha} \hat{g} \in L_{\text{loc}}^1(\mathbb{R}^d)$, so that the elements of $\dot{H}^{\alpha}(\mathbb{R}^d)$ have locally integrable Fourier transform.

With the restriction $0 \le \alpha < d/2$, this is a Hilbert space and $\mathcal{S}(\mathbb{R}^d) \subset \dot{H}^{\alpha}(\mathbb{R}^d) \subset \mathcal{S}'(\mathbb{R}^d)$ with continuous and dense inclusions. Elements of $\dot{H}^{\alpha}(\mathbb{R}^d)$ are in fact $L^q$-functions for $q = q_{\alpha} = \frac{2d}{d-2\alpha}$ the Sobolev conjugate exponent for the Sobolev inequality $\|f\|_q \le C(d,\alpha)\|f\|_{\dot{H}^{\alpha}}$ and $\dot{H}^{\alpha}(\mathbb{R}^d)$ agrees with $\{f \in L^q(\mathbb{R}^d); (-\Delta)^{\alpha/2} f \in L^2(\mathbb{R}^d)\}$. For $\alpha = 0$, $\dot{H}^0(\mathbb{R}^d) = L^2(\mathbb{R}^d)$.
Sobolev spaces are defined) with source term $\partial_t f + \Delta f$. As this integral is shown to belong to $C_0([0, \infty); L^2(\mathbb{R}^d))$, this yields the desired conclusion for the embedding. This can also be done on $\mathbb{R}$.

With another strategy using half-time derivatives, it was shown by the first author and M. Egert in [3] that the condition $f \in \dot{H}^1(\mathbb{R}; \dot{H}^{-1}(\mathbb{R}^d))$, that is, $\partial_t f \in L^2(\mathbb{R}; \dot{H}^{-1}(\mathbb{R}^d))$, can even be relaxed to larger spaces, with the aim of allowing more source terms in equations. Again, what was new was to work with homogeneous spaces.

Our proof of continuity in time in Theorem 1.16 will rely on using the strategy of [6] in a kinetic context: working on infinite intervals is required to obtain the scale-invariant estimates; and we allow as general source terms as possible, in the spirit of [3]. We also make it work for fractional diffusion, which is new even in the homogeneous/parabolic case.

**1.6. Review of literature.** This work is concerned with the study of kinetic equations with both local and integral diffusion. It is related to the existing literature when $0 < \beta \le 1$ stemming from various trends of research.

**1.6.1. Kolmogorov equations.** The Kolmogorov equation was proposed by A. Kolmogorov in 1934 in [26]. He derived explicitly the fundamental solution of $(\partial_t + v \cdot \nabla)_x) - \Delta_v$, which allows one to observe the regularising effect of the, at first glance, degenerate diffusion equation. This was the starting point of Hörmander's hypoellipticity theory [21]. One of the examples in his paper gave birth to a trend of research devoted to a class of equations now referred to as ultra-parabolic equations or equations of Kolmogorov type. It was launched by E. Lanconelli and S. Polidoro [27]. In this article, they first considered such equations with constant coefficients, exhibited a Lie group structure associated to the equation (extending the Galilean invariance of the original Kolmogorov equation) and established a Harnack inequality.

**1.6.2. Construction of weak solutions.** Existence (and uniqueness) of weak solutions has been addressed in the literature devoted to kinetic theory in various settings.

P. Degond used Lions' existence theorem to treat the kinetic Cauchy problem on $[0, T]$ for the Fokker-Planck equation ($\beta = 1$) with constant diffusion coefficients and a bounded drift term with bounded divergence and source terms in $L_{t,x}^2 H_v^{-1}$ [14]. He attributes the idea to a previous work of S. Baouendi and P. Grisvard for the related stationary problem in one dimension [8]. J. A. Carrillo used the same idea in bounded domains [13]. More recently, D. Albritton, S. Armstrong, J. C. Mourrat and M. Novack [1] used this idea in a context of equations with confinement, and they worked with a Gaussian weight $\gamma$ in the velocity variable. More precisely, they defined a kinetic space made of functions $f \in L_{t,x}^2 H_v^1(\gamma dv)$ such that $(\partial_t + v \cdot \nabla_x)f \in L_{t,x}^2 H_v^{-1}(\gamma dv)$. It is proven that these functions are continuous in time with values in $L_{x,v}^2(\gamma dv)$. They also used another route via a minimisation procedure.

K. Nyström and M. Litsgård considered the Cauchy-Dirichlet problem for the Fokker-Planck equation ($\beta = 1$) in possibly unbounded domains in [29]. Weak solutions are constructed on the full space in the case where coefficients are allowed to be rough using a minimisation procedure of a convex functional. For the uniqueness result, coefficients are allowed to be rough only in a compact set; they are assumed to
**Remark 2.4.** Literature on kinetic equations is mostly concerned with $\alpha \in (0, 1]$. We observe that the requirement $\alpha < d/2$ is only to have a space of functions. When $\alpha \ge d/2$ one would need to use distributions modulo polynomials, and it is not clear what to do unless we add some further qualitative conditions, see Section 7. It forces $\alpha < 1$ when $d = 2$ and $\alpha < 1/2$ when $d = 1$. Such a condition is also removed by extending our approach to inhomogeneous functional spaces, see Section 5.

**Remark 2.5.** When $0 < \alpha < 1$, an equivalent norm on $\dot{H}^{\alpha}(\mathbb{R}^d)$ is given by the Gagliardo or Beppo-Levi norm (see [6] for example) but we shall not use it. When $\alpha = 1$, then this definition differs from the space $\dot{W}^{1,2}(\mathbb{R}^d) = \{f \in \mathcal{D}'(\mathbb{R}^d); \nabla f \in L^2(\mathbb{R}^d)\}$. We will indicate how this changes the statement and proof of some of our results for kinetic equations, see Section 6.

Define for $\alpha > 0$,

$$ \dot{H}^{-\alpha}(\mathbb{R}^d) = \{f \in \mathcal{S}'(\mathbb{R}^d); \exists g \in L^2(\mathbb{R}^d), \hat{f} = |\xi|^{\alpha} \hat{g}\} $$

with norm $\|f\|_{\dot{H}^{-\alpha}} = \|g\|_2$. This is also a Hilbert space, contained in $\mathcal{S}'(\mathbb{R}^d)$ and with elements having $L_{\text{loc}}^2(\mathbb{R}^d)$ Fourier transforms. When $\alpha < d/2$, it identifies to the dual of $\dot{H}^{\alpha}(\mathbb{R}^d)$ for the duality induced by $L^2$ inner product and also $\mathcal{S}(\mathbb{R}^d) \subset \dot{H}^{-\alpha}(\mathbb{R}^d)$.

If $f \in \dot{H}^{\alpha}(\mathbb{R}^d)$, $\alpha < d/2$, and $\gamma \in \mathbb{R}$ with $\alpha - 2\gamma < d/2$, then $(-\Delta)^{\gamma}f$ is the tempered distribution whose Fourier transform agrees with $|\xi|^{2\gamma-\alpha}\hat{g}$ where $\hat{f} = |\xi|^{-\alpha}\hat{g}$ and $g \in L^2(\mathbb{R}^d)$. Thus, $(-\Delta)^{\gamma}f \in \dot{H}^{\alpha-2\gamma}(\mathbb{R}^d)$ and has Fourier transform $|\xi|^{2\gamma}\hat{f}$.

**2.2.2. Homogeneous weak kinetic spaces.** We work with time, position and velocity variables $(t, x, v) \in \Omega = \mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$. As we use mixed spaces, we shall put the variables in the index, e.g. $L_{t,x}^2 \dot{H}_v^{-\beta}$ for $L^2(\mathbb{R}_t \times \mathbb{R}_x^d; \dot{H}^{-\beta}(\mathbb{R}_v^d))$. Without indication, Lebesgue spaces are with respect to the Lebesgue measure. We also indicate the variables on differentiation operators. We recall that we use the notation $D_v$ for $(-\Delta_v)^{1/2}$ and $D_x$ for $(-\Delta_x)^{1/2}$. In what follows, $\beta > 0$ is fixed.

We introduce next for $\gamma \in [0, d/2)$,

$$ (2.2) \qquad \dot{X}_{\beta}^{\gamma} = L_x^2 \dot{H}_v^{\gamma} \cap L_v^2 \dot{H}_x^{\gamma} $$

with Hilbertian norm $\|f\|_{\dot{X}_{\beta}^{\gamma}}$ given by

$$ \|f\|_{\dot{X}_{\beta}^{\gamma}}^2 = \|f\|_{L_x^2 \dot{H}_v^{\gamma}}^2 + \|f\|_{L_v^2 \dot{H}_x^{\gamma}}^2 $$

If $\gamma < 0$, we set

$$ (2.3) \qquad \dot{X}_{\beta}^{\gamma} = L_x^2 \dot{H}_v^{\gamma} + L_v^2 \dot{H}_x^{\gamma} $$

with Hilbertian norm $\|f\|_{\dot{X}_{\beta}^{\gamma}}$ given by

$$ \|f\|_{\dot{X}_{\beta}^{\gamma}}^2 = \inf_{f=f_1+f_2} \|f_1\|_{L_x^2 \dot{H}_v^{\gamma}}^2 + \|f_2\|_{L_v^2 \dot{H}_x^{\gamma}}^2 $$

**Remark 2.6.** All these spaces are complete and are subspaces of $\mathcal{S}'_{x,v} = \mathcal{S}'(\mathbb{R}^{2d})$, and of $L_{\text{loc},x,v}^2 = L_{\text{loc}}^2(\mathbb{R}^{2d})$ when $\gamma \ge 0$.
be constant in the rest of the domain. See also the work of F. Anceschi and A. Rebucci in [2] for a follow-up on an obstacle problem.

In [32], the third author together with R. Zacher introduced and developed a notion called kinetic maximal L²-regularity. This concept is applied to distributional solutions of the Kolmogorov operators with constant diffusion coefficients when $0 < \beta \le 1$. The Cauchy problem is studied on $[0, T]$ and in inhomogeneous spaces in $(x, v)$-variables on the full space. Continuity in time is proven for the space of solutions, and the temporal trace space is characterised in terms of an anisotropic Sobolev space. There are similarities and differences with our approach of kinetic embedding that we shall explain in the last section.

1.6.3. *Transfer of regularity.* The regularising effect of equations combining free transport $(\partial_t + v \cdot \nabla)_x)$ and diffusion $-\Delta_v$ in the velocity variable $v$ was first exhibited by A. Kolmogorov in [26]. It can be seen from the explicit computation of the fundamental solution. This transfer of regularity has been quantified in Hörmander's work, although only in a crude way for the regularity exponents.

An important step in the study of the regularity of kinetic equations is the observation that the means of solutions in the velocity variable $v$ are more regular than the solutions themselves. This was observed in particular by F. Golse, B. Perthame and R. Sentis [18] (see also [17]). Results of this type are nowadays called *averaging lemmas*. The theory was further developed by the study of the regularity of the solution itself, exhibiting the *transfer of regularity* property of the free transport. In this perspective, F. Bouchut's article [10] was influential as it was among the first to provide optimal and global estimates for this phenomenon, which is sometimes also called *kinetic regularisation*. We emphasise again that the results of [10] apply to weak solutions only when $\beta = 1$ and in inhomogeneous spaces.

This regularising effect can also be seen for the weak solutions constructed in [1] by working with a commutator method inspired by the work of Hörmander. Sharp estimates are proven in adapted Besov spaces.

1.6.4. *Regularity of weak solutions.* Although we do not use the local regularity of weak solutions at all, recent developments in such topics have been crucial to our understanding and definition of a weak solution. In the case of rough and real coefficients satisfying an ellipticity condition, but without any smoothness assumptions on coefficients, a local $L^\infty$ bound for weak solutions was obtained through Moser's iterative method by A. Pascucci and S. Polidoro [35]. Then W. Wang and L. Zhang [37] proved a local Hölder regularity thanks to a weak Poincaré inequality satisfied by weak subsolutions. In both works, weak solutions $f$ together with $\nabla_v f$ and $(\partial_t + v \cdot \nabla_x)f$ are assumed to be locally square integrable. It makes sense for $f$ and $\nabla_v f$, because of the natural energy estimate, but the free transport term has no reason to be square integrable in the case of rough diffusion coefficients. Another proof of the local Hölder estimate was proposed in [16] by F. Golse, the second author, C. Mouhot and A. Vasseur, where a Harnack inequality is established. This latter work focuses on (1.1) with $Af = -\nabla_v \cdot (\mathbf{A}\nabla_v f)$ and weak solutions are assumed to be locally in $L_t^\infty L_{x,v}^2 \cap L_{t,x}^2 H_v^1$ and such that $(\partial_t + v \cdot \nabla_x)f \in L_{t,x}^2 H_v^{-1}$. See also the works of J. Guerand with the second author and with C. Mouhot, and also the work
**Remark 2.7.** For $\gamma \in [0, d/2)$, the space $\dot{X}_{\beta}^{-\gamma}$ identifies with the dual of $\dot{X}_{\beta}^{\gamma}$ for the duality extending the $L_{x,v}^2$ inner product.

We introduce next for $\gamma < d/2$,

$$ (2.4) \qquad \dot{\mathcal{F}}_{\beta}^{\gamma} = \{f \in \mathcal{D}'(\Omega); f \in L_{t,x}^2 \dot{H}_v^{\gamma} \ \& \ (\partial_t + v \cdot \nabla_x) f \in L_{t,x}^2 \dot{H}_v^{\gamma-2\beta}\} $$

with Hilbertian norm $\|f\|_{\dot{\mathcal{F}}_{\beta}^{\gamma}}$ defined by

$$ (2.5) \qquad \|f\|_{\dot{\mathcal{F}}_{\beta}^{\gamma}}^2 = \|D_v^{\gamma} f\|_{L_{t,x,v}^2}^2 + \|(\partial_t + v \cdot \nabla_x) f\|_{L_{t,x}^2 \dot{H}_v^{\gamma-2\beta}}^2, $$

$$ (2.6) \qquad \dot{\mathcal{G}}_{\beta}^{\gamma} = \{f \in \mathcal{D}'(\Omega); f \in L_{t,x}^2 \dot{H}_v^{\gamma} \ \& \ (\partial_t + v \cdot \nabla_x) f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta}\} $$

with Hilbertian norm $\|f\|_{\dot{\mathcal{G}}_{\beta}^{\gamma}}$ defined by

$$ \|f\|_{\dot{\mathcal{G}}_{\beta}^{\gamma}}^2 = \|D_v^{\gamma} f\|_{L_{t,x,v}^2}^2 + \|(\partial_t + v \cdot \nabla_x) f\|_{L_t^2 \dot{X}_{\beta}^{\gamma-2\beta}}^2, $$

and

$$ (2.7) \qquad \dot{\mathcal{L}}_{\beta}^{\gamma} = \{f \in \mathcal{D}'(\Omega); f \in L_{t,x}^2 \dot{H}_v^{\gamma} \ \& \ (\partial_t + v \cdot \nabla_x) f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}\} $$

with norm $\|f\|_{\dot{\mathcal{L}}_{\beta}^{\gamma}}$ defined by

$$ \|f\|_{\dot{\mathcal{L}}_{\beta}^{\gamma}} = \|D_v^{\gamma} f\|_{L_{t,x,v}^2} + \|(\partial_t + v \cdot \nabla_x) f\|_{L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}} $$

The action of the field $\partial_t + v \cdot \nabla_x$ is taken in the sense of distributions on $\Omega$.

**Remark 2.8.** These spaces are complete and $\dot{\mathcal{F}}_{\beta}^{\gamma} \subset \dot{\mathcal{G}}_{\beta}^{\gamma} \subset \dot{\mathcal{L}}_{\beta}^{\gamma} \subset L_t^2 S'_{x,v} \subset S'(\Omega)$.

Eventually, we can repeat *verbatim* the definitions on $\Omega_+ = (0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d$. But only the first three inclusions of Remark 2.8 continue to hold as there is no natural definition of $S'(\Omega_+)$. Yet, to simplify the presentation, we shall use the same notation and make it clear that we work on $\Omega_+$.

## 2.3. Kolmogorov operators.

### 2.3.1. Galilean change of variables.
The Galilean change of variables is defined by

$$ \Gamma f(t, x, v) = f(t, x + tv, v) $$

and also for fixed $t$,

$$ \Gamma(t)f(x, v) = f(x + tv, v) $$

which are isometries on $L_{t,x,v}^2$ and $L_{x,v}^2$ respectively. Furthermore, $(\Gamma(t))_{t \in \mathbb{R}}$ is a strongly continuous group on $L_{x,v}^2$ [28, Lemma 5.2]. These two properties imply that the mapping $\Gamma: C_0(\mathbb{R}_t; L_{x,v}^2) \to C_0(\mathbb{R}_t; L_{x,v}^2)$ is continuous.
of the third author with R. Zacher [19, 20, 34] for some further developments. An ex-
planation on how to prove estimates for weak (sub-) supersolutions belonging to the
space $L_{\text{loc},t,x}^2 L_v^2 \cap L_{t,x}^2 \dot{H}_v^1$ can be found in the recent work of H. Dietert, C. Mouhot,
the last author and R. Zacher [15].

We conclude this review of the literature by mentioning that we focused mainly on literature concerning weak solutions. For an overview of the literature on strong $L^p$ solutions or strong Hölder solutions, we refer to [31,33] and [22,24] and the references therein. See also the pioneering work of L. Rothschild and E. Stein [36].

1.7. **Future developments.** We announce some subsequent works. We have used the material of this article to provide a construction of the fundamental solution for non-constant diffusion coefficients, see [4]. In the local case, we have also obtained sharp decay estimates. We shall establish self-improvement of the regularity properties of local weak solutions with improved source terms, both in the local and non-local cases. The authors also plan to treat $L^p$ estimates in the spirit of kinetic maximal $L^p$-regularity as in [33] for weak solutions of the (fractional) Kolmogorov equation with constant (and even possibly rough) coefficients. We will also develop further regularity theory along the characteristics with some new variational methods, with consequences for weak solutions. Finally, the setup here will be adapted to allow lower-order terms with unbounded coefficients.

Among possible other developments is the construction of weak solutions on bounded
domains, but this would require other methods, as we use the Fourier transform.

It is an interesting question how far one can stretch the methods of the article.
While it is natural to apply them to operators of Kolmogorov type as considered in
[27], another problem is to tackle Hörmander sum of squares with a drift term.

1.8. **Organisation of the article.** In Section 2, homogeneous kinetic spaces are introduced, and estimates for Kolmogorov operators are obtained. Section 3 concerns kinetic embeddings, which rely on an appropriate uniqueness result for the equation associated with Kolmogorov operators. Weak solutions for equations with rough coefficients are constructed in Section 4. The kinetic Cauchy problem is treated in Section 5. In particular, inhomogeneous kinetic spaces are introduced, and they are used to solve the kinetic Cauchy problem on finite time intervals. In Section 8, we focus specifically on the case of local diffusion operators ($\beta = 1$). Section 9 contains an alternative definition of homogeneous kinetic spaces that permits us to relax the constraint $\gamma \le 2\beta$ and allows us to compare our findings with previously known kinetic embeddings. We also explain in this final section how to lift the limitation $\beta < d/2$.

1.9. **Notation.** The full space $\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$ is denoted by $\Omega$ while $\Omega_+$ denotes $(0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d$. The set of distributions in $\Omega$ (resp. tempered distributions in $\Omega$) is denoted by $\mathcal{D}'(\Omega)$ (resp. $\mathcal{S}'(\Omega)$). The operators $(-\Delta_v)^{1/2}$ and $(-\Delta_x)^{1/2}$ are denoted by $D_v$ and $D_x$ respectively. We work with functions of $(t, x, v)$. As we use mixed spaces, we shall put variables in the index, e.g. $L_{t,x}^2 \dot{H}_v^\beta$. The Galilean change of variables is denoted by $\Gamma f(t, x, v) = f(t, x + tv, v)$. Given a Banach space B, $C_0(\mathbb{R}; B)$ denotes the space of continuous functions valued into B, with limit zero at $\pm\infty$. Given two Banach spaces X and Y, $X \hookrightarrow Y$ means that X is continuously embedded in Y.
2.3.2. Partial Fourier transform. We use (partial) Fourier transform in the variables $(\varphi, \xi)$ dual to $(x, v)$, defined by

$$ \hat{f}(t, \varphi, \xi) = \iint_{\mathbb{R}^d \times \mathbb{R}^d} e^{-i(x \cdot \varphi + v \cdot \xi)} f(t, x, v) \, dx \, dv. $$

This technical lemma relating the Fourier transform and the Galilean change of variables $\Gamma$ will be used frequently. To simplify the presentation we introduce the weights

$$ (2.8) \qquad W(t, \varphi, \xi) = \sup(|\xi - t\varphi|, |\varphi|^{\frac{1}{1+2\beta}}), $$

and

$$ w(\varphi, \xi) = \sup(|\xi|, |\varphi|^{\frac{1}{1+2\beta}}) $$

which depend on the fixed parameter $\beta > 0$.

**Lemma 2.9.** Let $\gamma \in \mathbb{R}$ with $\gamma < d/2$. Then the map $f \mapsto \hat{\Gamma}f$ is an isomorphism

(i) $S(\Omega)$ onto $S(\Omega)$.

(ii) $S'(\Omega)$ onto $S'(\Omega)$.

(iii) $L_t^p \dot{X}_\beta^\gamma$ onto $L_t^p L_{\varphi, \xi}^2 (W^{2\gamma} d\varphi d\xi)$, $1 \le p < \infty$.

(iv) $C_0(\mathbb{R}_t; \dot{X}_\beta^\gamma)$ onto $C_0(\mathbb{R}_t; L_{\varphi, \xi}^2 (W^{2\gamma} d\varphi d\xi))$.

(v) $L_t^2 \dot{X}_\beta^\gamma$ onto $L_{t, \varphi, \xi}^2 (W^{2\gamma} dt d\varphi d\xi)$.

*Proof.* We have $\hat{\Gamma}f(t, \varphi, \xi) = \hat{f}(t, \varphi, \xi - t\varphi)$. The first item is rather easy and we skip details. The second item follows by duality using the first one replacing $\Gamma$ by $\Gamma^* = \Gamma^{-1}$. For the third and fourth items, Plancherel's theorem tells us that $u \in \dot{X}_\beta^\gamma$ if and only if $\hat{u} \in L_{\varphi, \xi}^2 (w^{2\gamma} d\varphi d\xi)$. Apply this to $u(x, v) = f(t, x, v)$ for fixed $t$ and then use the change of variables induced by $\Gamma(t)$. This proves the third item. For the fifth item apply further Fubini's theorem when $p = 2$. Concerning the fourth item, the right-hand space should be understood as the image under the change of variable $\hat{\Gamma}$ of $C_0(\mathbb{R}_t; L_{\varphi, \xi}^2 (w^{2\gamma} d\varphi d\xi))$ where $\hat{\Gamma}$ is the conjugate of the Galilean change of variables through the Fourier transform, that is $\hat{\Gamma}h(t, \varphi, \xi) = h(t, \varphi, \xi - t\varphi)$. $\square$

2.3.3. Forward and backward Kolmogorov operators of order $\beta$. The forward and backward Kolmogorov operators of order $\beta$ are denoted by $K_\beta^\pm$. They are formally defined as inverse maps of $\pm(\partial_t + v \cdot \nabla_x) + (-\Delta_v)^\beta$. Rigorously, we use Lemma 2.9 and for $S \in S'(\Omega)$, we set

$$ f = K_\beta^+ S $$

whenever

$$ (2.9) \qquad \hat{\Gamma}f = T(\hat{\Gamma}S) \in S'(\Omega), $$

with

$$ (2.10) \qquad Th(t, \varphi, \xi) = \int_{-\infty}^{t} K(t, s, \varphi, \xi)h(s, \varphi, \xi) \, ds, $$

where for $s, t \in \mathbb{R}$ and $(\varphi, \xi) \in \mathbb{R}^{2d}$,

$$ (2.11) \qquad K(t, s, \varphi, \xi) = \exp \left( - \int_s^t |\xi - \tau\varphi|^{2\beta} d\tau \right). $$
We use the same bracket notation for the duality pairing in different contexts with subsequent explanations, hoping this does not create any confusion.

## 2. HOMOGENEOUS KINETIC SPACES AND KOLMOGOROV OPERATORS

### 2.1. Main functional spaces.
Before defining rigorously the homogeneous kinetic spaces that we will work with, we give here a short list and state the main result for the reader's convenience. The ranges of $\gamma, \beta$ will be specified below.

* The space $\dot{H}^{\alpha}$ is the homogeneous Sobolev space on $\mathbb{R}^d$.

* The space $\dot{F}^{\beta}$ is $L_{t,x}^2 \dot{H}_v^{\beta}$ and $\dot{U}^{\beta}$ is $L_{\text{loc},t}^2 L_{x,v}^2 \cap L_{t,x}^2 \dot{H}_v^{\beta}$.

* The space $\dot{X}_{\beta}^{\gamma}$ equals $L_x^2 \dot{H}_v^{\gamma} \cap L_v^2 \dot{H}_x^{\frac{\gamma}{2\beta+1}}$ when $\gamma \ge 0$ and $\dot{X}_{\beta}^{\gamma} = L_x^2 \dot{H}_v^{\gamma} + L_v^2 \dot{H}_x^{\frac{\gamma}{2\beta+1}}$ if $\gamma \le 0$.

* The space $\dot{F}_{\beta}^{\gamma}$ is made of $f \in L_{t,x}^2 \dot{H}_v^{\gamma}$ such that $(\partial_t + v \cdot \nabla_x)f \in L_{t,x}^2 \dot{H}_v^{\gamma-2\beta}$.

* The space $\dot{G}_{\beta}^{\gamma}$ is made of $f \in L_{t,x}^2 \dot{H}_v^{\gamma}$ such that $(\partial_t + v \cdot \nabla_x)f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta}$.

* The space $\dot{L}_{\beta}^{\gamma}$ is made of $f \in L_{t,x}^2 \dot{H}_v^{\gamma}$ such that $(\partial_t + v \cdot \nabla_x)f \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}$.

We use calligraphic letters whenever we are concerned with kinetic spaces, that is, spaces which also describe the regularity in terms of the kinetic operator $\partial_t + v \cdot \nabla_x$.

**Remark 2.1.** The calligraphic scales are adapted to the fractional Laplacian of order $\beta$: formally, $(-\Delta_v)^\beta$ maps $\dot{L}_\beta^\gamma$ into $L_{t,x}^2 \dot{H}_v^{\gamma-2\beta} \subset L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ when $\gamma-2\beta \le 0$. The spaces $\dot{F}_\beta^\beta$ and $\dot{L}_\beta^\beta$ are the most important ones here. Having full scales is important for the potential extension of the theory (and proofs are not much different anyway). The spaces $\dot{G}_\beta^\gamma$ are also related to other kinetic spaces defined in Section 9.

**Theorem 2.2 (Kinetic embeddings and transfers of regularity: homogeneous case).**
Assume $\beta > 0$, $0 \le \gamma \le 2\beta$ with $\gamma < d/2$.

(i) $\dot{L}_\beta^\gamma \hookrightarrow L_{t,v}^2 \dot{H}_x^{\frac{\gamma}{2\beta+1}}$ with $\|D_x^{\frac{\gamma}{2\beta+1}} f\|_{L_{t,x,v}^2} \lesssim_{d,\beta,\gamma} \|f\|_{\dot{L}_\beta^\gamma}$.

(ii) $\dot{L}_\beta^\gamma \hookrightarrow C_0(\mathbb{R}_t; \dot{X}_\beta^{\gamma-\beta})$ with $\sup_{t \in \mathbb{R}} \|f(t)\|_{\dot{X}_\beta^{\gamma-\beta}} \lesssim_{d,\beta,\gamma} \|f\|_{\dot{L}_\beta^\gamma}$.

(iii) There is a subspace of $L_{t,x,v}^2$ (even of $S(\Omega)$ if $d > 2$) dense in all $\dot{F}_\beta^\gamma, \dot{G}_\beta^\gamma$ and $\dot{L}_\beta^\gamma$.

(iv) The families of spaces $(\dot{F}_\beta^\gamma)_\gamma$ and $(\dot{G}_\beta^\gamma)_\gamma$ have the complex interpolation property.

When $\gamma = \beta \ge d/2$, (i), (ii), (iii) hold provided we define the kinetic space $\dot{L}_\beta^\gamma$ with $\dot{U}^\beta$ replacing $L_{t,x}^2 \dot{H}_v^\beta$.

**Remark 2.3.** It is not clear whether the complex interpolation property is satisfied by $(\dot{L}_\beta^\gamma)_\gamma$ or not.

**Remark 2.4.** If we impose $f \in \dot{U}^\gamma$ in the definition of the kinetic spaces when $\gamma \ge d/2$ and $\gamma \ne \beta$, then we can keep (i), (ii), and (iii), but it is not clear whether they are Banach spaces, and we may lose property (iv).

### 2.2. Homogeneous kinetic spaces.
We come to precise definitions.
The operator $K_{\beta}^{-}$ is the formal adjoint to $K_{\beta}^{+}$. It is associated in the same fashion to the adjoint of $T$ given by the following formula,

$$ (2.12) \qquad T^*h(s, \varphi, \xi) = \int_s^{+\infty} K(t, s, \varphi, \xi)h(t, \varphi, \xi) dt. $$

Of course, the integral operator $T$ above makes sense only for certain classes of functions and is thought of as the Duhamel formula for the ordinary differential equation,

$$ g'(t) + |\xi - t\varphi|^{2\beta} g(t) = h(t), \quad t \in \mathbb{R}. $$

### 2.3.4. Density.
In order to establish for instance the energy equality (see Lemma 4.2), it is useful to have a density result at hand. This will also be needed to prove (iii) in Theorem 2.2.

**Lemma 2.10 (Density).** There exists a subset $S_K$ of $S(\Omega)$ which is dense in $L_t^p \dot{X}_\beta^\gamma$ and in $C_0(\mathbb{R}_t; \dot{X}_\beta^\gamma)$ for all $1 \le p < \infty$ and $\gamma < d/2$ and such that $K_\beta^\pm(S_K) \subset S(\Omega)$.

*Proof.* We begin with the construction of $S_K$. Recall that $\Omega = \mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$. Again, we argue on the Fourier side. Let $d(\xi, \mathbb{R}\varphi)$ denote the distance from $\xi$ to the line $\mathbb{R}\varphi$. When $\varphi \ne 0$, we have $d(\xi, \mathbb{R}\varphi) = |\xi - (\xi \cdot \varphi) \frac{\varphi}{|\varphi|^2}|$, so it is a continuous function of $(\varphi, \xi)$ on $\mathbb{R}^d \setminus \{0\} \times \mathbb{R}^d$. Set

$$ \Omega_K = \{(t, \varphi, \xi) \in \Omega; |\varphi| > 0, d(\xi, \mathbb{R}\varphi) > 0\}, $$

which is an open set. We say that $S \in S_K$ if and only if $h = \widehat{\Gamma S} \in S(\Omega)$ and has compact support in $\Omega_K$.

Let us prove the density of $S_K$ in $L_t^p \dot{X}_\beta^\gamma$ and $C_0(\mathbb{R}_t; \dot{X}_\beta^\gamma)$. Assume $f$ is in one of these spaces. Thanks to Lemma 2.9, we may argue on the Fourier side. We first approximate $\hat{f}$ with a function in $S(\Omega)$ with compact support in $\Omega$, and away from $|\varphi| = 0$ and $|\xi| = 0$. This follows right away from the definition of homogeneous Sobolev spaces and density in Lebesgue spaces or $C_0$. Thus, we may assume that $h = \widehat{\Gamma}f$ has support in $\{(t, \varphi, \xi) \in \Omega; |t| \le M, m \le |\varphi| \le M, m \le |\xi - t\varphi| \le M\}$ for some $0 < m \le M < \infty$. On this set, the function $(t, \varphi, \xi) \mapsto d(\xi, \mathbb{R}\varphi)^2$ is $C^\infty$ and if we let $\chi \in C^\infty(0, \infty)$ with $\chi(y) = 1$ if $y \ge 4$ and $\chi(y) = 0$ if $y \le 1$, then $h_\varepsilon : (t, \varphi, \xi) \mapsto \chi(d(\xi, \mathbb{R}\varphi)^2/\varepsilon^2)h(t, \varphi, \xi)$ is $C^\infty$ with compact support in $\Omega_K$. Moreover, for the convergence, we have that $h$ has support in $[-M, M] \times \{m \le |\varphi| \le M\} \times \{|\xi| \le M + M^2\}$. In addition $h - h_\varepsilon = 0$ when $d(\xi, \mathbb{R}\varphi) \ge 2\varepsilon$. If $d(\xi, \mathbb{R}\varphi) \le 2\varepsilon$, then $\xi$ belongs to the cylinder $\mathbb{R}\varphi + B(0, 2\varepsilon)$. Thus, using all the support information,

$$ \left\| \left( \iint |h - h_{\varepsilon}|^2 W^{2\gamma} d\xi d\varphi \right)^{1/2} \right\|_{L_t^p} \lesssim_{m,M,\gamma,\beta,p} \|h\|_{L^\infty} \varepsilon^{1/2}. $$

We turn to proving that the Kolmogorov operators map $S_K$ to $S(\Omega)$. We do it for $K_\beta^+$, the proof for $K_\beta^-$ being the same. Let $S \in S_K$ and set $h = \widehat{\Gamma S}$. We have to show that $g = Th \in S(\Omega)$. Since $h$ has compact support in $\Omega_K$, there exists $0 < m < M < \infty$ such that $h$ is supported in $[-M, M] \times F$, with

$$ F = \{(\varphi, \xi) \in \mathbb{R}^{2d}; m \le |\varphi| \le M, m \le d(\xi, \mathbb{R}\varphi), |\xi| \le M\}. $$

On $\mathbb{R} \times (\mathbb{R}^{2d} \setminus F) \cup (-\infty, -M] \times F$, $g$ is clearly 0, that is, $g$ is supported in $[-M, \infty) \times F$. Let $0 < m' < m$ and $M < M' < \infty$ and $O = \{(\varphi, \xi) \in \mathbb{R}^{2d}; m' < |\varphi| < M', m' <$
2.2.1. *Homogeneous Sobolev spaces* $\dot{H}^\alpha(\mathbb{R}^d)$. To define homogeneous Sobolev spaces, we use the Fourier transform on tempered distributions in $\mathbb{R}^d$, equipped with Euclidean norm $|x|$ and inner product $x \cdot y$.

Assume first $0 \le \alpha < d/2$. We define the homogeneous Sobolev space $\dot{H}^\alpha(\mathbb{R}^d)$ as

$$ (2.1) \qquad \dot{H}^{\alpha}(\mathbb{R}^{d}) = \{f \in S'(\mathbb{R}^{d}); \exists g \in L^{2}(\mathbb{R}^{d}), \hat{f} = |\xi|^{-\alpha} \hat{g}\} $$

and we set $(-\Delta)^{\alpha/2} f = g$ and $\|f\|_{\dot{H}^\alpha} = \|g\|_2$. The key point is not only $|\xi|^{-\alpha} \hat{g} \in S'(\mathbb{R}^d)$, it is also that $|\xi|^{-\alpha} \hat{g} \in L_{\text{loc}}^1(\mathbb{R}^d)$, so that the elements of $\dot{H}^\alpha(\mathbb{R}^d)$ have locally integrable Fourier transform.

With this definition, $\dot{H}^\alpha(\mathbb{R}^d)$ is a Hilbert space and $S(\mathbb{R}^d) \subset \dot{H}^\alpha(\mathbb{R}^d) \subset S'(\mathbb{R}^d)$ with continuous and dense inclusions. Elements of $\dot{H}^\alpha(\mathbb{R}^d)$ are in fact $L^q$-functions for $q = q_\alpha = \frac{2d}{d-2\alpha}$ the Sobolev conjugate exponent for the Sobolev inequality $\|f\|_q \le C(d, \alpha) \|f\|_{\dot{H}^\alpha}$ and $\dot{H}^\alpha(\mathbb{R}^d)$ agrees with $\{f \in L^q(\mathbb{R}^d); (-\Delta)^{\alpha/2} f \in L^2(\mathbb{R}^d)\}$. For $\alpha = 0$, $\dot{H}^0(\mathbb{R}^d) = L^2(\mathbb{R}^d)$.

**Remark 2.5.** Literature on kinetic equations is mostly concerned with $\alpha \in (0, 1]$. We observe that the requirement $0 < \alpha < d/2$ is only to have a space of functions that is a Hilbert space. It forces $\alpha < 1$ when $d=2$ and $\alpha < 1/2$ when $d=1$. But we can also manage the case $\alpha \ge d/2$ differently. We refer the reader to forthcoming work [5] for further explanations. The $d/2$ threshold never shows up when using inhomogeneous functional spaces, see Section 5.

**Remark 2.6.** When $0 < \alpha < 1$, an equivalent norm on $\dot{H}^\alpha(\mathbb{R}^d)$ is given by the Gagliardo or Beppo-Levi (semi-)norm (see [7] for example), but we shall not use it. When $\alpha = 1$, then our definition differs from the space $\dot{W}^{1,2}(\mathbb{R}^d) = \{f \in D'(\mathbb{R}^d); \nabla f \in L^2(\mathbb{R}^d)\}$ as this space contains constants. We will indicate how this changes the statement and proof of some of our results for kinetic equations, see Section 8.

Define for $\alpha < 0$,

$$ \dot{H}^{\alpha}(\mathbb{R}^{d}) = \{f \in S'(\mathbb{R}^{d}); \exists g \in L^{2}(\mathbb{R}^{d}), \hat{f} = |\xi|^{-\alpha} \hat{g}\} $$

with norm $\|f\|_{\dot{H}^\alpha} = \|g\|_2$. This is also a Hilbert space, contained in $S'(\mathbb{R}^d)$ and with elements having $L_{\text{loc}}^2(\mathbb{R}^d)$ Fourier transforms. When $-d/2 < \alpha < 0$, $\dot{H}^\alpha(\mathbb{R}^d)$ identifies to the dual of $\dot{H}^{-\alpha}(\mathbb{R}^d)$ for the duality induced by the $L^2(\mathbb{R}^d)$ inner product and we also have $S(\mathbb{R}^d) \subset \dot{H}^\alpha(\mathbb{R}^d)$ as a dense subset. If $\alpha \le -d/2$, then Schwartz functions whose Fourier transforms are supported away from the origin form a dense set.

We eventually come to the proper interpretation of fractional powers of the Laplacian. If $f \in \dot{H}^\alpha(\mathbb{R}^d)$, $\alpha \in \mathbb{R}$ and $\alpha < d/2$, and $\gamma \in \mathbb{R}$ with $\alpha - 2\gamma < d/2$, then $(-\Delta)^\gamma f$ is the tempered distribution whose Fourier transform agrees with $|\xi|^{2\gamma-\alpha} \hat{g}$ where $\hat{f} = |\xi|^{-\alpha} \hat{g}$ and $g \in L^2(\mathbb{R}^d)$. Thus, $(-\Delta)^\gamma f \in \dot{H}^{\alpha-2\gamma}(\mathbb{R}^d)$ and has Fourier transform $|\xi|^{2\gamma} \hat{f}$.

2.2.2. *Homogeneous weak kinetic spaces.* We work with time, position and velocity variables $(t, x, v) \in \Omega = \mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$. As we use mixed spaces, we shall put the variables in the index, e.g. $L_{t,x}^2 \dot{H}_v^{-\beta}$ for $L^2(\mathbb{R}_t \times \mathbb{R}_x^d; \dot{H}^{-\beta}(\mathbb{R}_v^d))$. Without indication, Lebesgue spaces are with respect to the Lebesgue measure. We also indicate the
$d(\xi, \mathbb{R}\varphi), |\xi| < M'$}. When $m' \le d(\xi, \mathbb{R}\varphi)$ and $t \ge s$, $\int_s^t |\xi - \tau\varphi|^{2\beta} d\tau \ge (t-s)(m')^{2\beta}$. In particular $(\varphi, \xi) \mapsto \exp(-\int_s^t |\xi - \tau\varphi|^{2\beta} d\tau)$ is $C^\infty$ on $O$ and $g$ is $C^\infty$ with respect to $(\varphi, \xi) \in O$ for any $t > -M'$, and all its partial derivatives with respect to $(\varphi, \xi) \in O$ have exponential decay in $t \to \infty$, uniformly. Next, as $\partial_t g = -|\xi - t\varphi|^{2\beta}g + h$, the same is true for $\partial_t g$. By induction, we obtain that $g$ is $C^\infty$ with respect to $(t, \varphi, \xi)$. Since $g$ has support in $[-M, \infty) \times F$, with all partial derivatives having exponential decay in time, uniformly in $(\varphi, \xi)$, we conclude that $g \in S(\Omega)$. $\square$

2.3.5. *Estimates.* The main technical result related to Kolmogorov operators is the following proposition. It makes precise how Kolmogorov operators act on various functional spaces in $(t, x, v)$, in particular on $L_t^2 \dot{X}_\beta^{\gamma-2\beta}$.

**Proposition 2.11 (Estimates for Kolmogorov operators).** We have the following bounds.

(i) For $\gamma < d/2$, the operators $K_\beta^\pm$ map boundedly $L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ to $L_t^2 \dot{X}_\beta^\gamma$.

(ii) For $\gamma - \beta < d/2$, the operators $K_\beta^\pm$ map boundedly $L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ to $C_0(\mathbb{R}_t; \dot{X}_\beta^{\gamma-\beta})$.

(iii) For $\gamma < d/2$, the operators $K_\beta^\pm$ map boundedly $L_t^1 \dot{X}_\beta^{\gamma-\beta}$ to $L_t^2 \dot{X}_\beta^\gamma$.

(iv) For $\gamma - \beta < d/2$, $K_\beta^\pm$ map boundedly $L_t^1 \dot{X}_\beta^{\gamma-\beta}$ to $C_0(\mathbb{R}_t; \dot{X}_\beta^{\gamma-\beta})$.

**Remark 2.12.** The continuity is a key aspect of our approach. The only restriction on $\gamma$ is the one imposed by our definition of Sobolev spaces. See Proposition 2.13 below where the Fourier estimates are proven without restriction.

Thanks to (2.9) and Lemma 2.9, the proof of Proposition 2.11 is done here on the Fourier side. The method consists of working with fixed Fourier variables $(\varphi, \xi)$ and proving weighted temporal estimates by integrating with respect to $t$. It makes explicitly clear two facts: on the one hand, the uniformity (or dependency) of the constants with respect to $(\varphi, \xi)$, yielding in the end homogeneous estimates, and on the other hand that time intervals can be infinite.

Making the variables $(\varphi, \xi)$ implicit, we set

$$K(t, s) = K(t, s, \varphi, \xi), \quad Th(t) = \int_{-\infty}^{t} K(t, s)h(s) \, ds, \quad W(t) = W(t, \varphi, \xi)$$

for the Kolmogorov kernel, the associated operator and the weight defined in (2.11), (2.10) and (2.8).

2.3.6. *Weighted temporal estimates.* In order to prove Proposition 2.11, we fix $(\varphi, \xi) \ne (0, 0)$, define weighted temporal spaces and study how the map $T$ acts on them.

For $\varepsilon \in \mathbb{R}$ and $1 \le p < \infty$, we use the notation $L_t^p(W^{\varepsilon p})$ for the space $L^p(\mathbb{R}; W^{\varepsilon p} dt)$ of measurable functions $g$ such that $W^\varepsilon g \in L_t^p$ with norm $\|g\|_{L_t^p(W^{\varepsilon p})} = \|W^\varepsilon g\|_{L_t^p}$, that is

$$\|g\|_{L_t^p(W^{\varepsilon p})} := \int_{\mathbb{R}} |g(t)|^p \left( \sup(|\xi - t\varphi|, |\varphi|^{\frac{1}{1+2\beta}}) \right)^{\varepsilon p} dt.$$

The following proposition will use elementary estimates on the kernel $K$. By abuse of notation, we also denote by $W^\varepsilon$ the operator of multiplication by $W^\varepsilon(t)$.
variables on differentiation operators. We recall that we use the notation $D_v$ for $(-\Delta_v)^{1/2}$ and $D_x$ for $(-\Delta_x)^{1/2}$. In what follows, $\beta > 0$ is fixed.

We introduce next for $\gamma \in [0, d/2)$,

$$ (2.2) \qquad \dot{X}_{\beta}^{\gamma} = L_{x}^{2} \dot{H}_{v}^{\gamma} \cap L_{v}^{2} \dot{H}_{x}^{\frac{\gamma}{2\beta+1}} $$

with Hilbertian norm $\|f\|_{\dot{X}_{\beta}^{\gamma}}$ given by

$$ \|f\|_{\dot{X}_{\beta}^{\gamma}}^{2} = \|f\|_{L_{x}^{2} \dot{H}_{v}^{\gamma}}^{2} + \|f\|_{L_{v}^{2} \dot{H}_{x}^{\frac{\gamma}{2\beta+1}}}^{2}. $$

If $\gamma < 0$, we set

$$ (2.3) \qquad \dot{X}_{\beta}^{\gamma} = L_{x}^{2} \dot{H}_{v}^{\gamma} + L_{v}^{2} \dot{H}_{x}^{\frac{\gamma}{2\beta+1}} $$

with Hilbertian norm $\|f\|_{\dot{X}_{\beta}^{\gamma}}$ given by

$$ \|f\|_{\dot{X}_{\beta}^{\gamma}}^{2} = \inf_{f=f_{1}+f_{2}} \|f_{1}\|_{L_{x}^{2} \dot{H}_{v}^{\gamma}}^{2} + \|f_{2}\|_{L_{v}^{2} \dot{H}_{x}^{\frac{\gamma}{2\beta+1}}}^{2}. $$

**Remark 2.7.** For $\gamma < d/2$, $\dot{X}_{\beta}^{\gamma}$ are Hilbert spaces. They are subspaces of $S'_{x,v} = S'(\mathbb{R}^{2d})$, and subspaces of $L_{\text{loc},x,v}^{2} = L_{\text{loc}}^{2}(\mathbb{R}^{2d})$ when $\gamma \ge 0$. The subspace of $S_{x,v} = S(\mathbb{R}^{2d})$ of Schwartz functions whose Fourier transforms (in $S_{\varphi,\xi}$) have compact support away from $\varphi = 0$ and $\xi = 0$ is a dense subspace for $-\infty < \gamma < d/2$.

**Remark 2.8.** For $\gamma \in [0, d/2)$, the space $\dot{X}_{\beta}^{-\gamma}$ identifies with the dual of $\dot{X}_{\beta}^{\gamma}$ for the duality extending the $L_{x,v}^{2}$ inner product.

We introduce next for $\gamma < d/2$,

$$ (2.4) \qquad \dot{\mathcal{F}}_{\beta}^{\gamma} = \{f \in \mathcal{D}'(\Omega); f \in L_{t,x}^{2} \dot{H}_{v}^{\gamma} \ \& \ (\partial_{t} + v \cdot \nabla_{x})f \in L_{t,x}^{2} \dot{H}_{v}^{\gamma-2\beta}\} $$

with Hilbertian norm $\|f\|_{\dot{\mathcal{F}}_{\beta}^{\gamma}}$ defined by

$$ (2.5) \qquad \|f\|_{\dot{\mathcal{F}}_{\beta}^{\gamma}}^{2} = \|D_{v}^{\gamma}f\|_{L_{t,x,v}^{2}}^{2} + \|(\partial_{t} + v \cdot \nabla_{x})f\|_{L_{t,x}^{2} \dot{H}_{v}^{\gamma-2\beta}}^{2}, $$

$$ (2.6) \qquad \dot{\mathcal{G}}_{\beta}^{\gamma} = \{f \in \mathcal{D}'(\Omega); f \in L_{t,x}^{2} \dot{H}_{v}^{\gamma} \ \& \ (\partial_{t} + v \cdot \nabla_{x})f \in L_{t}^{2} \dot{X}_{\beta}^{\gamma-2\beta}\} $$

with Hilbertian norm $\|f\|_{\dot{\mathcal{G}}_{\beta}^{\gamma}}$ defined by

$$ \|f\|_{\dot{\mathcal{G}}_{\beta}^{\gamma}}^{2} = \|D_{v}^{\gamma}f\|_{L_{t,x,v}^{2}}^{2} + \|(\partial_{t} + v \cdot \nabla_{x})f\|_{L_{t}^{2} \dot{X}_{\beta}^{\gamma-2\beta}}^{2}, $$

and

$$ (2.7) \qquad \dot{\mathcal{L}}_{\beta}^{\gamma} = \{f \in \mathcal{D}'(\Omega); f \in L_{t,x}^{2} \dot{H}_{v}^{\gamma} \ \& \ (\partial_{t} + v \cdot \nabla_{x})f \in L_{t}^{2} \dot{X}_{\beta}^{\gamma-2\beta} + L_{t}^{1} \dot{X}_{\beta}^{\gamma-\beta}\} $$

with norm $\|f\|_{\dot{\mathcal{L}}_{\beta}^{\gamma}}$ defined by

$$ \|f\|_{\dot{\mathcal{L}}_{\beta}^{\gamma}} = \|D_{v}^{\gamma}f\|_{L_{t,x,v}^{2}} + \|(\partial_{t} + v \cdot \nabla_{x})f\|_{L_{t}^{2} \dot{X}_{\beta}^{\gamma-2\beta} + L_{t}^{1} \dot{X}_{\beta}^{\gamma-\beta}} $$

The action of the field $\partial_t + v \cdot \nabla_x$ is taken in the sense of distributions on $\Omega$.

**Remark 2.9.** $\dot{\mathcal{F}}_{\beta}^{\gamma} \subset \dot{\mathcal{G}}_{\beta}^{\gamma} \subset \dot{\mathcal{L}}_{\beta}^{\gamma} \subset L_{\text{loc},t}^{1} S'_{x,v}$.

Eventually, we can repeat *verbatim* the definitions on $\Omega_{+} = (0, \infty) \times \mathbb{R}^{d} \times \mathbb{R}^{d}$.
**Proposition 2.13** (Weighted temporal estimates). Let $\beta > 0$ and $\gamma \in \mathbb{R}$. Then with bounds for the norms that only depend on $\beta$ and $\gamma$,

$T: L_t^2(W^{2\gamma-4\beta}) \to L_t^2(W^{2\gamma})$ is bounded.

$T: L_t^1(W^{\gamma-\beta}) \to L_t^2(W^{2\gamma})$ is bounded.

$W^{\gamma-\beta}T: L_t^2(W^{2\gamma-4\beta}) \to C_0(\mathbb{R}_t)$ is bounded.

$W^{\gamma-\beta}T: L_t^1(W^{\gamma-\beta}) \to C_0(\mathbb{R}_t)$ is bounded.

The same estimates are valid for $T^*$.

2.3.7. *Elementary estimates on the function K.* To prove Proposition 2.13, we begin with two elementary lemmas about the function K.

**Lemma 2.14.** Let $\beta > 0$. There are constants $0 < c_\beta < C_\beta < \infty$ such that for all $s, t \in \mathbb{R}$ with $s < t$ and $(\varphi, \xi) \neq (0, 0)$,

$$c_{\beta} \le \frac{\int_{s}^{t} |\xi - \tau\varphi|^{2\beta} d\tau}{(t-s)(|\xi - s\varphi|^{2\beta} + ((t-s)|\varphi|)^{2\beta})} \le C_{\beta}$$

and

$$c_{\beta} \le \frac{\int_{s}^{t} |\xi - \tau\varphi|^{2\beta} d\tau}{(t-s)(|\xi - t\varphi|^{2\beta} + ((t-s)|\varphi|)^{2\beta})} \le C_{\beta}.$$

*Proof.* Set $\xi' = \xi - s\varphi$ and $\varphi' = (t-s)\varphi$, then a change of variable yields

$$\int_{s}^{t} |\xi - \tau\varphi|^{2\beta} d\tau = (t-s) \int_{0}^{1} |\xi' - \tau\varphi'|^{2\beta} d\tau.$$

Now the function

$$(\xi', \varphi') \mapsto \int_{0}^{1} |\xi' - \tau\varphi'|^{2\beta} d\tau$$

is continuous on $\mathbb{R}^{2d}$, hence bounded above and below by positive numbers $C_\beta, c_\beta$ on the compact set defined by $|\xi'|^{2\beta} + |\varphi'|^{2\beta} = 1$. One can conclude the first inequality by using that it is also homogeneous of order $2\beta$. One can do similarly with $\xi' = \xi - t\varphi$ writing

$$\int_{s}^{t} |\xi - \tau\varphi|^{2\beta} d\tau = (t-s) \int_{0}^{1} |\xi' + \tau\varphi'|^{2\beta} d\tau$$

and conclude as before.

□
## 2.3. Kolmogorov operators.

### 2.3.1. Galilean change of variables.
The Galilean change of variables is defined by

$$ \Gamma f(t, x, v) = f(t, x + tv, v) $$

and also for fixed $t$,

$$ \Gamma(t)f(x, v) = f(x + tv, v) $$

which are isometries on $L_{t,x,v}^2$ and $L_{x,v}^2$ respectively. Furthermore, $(\Gamma(t))_{t \in \mathbb{R}}$ is a strongly continuous group on $L_{x,v}^2$ [32, Lemma 5.2]. These two properties imply that the mapping $\Gamma: C_0(\mathbb{R}_t; L_{x,v}^2) \to C_0(\mathbb{R}_t; L_{x,v}^2)$ is continuous.

### 2.3.2. Partial Fourier transform.
Our definition of (partial) Fourier transform in the variables $(\varphi, \xi)$ dual to $(x, v)$ is

$$ \hat{f}(t, \varphi, \xi) = \iint_{\mathbb{R}^d \times \mathbb{R}^d} e^{-i(x \cdot \varphi + v \cdot \xi)} f(t, x, v) \, dx \, dv. $$

This technical lemma relating the Fourier transform and the Galilean change of variables $\Gamma$ will be used frequently. To simplify the presentation we introduce the weights

$$ (2.8) \qquad W(t, \varphi, \xi) = \sup(|\xi - t\varphi|, |\varphi|^{\frac{1}{1+2\beta}}), $$

and

$$ w(\varphi, \xi) = \sup(|\xi|, |\varphi|^{\frac{1}{1+2\beta}}) $$

which depend on the fixed parameter $\beta > 0$.

We define $\mathcal{D}_t S_{x,v}$ as the subspace of functions in $S(\Omega)$ having compact support in the $t$-variable, identified to compactly supported functions in $t$ valued in the Schwartz space in $(x, v)$ variables. Its dual is denoted $\mathcal{D}'_t S'_{x,v}$ and identified with the space of distributions valued in the space of tempered distributions.

**Lemma 2.10.** Let $\gamma \in \mathbb{R}$ with $\gamma < d/2$. Then the map $f \mapsto \hat{\Gamma}f$ is an isomorphism

(i) $S(\Omega)$ onto $S(\Omega)$, and $\mathcal{D}_t S_{x,v}$ onto $\mathcal{D}_t S_{\varphi,\xi}$.

(ii) $S'(\Omega)$ onto $S'(\Omega)$, and $\mathcal{D}'_t S'_{x,v}$ onto $\mathcal{D}'_t S'_{\varphi,\xi}$.

(iii) $L_t^p \dot{X}_\beta^\gamma$ onto $L_t^p L_{\varphi,\xi}^2 (W^{2\gamma} d\varphi d\xi)$, $1 \le p < \infty$.

(iv) $C_0(\mathbb{R}_t; \dot{X}_\beta^\gamma)$ onto $C_0(\mathbb{R}_t; L_{\varphi,\xi}^2 (W^{2\gamma} d\varphi d\xi))$.

(v) $L_t^2 \dot{X}_\beta^\gamma$ onto $L_{t,\varphi,\xi}^2 (W^{2\gamma} dt d\varphi d\xi)$.

*Proof.* We have $\hat{\Gamma}f(t, \varphi, \xi) = \hat{f}(t, \varphi, \xi - t\varphi)$. The first item is rather easy, and we skip the details. The second item follows by duality using the first one, replacing $\Gamma$ by $\Gamma^* = \Gamma^{-1}$. For the third and fourth items, Plancherel's theorem tells us that $u \in \dot{X}_\beta^\gamma$ if and only if $\hat{u} \in L_{\varphi,\xi}^2(w^{2\gamma} d\varphi d\xi)$. Apply this to $u(x, v) = f(t, x, v)$ for fixed $t$ and then use the change of variables induced by $\Gamma(t)$. This proves the third item. For the fifth item, apply further Fubini's theorem when $p = 2$. Concerning the fourth item, the right-hand space should be understood as the image under the change of variable $\hat{\Gamma}$ of $C_0(\mathbb{R}_t; L_{\varphi,\xi}^2(w^{2\gamma} d\varphi d\xi))$ where $\hat{\Gamma}$ is the conjugate of the Galilean change of variables through the Fourier transform, that is $\hat{\Gamma}h(t, \varphi, \xi) = h(t, \varphi, \xi - t\varphi)$. $\square$
**Lemma 2.15.** Let $\beta > 0$, $\varepsilon \in \mathbb{R}$, $\alpha > 0$ and $(\varphi, \xi) \neq (0,0)$. One has the following estimates with implicit constants depending on $\beta, \varepsilon, \alpha$ and independent of $s, t, \varphi, \xi$.

$$ (2.13) \qquad \int_{-\infty}^{t} K(t, s)^{\alpha} ds \approx_{\beta, \alpha} W^{-2\beta}(t), $$

$$ (2.14) \qquad \int_{s}^{\infty} K(t, s)^{\alpha} dt \approx_{\beta, \alpha} W^{-2\beta}(s), $$

$$ (2.15) \qquad \sup(W^{\varepsilon}(t), W^{\varepsilon}(s)) K(t, s) \lesssim_{\beta, \varepsilon} \inf(W^{\varepsilon}(t), W^{\varepsilon}(s)), \quad s \le t. $$

$$ (2.16) \qquad \int_{-\infty}^{t} K(t, s)W^{2\beta-\varepsilon}(s) ds \lesssim_{\beta, \varepsilon} W^{-\varepsilon}(t), $$

$$ (2.17) \qquad \int_{s}^{\infty} K(t, s)W^{2\beta-\varepsilon}(t) dt \lesssim_{\beta, \varepsilon} W^{-\varepsilon}(s). $$

*Proof.* Proof of (2.13) and (2.14). It is easily checked for all $A, B \ge 0$, and $c > 0$,

$$ (2.18) \qquad \int_{0}^{\infty} e^{-cu^{1+2\beta}A^{2\beta}} e^{-cuB^{2\beta}} du \approx_{c, \beta} \inf(B^{-2\beta}, A^{-\frac{2\beta}{1+2\beta}}) = (\sup(B, A^{\frac{1}{1+2\beta}}))^{-2\beta}. $$

Thus, use the comparisons in Lemma 2.14, the change of variable $t-s = u$ and (2.18).

Proof of (2.15). First, there is nothing to do if $s = t$ and we assume $s < t$. Also for $\varepsilon = 0$, we see that $0 < K(t, s) \le 1$, so we are done. Assume next $\varepsilon \ne 0$. Set $A = (t-s)^{\frac{1}{2\beta}}|\xi - t\varphi|$, $B = (t-s)^{\frac{1}{2\beta}}|\xi - s\varphi|$ and $C = (t-s)^{\frac{2\beta+1}{2\beta}}|\varphi|$. We have $C > 0$, $A, B \ge 0$ and $|A - B| \le C$. Observe that using both inequalities in Lemma 2.14 yields

$$ K(t, s) \le \exp \left( - \frac{c_{\beta}}{2} \left( A^{2\beta} + B^{2\beta} + 2C^{2\beta} \right) \right) $$

so that after multiplication by $(t-s)^{\frac{1}{2\beta}}$, the desired inequality follows from

$$ \sup(\rho^{\varepsilon}, \rho^{-\varepsilon}) \exp \left( - \frac{c_{\beta}}{2} \left( A^{2\beta} + B^{2\beta} + 2C^{2\beta} \right) \right) \le C_{\beta, \varepsilon}, $$

where

$$ \rho = \frac{\sup(A, C^{\frac{1}{2\beta+1}})}{\sup(B, C^{\frac{1}{2\beta+1}})}. $$

The case $\varepsilon < 0$ reduces to the case $\varepsilon > 0$ by symmetry. Thus, we assume now $\varepsilon > 0$. If $\sup(A, C^{\frac{1}{2\beta+1}}) = C^{\frac{1}{2\beta+1}}$, then $\rho \le 1$. If $\sup(A, C^{\frac{1}{2\beta+1}}) = A$, then, using $A \le B+C$,

$$ \rho \le 1 + \frac{C}{\sup(B, C^{\frac{1}{2\beta+1}})} \le 1 + \left( \sup(B, C^{\frac{1}{2\beta+1}}) \right)^{2\beta}. $$

Similarly, we have $\rho^{-1} \le 1 + \left( \sup(A, C^{\frac{1}{2\beta+1}}) \right)^{2\beta}$. Hence, the term $\sup(\rho^{\varepsilon}, \rho^{-\varepsilon})$ can be absorbed by the exponential and this finishes the proof of (2.15).
2.3.3. Forward and backward Kolmogorov operators of order $\beta$. The forward and backward Kolmogorov (integral) operators of order $\beta$ are denoted by $K_{\beta}^{\pm}$. They are formally defined as inverse maps of $\pm(\partial_t + v \cdot \nabla_x) + (-\Delta_v)^{\beta}$, and this is what we rigorously prove under specific assumptions. To define them for now, we use Lemma 2.10 and for $S \in \mathcal{D}'_t S_{x,v}$ (in particular $S \in S'(\Omega)$ is possible) we set

$$f = K_{\beta}^{\pm} S$$

whenever

$$(2.9) \qquad \widehat{\Gamma}f = T(\widehat{\Gamma}S) \in \mathcal{D}'_t S'_{\varphi, \xi},$$

with

$$(2.10) \qquad Th(t, \varphi, \xi) = \int_{-\infty}^{t} K(t, s, \varphi, \xi)h(s, \varphi, \xi) ds,$$

where for $s, t \in \mathbb{R}$ and $(\varphi, \xi) \in \mathbb{R}^{2d}$,

$$(2.11) \qquad K(t, s, \varphi, \xi) = \exp \left( - \int_s^t |\xi - \tau\varphi|^{2\beta} d\tau \right).$$

Of course, the integral for $Th$ above makes sense only on certain classes of measurable functions $h$ of $(t, \varphi, \xi)$ and is thought of as the Duhamel formula for the ordinary differential equation,

$$g'(t) + |\xi - t\varphi|^{2\beta}g(t) = h(t), \quad t \in \mathbb{R}.$$

But the corresponding $S$ are distributions.

The operator $K_{\beta}^{-}$ is the formal adjoint to $K_{\beta}^{+}$. It is associated in the same fashion to the adjoint of $T$ given by the following formula,

$$(2.12) \qquad T^*h(s, \varphi, \xi) = \int_s^{+\infty} K(t, s, \varphi, \xi)h(t, \varphi, \xi) dt.$$

**Remark 2.11.** We insist that at this stage that $K_{\beta}^{\pm}$ are defined through integral representations, that is, with the help of what is usually called the Kolmogorov fundamental solution in Fourier variables. That $K_{\beta}^{\pm}S$ are distributional solutions of the corresponding PDE under appropriate assumptions needs to be proved. It is after we prove uniqueness results for such PDEs that we will be able to connect these solutions to weak solutions.

2.3.4. *Density.* In order to establish, for instance, the energy equality (see Lemma 4.2), it is useful to have density results at hand. This will also be needed to prove (iii) in Theorem 2.2 and for various estimates. As we use a homogeneous setup, we cannot use convolution techniques in the spatial variables. We use Fourier techniques instead.

**Lemma 2.12 (Density).** Assume $d \ge 1$. Let $S_K$ be the subset of those $S \in \mathcal{D}_t S_{x,v}$ whose partial Fourier transform has compact support in $\mathbb{R}_t \times (\mathbb{R}_{\varphi}^d \setminus \{0\}) \times (\mathbb{R}_{\xi}^d \setminus \{0\})$. Then $S_K$ is dense in $L_t^p \dot{X}_{\beta}^{\gamma}$ and in $C_0(\mathbb{R}_t; \dot{X}_{\beta}^{\gamma})$ for all $1 \le p < \infty$ and $-\infty < \gamma < d/2$ and $\Gamma K_{\beta}^{\pm}(\tilde{S}_K) \subset H_t^1(L_{x,v}^2)$.
Proof of (2.16). Using the pointwise inequality (2.15) and the integral one (2.14),

$$
\begin{align*}
\int_{-\infty}^{t} K(t, s)W^{2\beta-\varepsilon}(s) \, ds &= \int_{-\infty}^{t} K(t, s)^{\frac{1}{2}}(W^{4\beta-2\varepsilon}(s)K(t, s))^{\frac{1}{2}} \, ds \\
&\lesssim_{\beta, 2\beta-\varepsilon} W^{2\beta-\varepsilon}(t) \int_{-\infty}^{t} K(t, s)^{\frac{1}{2}} \, ds \\
&\lesssim_{\beta, \varepsilon} W^{-\varepsilon}(t).
\end{align*}
$$

The proof of (2.17) is the same.

<figure><img src="image_3.png" /></figure>

2.3.8. *An intermediate result.* Before proving Proposition 2.13, we prove estimates for $T$ in $L^1$ and $L^\infty$ spaces.

**Proposition 2.16** ($L^1$ and $L^\infty$ estimates for $T$). Let $\beta > 0$ and $\varepsilon \in \mathbb{R}$. Then with bounds for the norms that only depend on $\beta$ and $\varepsilon$,

$W^\varepsilon T W^{-\varepsilon} : L_t^1 \to C_0(\mathbb{R}_t)$ is bounded.

$W^\varepsilon T W^{2\beta-\varepsilon} : L_t^1 \to L_t^1$ is bounded.

$W^\varepsilon T W^{2\beta-\varepsilon} : L_t^\infty \to L_t^\infty$ is bounded.

The same estimates are valid for $T^*$.

*Proof.* We only treat $T$ as the proofs for $T^*$ are similar. We start with the $L_t^\infty$ bound for $W^\varepsilon T(W^{-\varepsilon}f)$ when $f \in L_t^1$. We have by (2.15),

$$
W^{\varepsilon}(t)|T(W^{-\varepsilon}f)(t)| \leq \int_{-\infty}^{t} W^{\varepsilon}(t)K(t,s)W^{-\varepsilon}(s)|f(s)| \, ds \lesssim_{\beta, \varepsilon} \int_{-\infty}^{t} |f(s)| \, ds,
$$

which is controlled by $\|f\|_{L_t^1}$. This also proves the fact $W^\varepsilon(t)|T(W^{-\varepsilon}f)(t)| \to 0$ when $t \to -\infty$.

By a density argument and closedness of $C_0(\mathbb{R}_t)$ in $L_t^\infty$, it is now enough to prove the continuity of $W^\varepsilon T(W^{-\varepsilon}f)$ and limit $0$ at $+\infty$ assuming $f$ to be compactly supported in $[-R, R]$ for some fixed $R > 0$ and continuous. For the limit, the above estimate rewrites for $t \ge R$,

$$
W^{\epsilon}(t)|T(W^{-\epsilon}f)(t)| \leq \int_{-R}^{R} W^{\epsilon}(t)K(t,s)W^{-\epsilon}(s)|f(s)| \, ds
$$

and clearly the integral converges to 0 by dominated convergence as $t \to +\infty$.

For the continuity, we use that $W^\epsilon$ and $W^{-\epsilon}$ are continuous and also that $T(W^{-\epsilon}f)$ is locally Lipschitz in time since, as $f$ is continuous with compact support, it is easy to see that $g = T(W^{-\epsilon}f)$ solves the ordinary differential equation $g'(t) + |\xi - t\varphi|^{2\beta}g(t) = (W^{-\epsilon}f)(t)$ in the classical sense.

Next, the $L_t^1 - L_t^1$ and $L_t^\infty - L_t^\infty$ bounds of $W^\epsilon T W^{2\beta-\epsilon}$ follow from (2.16) and (2.17) by Fubini's theorem. $\square$

<figure><img src="image_4.png" /></figure>

2.3.9. *Proof of Proposition 2.13.* We are now ready to prove the temporal weighted estimates for the operator $T$ announced in Proposition 2.11.

*Proof of Proposition 2.13.* The last boundedness statement is already contained in Proposition 2.16. We prove the other three properties in the order they are stated.
*Proof.* By Remark 2.7 after the definition of the spaces $\dot{X}_{\beta}^{\gamma}$, one can see that any element of the listed spaces can be approximated by a function $S \in S_K$.

It remains to show that for $S \in S_K$, $\Gamma K_{\beta}^{\pm} S$ and $\partial_t \Gamma K_{\beta}^{\pm} S$ belong to $L_{t,x,v}^2$. We do it for $K_{\beta}^{+}$, the proof for $K_{\beta}^{-}$ being the same.

Let $h = \widehat{\Gamma S}$. This function is $C^{\infty}$ and has support in

$$ \{(t, \varphi, \xi) \in \Omega; |t| \le M, m \le |\varphi| \le M, m \le |\xi - t\varphi| \le M\} $$

for some $0 < m \le M < \infty$.

We now look at $g = Th$ which is to be $g = \widehat{\Gamma K_{\beta}^{+} S}$. Hence it is enough to show that $g$ and $\partial_t g$ belong to $L_{t,\varphi,\xi}^2$.

Since

$$ g(t, \varphi, \xi) = \int_{-M}^{t} K(t, s, \varphi, \xi)h(s, \varphi, \xi) \, ds $$

and $|\xi| \le M + M^2$ on the support of $h$, we have that the support of $g$ is contained in $\{(t, \varphi, \xi) \in \mathbb{R}^{2d+1}; t \ge -M, m \le |\varphi| \le M, |\xi| \le M + M^2\}$, and that $g$ is continuous and bounded. In addition, if $t \ge 2M$ and using $|\xi - s\varphi| \ge m$ and $s \le M$ on the support of $h(s, \varphi, \xi)$, we have $|K(t, s, \varphi, \xi)| \lesssim e^{-c_{\beta}m^{2\beta}(t-M)}$ (see Lemma 2.20 below). Hence $e^{c|t-M|}g$ is also bounded for some $c > 0$. All this implies that $g, |\xi - t\varphi|^{2\beta}g \in L_{t,\varphi,\xi}^2$.

Eventually, using the support of $h$ and the estimates for $\partial_t K$ allows us to differentiate $g$ under the integral sign and we find the equation $\partial_t g + |\xi - t\varphi|^{2\beta}g = h$. From the above properties of $g$, we conclude that $\partial_t g \in L_{t,\varphi,\xi}^2$ as well. $\square$

**Remark 2.13.** For $S \in S_K$, $K_{\beta}^{\pm} S$ are strong $L^2$ solutions of the forward and backward Kolmogorov equations respectively.

We remark that we can reach a stronger density result when $d \ge 2$. We include this result for its own sake.

**Lemma 2.14** (Stronger density). Assume $d \ge 2$. There exists a subset $\tilde{S}_K$ of $S(\Omega)$ which is dense in $L_t^p \dot{X}_{\beta}^{\gamma}$ and in $C_0(\mathbb{R}_t; \dot{X}_{\beta}^{\gamma})$ for all $1 \le p < \infty$ and $-\infty < \gamma < d/2$ and such that $K_{\beta}^{\pm}(S_K) \subset S(\Omega)$.

*Proof.* We begin with the construction of $\tilde{S}_K$, arguing again on the Fourier side. Let $d(\xi, \mathbb{R}\varphi)$ denote the distance from $\xi$ to the line $\mathbb{R}\varphi$. When $\varphi \ne 0$, we have $d(\xi, \mathbb{R}\varphi)^2 = |\xi - (\xi \cdot \varphi) \frac{\varphi}{|\varphi|^2}|^2$ is a $C^{\infty}$ function of $(\varphi, \xi)$ on $\mathbb{R}_{\varphi}^d \setminus \{0\} \times \mathbb{R}_{\xi}^d$. It is not identically zero as $d \ge 2$. Define the non-empty open set

$$ \Omega_K = \{(t, \varphi, \xi) \in \Omega; |\varphi| > 0, d(\xi, \mathbb{R}\varphi) > 0\}. $$

We say that $S \in \tilde{S}_K$ if and only if $h = \widehat{\Gamma S} \in S(\Omega)$ and has compact support in $\Omega_K$.

Let us prove the density of $\tilde{S}_K$ in $L_t^p \dot{X}_{\beta}^{\gamma}$ and $C_0(\mathbb{R}_t; \dot{X}_{\beta}^{\gamma})$. By Lemma 2.12, we may assume that $S \in S_K$, so that $h = \widehat{\Gamma S}$ has support in $\{(t, \varphi, \xi) \in \Omega; |t| \le M, m \le |\varphi| \le M, m \le |\xi - t\varphi| \le M\}$ for some $0 < m \le M < \infty$. On this set, the function $(t, \varphi, \xi) \mapsto d(\xi, \mathbb{R}\varphi)^2$ is $C^{\infty}$ and if we let $\chi \in C^{\infty}(0, \infty)$ with $\chi(y) = 1$ if $y \ge 4$ and $\chi(y) = 0$ if $y \le 1$, then $h_{\varepsilon} : (t, \varphi, \xi) \mapsto \chi(d(\xi, \mathbb{R}\varphi)^2/\varepsilon^2)h(t, \varphi, \xi)$ is $C^{\infty}$ with compact support in $\Omega_K$. For the convergence of $h_{\varepsilon}$ to $h$, we have that $h$ has support in $[-M, M] \times \{m \le |\varphi| \le M\} \times \{|\xi| \le M + M^2\}$. In addition $h - h_{\varepsilon} = 0$ when
First, interpolating $L_t^1 - L_t^1$ and $L_t^\infty - L_t^\infty$ boundedness of $W^\gamma TW^{2\beta-\gamma}$, we obtain $L_t^2 - L_t^2$ boundedness which is equivalent to the first item.

Secondly, we have $W^\gamma T : L_t^1(W^{\gamma-2\beta}) \to L_t^1$ and $W^\gamma T : L_t^1(W^\gamma) \to L_t^\infty$, Using Stein-Weiss interpolation with change of measure (see [7, Theorem 5.4.1]), we obtain $W^\gamma T : L_t^1(W^{\gamma-\beta}) \to L_t^2$ is bounded, which is equivalent to the second item.

Thirdly, the second case applied to $T^*$ rewrites $W^\gamma T^* W^{\beta-\gamma} : L_t^1 \to L_t^2$ is bounded, so by duality and changing $\beta - \gamma$ to $\gamma - \beta$, we have $W^{\gamma-\beta}TW^{2\beta-\gamma}$ maps $L_t^2$ into $L_t^\infty$. Note that if we apply this to continuous and compactly supported functions $f$ then we also have $W^{\gamma-\beta}TW^{2\beta-\gamma}f = W^{\gamma-\beta}TW^{\beta-\gamma}(W^\beta f) \in C_0(\mathbb{R}_t)$ as $W^\beta f \in L_t^1$. By density of such functions $f$ in $L_t^2$ and closedness of $C_0(\mathbb{R}_t)$ in $L_t^\infty$, we conclude that $W^{\gamma-\beta}TW^{2\beta-\gamma}$ maps $L_t^2$ into $C_0(\mathbb{R}_t)$ which is equivalent to the third item. $\square$

2.3.10. *Proof of Proposition 2.11.* We can now obtain the desired estimates for the Kolmogorov operators.

*Proof of Proposition 2.11.* Let us concentrate on $K_\beta^+$, the proof for $K_\beta^-$ being similar. We come back to the definition of $T$ depending on the three variables and recall that the weight $W$ also depends on the three variables. By the Fourier transform correspondence $S \mapsto \hat{\Gamma}S$, Lemma 2.9 and Fubini's theorem, the first assertion for $K_\beta^+$ is equivalent to the weighted boundedness

$$T : L_{\varphi, \xi}^2 L_t^2(W^{2\gamma-4\beta}) \to L_{\varphi, \xi}^2 L_t^2(W^{2\gamma}).$$

Using the conjugate $\hat{\Gamma}$ of the Galilean change of variables through the Fourier transform, and using the fact that this map preserves $C_0(\mathbb{R}_t; L_{\varphi, \xi}^2)$, the second is equivalent to

$$W^{\gamma-\beta} T : L_{\varphi, \xi}^2 L_t^2(W^{2\gamma-4\beta}) \to C_0(\mathbb{R}_t; L_{\varphi, \xi}^2).$$

The third is equivalent to

$$T : L_t^1 L_{\varphi, \xi}^2 (W^{2\gamma-2\beta}) \to L_{\varphi, \xi}^2 L_t^2 (W^{2\gamma}),$$

where $L_{\varphi, \xi}^2(W^{2\gamma-2\beta})$ is here the weighted $L^2$ space with respect to $W^{2\gamma-2\beta}(t, \varphi, \xi) d\varphi d\xi$, for fixed $t$. With the same idea as for the second assertion, the fourth one is equivalent to

$$W^{\gamma-\beta} T : L_t^1 L_{\varphi, \xi}^2 (W^{2\gamma-2\beta}) \to C_0(\mathbb{R}_t; L_{\varphi, \xi}^2).$$

The first two assertions on $T$ are direct consequences of Proposition 2.13.

For the third one, we have by Proposition 2.13 and by Minkowski integral inequality

$$\|\| W^\gamma Th \|_{L_t^2} \|_{L_{\varphi, \xi}^2} \lesssim_{\beta, \gamma} \|\| W^{\gamma-\beta} h \|_{L_t^1} \|_{L_{\varphi, \xi}^2} \\ \le \|\| W^{\gamma-\beta} h \|_{L_{\varphi, \xi}^2} \|_{L_t^1}.$$

For the $L^\infty$ bound in the fourth one, write

$$\| W^{\gamma-\beta}(t) Th(t) \|_{L_{\varphi, \xi}^2} \le \int_{-\infty}^t \| W^{\gamma-\beta}(t) K(t, s) h(s) \|_{L_{\varphi, \xi}^2} \, ds \\ \lesssim_{\beta, \gamma} \int_{-\infty}^t \| W^{\gamma-\beta}(s) h(s) \|_{L_{\varphi, \xi}^2} \, ds$$
$d(\xi, \mathbb{R}\varphi) \ge 2\epsilon$. If $d(\xi, \mathbb{R}\varphi) \le 2\epsilon$, then $\xi$ belongs to the cylinder $\mathbb{R}\varphi + B(0, 2\epsilon)$. Thus, using all the support information,

$$ \left\| \left( \iint |h - h_{\epsilon}|^2 W^{2\gamma} d\xi d\varphi \right)^{1/2} \right\|_{L_t^p} \lesssim_{m,M,\gamma,\beta,p} \|h\|_{L^{\infty}} \epsilon^{1/2}. $$

We turn to proving that the Kolmogorov operators map $\tilde{S}_K$ to $S(\Omega)$. We do it for $K_\beta^+$, the proof for $K_\beta^-$ being the same. Let $S \in S_K$ and set $h = \Gamma_S$. We have to show that $g = Th \in S(\Omega)$. Since $h$ has compact support in $\Omega_K$, there exists $0 < m < M < \infty$ such that $h$ is supported in $[-M, M] \times F$, with

$$ F = \{(\varphi, \xi) \in \mathbb{R}^{2d}; m \le |\varphi| \le M, m \le d(\xi, \mathbb{R}\varphi), |\xi| \le M\}. $$

On $\mathbb{R} \times (\mathbb{R}^{2d} \setminus F) \cup ((-\infty, -M] \times F)$, $g$ is clearly 0, that is, $g$ is supported in $[-M, \infty) \times F$. Let $0 < m' < m$ and $M < M' < \infty$ and $O = \{(\varphi, \xi) \in \mathbb{R}^{2d}; m' < |\varphi| < M', m' < d(\xi, \mathbb{R}\varphi), |\xi| < M'\}$. When $m' \le d(\xi, \mathbb{R}\varphi)$ and $t \ge s$, $\int_s^t |\xi - \tau\varphi|^{2\beta} d\tau \ge (t-s)(m')^{2\beta}$. In particular $(\varphi, \xi) \mapsto \exp(-\int_s^t |\xi - \tau\varphi|^{2\beta} d\tau)$ is $C^\infty$ on $O$ and $g$ is $C^\infty$ with respect to $(\varphi, \xi) \in O$ for any $t > -M'$, and all its partial derivatives with respect to $(\varphi, \xi) \in O$ have exponential decay in $t \to \infty$, uniformly. Next, as $\partial_t g = -|\xi - t\varphi|^{2\beta}g + h$, the same is true for $\partial_t g$. By induction, we obtain that $g$ is $C^\infty$ with respect to $(t, \varphi, \xi)$. Since $g$ has support in $[-M, \infty) \times F$, with all partial derivatives having exponential decay in time, uniformly in $(\varphi, \xi)$, we conclude that $g \in S(\Omega)$. $\square$

Remark that $\tilde{S}_K$ is even contained in $D_t S_{x,v}$ but elements in $K_\beta^+(\tilde{S}_K)$ do not have compact support in $t$.

**2.3.5. Estimates.** The main technical result related to Kolmogorov operators is the following a priori estimates. It makes precise how Kolmogorov operators act on various functional spaces in $(t, x, v)$.

**Proposition 2.15** (A priori estimates for Kolmogorov operators). Let $\beta > 0$ and $\gamma \in \mathbb{R}$. Assume that $S \in S_K$ of Lemma 2.12. In particular, $K_\beta^+ S$ belongs to $L_{t,x,v}^2$, so all fractional derivatives below make sense allowing the $\dot{X}_\beta^\alpha$ and $\dot{H}_v^\alpha$ norms to be computed. Then $K_\beta^+ S \in L_t^2 \dot{X}_\beta^\gamma \cap C_0(\mathbb{R}_t; \dot{X}_\beta^{\gamma-\beta})$ with bound

$$ \max\{\|K_\beta^+ S\|_{L_t^2 \dot{X}_\beta^\gamma}, \sup_{t \in \mathbb{R}} \|K_\beta^+ S\|_{\dot{X}_\beta^{\gamma-\beta}}\} \lesssim \min\{\|S\|_{L_t^2 \dot{X}_\beta^{\gamma-2\beta}}, \|S\|_{L_t^1 \dot{X}_\beta^{\gamma-\beta}}\} $$

where the implicit constants depend only on $\beta, \gamma, d$. In particular, if $0 \le \gamma \le 2\beta$, we have the following four inequalities

$$ \max\{\|D_v^\gamma K_\beta^+ S\|_{L_{t,x,v}^2}, \|D_x^{\frac{\gamma}{2\beta+1}} K_\beta^+ S\|_{L_{t,x,v}^2}\} \lesssim \min\{\|D_v^{\gamma-2\beta} S\|_{L_{t,x,v}^2}, \|D_x^{\frac{\gamma-2\beta}{2\beta+1}} S\|_{L_{t,x,v}^2}\}, \\ \text{and if } \beta \le \gamma \le 2\beta, $$

$$ \max \left\{ \sup_{t \in \mathbb{R}} \| D_v^{\gamma-\beta} K_\beta^+ S(t) \|_{L_{x,v}^2}, \sup_{t \in \mathbb{R}} \| D_x^{\frac{\gamma-\beta}{2\beta+1}} K_\beta^+ S(t) \|_{L_{x,v}^2} \right\} \\ \lesssim \min \{ \| D_v^{\gamma-2\beta} S \|_{L_{t,x,v}^2}, \| D_x^{\frac{\gamma-2\beta}{2\beta+1}} S \|_{L_{t,x,v}^2}, \| S \|_{L_t^1 \dot{X}_\beta^{\gamma-\beta}} \}. $$

Using our definition of spaces in the restricted range of parameters and density, this immediately gives us the following boundedness results.
using Minkowski integral inequality and (2.15) directly, rather than Proposition 2.13. Continuity and limits follow the argument in Proposition 2.16. We leave details to the reader. □

### 3. UNIQUENESS AND KINETIC EMBEDDINGS

**3.1. Uniqueness.** The kinetic embeddings rely on a crucial uniqueness result.

**Lemma 3.1 (Uniqueness).** Let $\gamma < d/2$ and $\beta > 0$. If $f \in L_{t,x}^2 \dot{H}_v^\gamma$ satisfies $\pm(\partial_t + v \cdot \nabla_x) f + (-\Delta_v)^\beta f = 0$ in the sense of distributions, then $f = 0$.

*Proof.* We only consider $(\partial_t + v \cdot \nabla_x) f + (-\Delta_v)^\beta f = 0$, while the backward equation can be treated by a similar argument. When $f \in L_{t,x}^2 \dot{H}_v^\gamma$ and $\gamma < d/2$, $f$ and $(-\Delta_v)^\beta f$ are tempered distributions, with Fourier transform in $L_{\text{loc}}^1(\Omega)$, and the equation can be interpreted in $S'(\Omega)$. Setting $G = \hat{\Gamma}f \in S'(\Omega)$, we have $G \in L_{\text{loc}}^1(\Omega)$ and $\partial_t G + |\xi - t\varphi|^{2\beta}G = 0$ in $S'(\Omega)$. From $\partial_t G = -|\xi - t\varphi|^{2\beta}G \in L_{\text{loc}}^1(\Omega)$, we deduce that $G$ has a representative in $C(\mathbb{R}; L_{\text{loc}}^1(\mathbb{R}^{2d}))$ and then in $C^1(\mathbb{R}; L_{\text{loc}}^1(\mathbb{R}^{2d}))$. Identifying $G$ to this representative, the equation can be interpreted as a first-order differential equation valued in $L_{\text{loc}}^1(\mathbb{R}^{2d})$. Thus, there exists $G_0 \in L_{\text{loc}}^1(\mathbb{R}^{2d})$ such that $G = K(t, 0)G_0$ for all $t \in \mathbb{R}$, where $K(t, 0)$ is the kernel defined in (2.11). As $(-\Delta_v)^{\gamma/2}f \in L_{t,x,v}^2$, we have that $|\xi - t\varphi|^\gamma G \in L_{t,\varphi,\xi}^2$, that is

$$ (3.1) \quad \iiint_{\Omega} |\xi - t\varphi|^{2\gamma} \exp \left( -2 \int_0^t |\xi - \tau\varphi|^{2\beta} d\tau \right) |G_0(\varphi, \xi)|^2 dt d\varphi d\xi < \infty. $$

We can use Fubini's theorem and notice that the $t$-integral on $\mathbb{R}$ is infinite when $(\varphi, \xi) \neq (0, 0)$ because of the exponential growth when $t \to -\infty$. This proves that $G_0 = 0$ almost everywhere. Hence, $G = 0$ and so is $f$. □

**Remark 3.2.** The argument works for a larger class of $f$. It suffices that $f, (-\Delta_v)^\beta f \in S'(\Omega)$ with $\hat{f} \in L_{\text{loc}}^1(\Omega)$ such that there is some measurable function $m$ for which $mG \in L_t^1 L_{\text{loc}}^1(\mathbb{R}^{2d})$ and

$$ \int_{\mathbb{R}} |m(t, \varphi, \xi)| \exp \left( - \int_{0}^{t} |\xi - \tau\varphi|^{2\beta} d\tau \right) dt = \infty $$

almost everywhere. For example this works for $f \in L_{t,v}^2 \dot{H}_x^{\frac{\gamma}{2\beta+1}}$. This also works for sums of spaces for which these conditions hold.

**3.2. Isomorphism properties.** A consequence of uniqueness is the isomorphism properties of the Kolmogorov operators for our scales of spaces in an appropriate range of exponents.

**Lemma 3.3 (Isomorphisms).** Assume $\gamma < d/2$ with $0 \le \gamma \le 2\beta$. The Kolmogorov operators $K_\beta^\pm$ are isomorphisms from $L_{t,x}^2 \dot{H}_v^{\gamma-2\beta}$ onto $\dot{F}_\beta^\gamma$, from $L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ onto $\dot{G}_\beta^\gamma$ and from $L_t^2 \dot{X}_\beta^{\gamma-2\beta} + L_t^1 \dot{X}_\beta^{\gamma-\beta}$ onto $\dot{\mathcal{L}}_\beta^\gamma$.

*Proof.* The proofs are all the same. We do the proof of the last statement.

It follows from Proposition 2.11 that $K_\beta^+$ maps continuously $S \in L_t^2 \dot{X}_\beta^{\gamma-2\beta} + L_t^1 \dot{X}_\beta^{\gamma-\beta}$ into $\dot{\mathcal{L}}_\beta^\gamma$. Indeed, $K_\beta^+ S \in L_{t,x}^2 \dot{H}_v^\gamma$ as $\gamma \ge 0$, and $(\partial_t + v \cdot \nabla_x)K_\beta^+ S = S - (-\Delta_v)^\beta K_\beta^+ S \in L_t^2 \dot{X}_\beta^{\gamma-2\beta} + L_t^1 \dot{X}_\beta^{\gamma-\beta}$ since $(-\Delta_v)^\beta K_\beta^+ S \in L_{t,x}^2 \dot{H}_v^{\gamma-2\beta} \subset L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ when $\gamma - 2\beta \le 0$.
**Proposition 2.16** (Estimates for Kolmogorov operators). Let $\beta > 0$ and $\gamma \in \mathbb{R}$. We have the following bounds.

(i) For $\gamma < d/2$, the operators $K_{\beta}^{\pm}$ map boundedly $L_{t}^{2} \dot{X}_{\beta}^{\gamma-2\beta}$ to $L_{t}^{2} \dot{X}_{\beta}^{\gamma}$.

(ii) For $\gamma - \beta < d/2$, the operators $K_{\beta}^{\pm}$ map boundedly $L_{t}^{2} \dot{X}_{\beta}^{\gamma-2\beta}$ to $C_{0}(\mathbb{R}_{t}; \dot{X}_{\beta}^{\gamma-\beta})$.

(iii) For $\gamma < d/2$, the operators $K_{\beta}^{\pm}$ map boundedly $L_{t}^{1} \dot{X}_{\beta}^{\gamma-\beta}$ to $L_{t}^{2} \dot{X}_{\beta}^{\gamma}$.

(iv) For $\gamma - \beta < d/2$, $K_{\beta}^{\pm}$ map boundedly $L_{t}^{1} \dot{X}_{\beta}^{\gamma-\beta}$ to $C_{0}(\mathbb{R}_{t}; \dot{X}_{\beta}^{\gamma-\beta})$.

If $0 \le \gamma \le 2\beta$ and $\gamma < d/2$, then we have boundedness on $L_{t,x,v}^{2}$ of

$$D_{v}^{\gamma} K_{\beta}^{+} D_{v}^{2\beta-\gamma}, D_{x}^{\frac{\gamma}{2\beta+1}} K_{\beta}^{+} D_{v}^{2\beta-\gamma}, D_{v}^{\gamma} K_{\beta}^{+} D_{x}^{\frac{2\beta-\gamma}{2\beta+1}}, D_{x}^{\frac{\gamma}{2\beta+1}} K_{\beta}^{+} D_{x}^{\frac{2\beta-\gamma}{2\beta+1}}.$$

**Remark 2.17.** The bounds into $C_0$ spaces are a key aspect of our approach. The only restriction on $\gamma$ is the one imposed by our definition of Sobolev spaces. See Proposition 2.19 below, where the Fourier estimates are proven without restriction.

We isolate an important boundedness result for later use in the case $\gamma = \beta$, and that is valid when $\beta > 0$.

**Corollary 2.18.** If $\beta > 0$, the operator $K_{\beta}^{\pm}$ is a bounded linear map

$$(2.13) \quad K_{\beta}^{\pm} : L_{t,x}^{2} \dot{H}_{v}^{-\beta} + L_{t,v}^{2} \dot{H}_{x}^{-\frac{\beta}{2\beta+1}} + L_{t}^{1} L_{x,v}^{2} \to L_{t,x}^{2} \dot{H}_{v}^{\beta} \cap L_{t,v}^{2} \dot{H}_{x}^{\frac{\beta}{2\beta+1}} \cap C_{0}(\mathbb{R}_{t}; L_{x,v}^{2}).$$

*Proof.* Suppose that $S$ belongs to the source space. As $-\beta < 0 < d/2$, by Lemma 2.12 one can find $S_j \in S_K$ converging to $S$ in the source space norm. By Proposition 2.15, $K_{\beta}^{\pm}S_j$ satisfies the estimates in the sum of norms of the target space. In particular, $K_{\beta}^{\pm}S \in C_0(\mathbb{R}_t; L_{x,v}^2)$ at the limit. Hence, we can also take limits in the other norms and obtain the conclusion. $\square$

Our goal is now to establish Proposition 2.15. Thanks to (2.9) and Lemma 2.10, its proof is done in the Fourier side, using Plancherel in the end. The method consists of working with fixed Fourier variables $(\varphi, \xi)$ and proving weighted temporal estimates by integrating with respect to $t$. It makes explicitly clear two facts: on the one hand, the uniformity (or dependency) of the constants with respect to $(\varphi, \xi)$, yielding in the end homogeneous estimates, and on the other hand, that time intervals can be infinite.

Making the variables $(\varphi, \xi)$ implicit, we set

$$K(t, s) = K(t, s, \varphi, \xi), \quad Th(t) = \int_{-\infty}^{t} K(t, s)h(s) \, ds, \quad W(t) = W(t, \varphi, \xi)$$

for the Kolmogorov kernel, the associated operator and the weight defined in (2.11), (2.10) and (2.8).

2.3.6. *Weighted temporal estimates.* In order to prove Proposition 2.15, we fix $(\varphi, \xi) \neq (0, 0)$, define weighted temporal spaces and study how the map $T$ acts on them.

For $\varepsilon \in \mathbb{R}$ and $1 \le p < \infty$, we use the notation $L_t^p(W^{\varepsilon p})$ for the space $L^p(\mathbb{R}_t; W^{\varepsilon p} dt)$ of measurable functions $g$ such that $W^{\varepsilon} g \in L_t^p$ with norm $\|g\|_{L_t^p(W^{\varepsilon p})} = \|W^{\varepsilon} g\|_{L_t^p}$, that is

$$\|g\|_{L_t^p(W^{\varepsilon p})} := \int_{\mathbb{R}} |g(t)|^p \left( \sup(|\xi - t\varphi|, |\varphi|^{\frac{1}{1+2\beta}}) \right)^{\varepsilon p} dt.$$
Next, we show the ontoness. Consider $f \in \dot{\mathcal{L}}_{\beta}^{\gamma}$ and set $S = (\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^{\beta}f$. By definitions of the spaces $\dot{\mathcal{L}}_{\beta}^{\gamma}$ and $\dot{X}_{\beta}^{\gamma-2\beta}$ with $\gamma - 2\beta \le 0$, we have $S \in L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}$ with the norm controlled by that of $f$ in $\dot{\mathcal{L}}_{\beta}^{\gamma}$. We observe that $K_{\beta}^{+}S \in L_t^2 \dot{X}_{\beta}^{\gamma} \subset L_{t,x}^2 \dot{H}_v^{\gamma}$ by Proposition 2.11 and $\gamma \ge 0$. As $K_{\beta}^{+}S$ is also a distributional solution of the equation satisfied by $f$, it agrees with $f$ from Lemma 3.1.

This shows bijectivity together with the continuity estimates in both directions, hence the isomorphism property is proved.

The proof for $K_{\beta}^{-}$ is the same. $\square$

### 3.2.1. *Proof of the embeddings and transfers of regularity.*

*Proof of Theorem 2.2.* The first two items (i) and (ii) follow from Lemma 3.3 and Proposition 2.11.

Proof of (iii): Lemma 2.10 asserts that one can construct a subspace $S_K$ of $S(\Omega)$ which is dense in any $L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta}$, and such that $K_{\beta}^{\pm}(S_K) \subset S(\Omega)$. We conclude using the isomorphism property in Lemma 3.3 that (each of) $K_{\beta}^{\pm}(S_K)$ is dense in each of our kinetic spaces.

Proof of (iv): When $\gamma - 2\beta < d/2$, the Fourier transform in the $(x, v)$ variables provides isomorphisms

$$ \begin{aligned} L_{t,x}^2 \dot{H}_v^{\gamma-2\beta} &\to L_{t,\varphi,\xi}^2 (|\xi|^{2\gamma-4\beta} dt d\varphi d\xi), \\ L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} &\to L_{t,\varphi,\xi}^2 (w^{2\gamma-4\beta} dt d\varphi d\xi), \end{aligned} $$

where we recall that $w(\varphi, \xi) = \sup(|\xi|, |\varphi|^{1/(1+2\beta)})$. Hence, complex interpolation for weighted $L^2$ spaces, see [7], implies that the right-hand spaces in the first two lines have the complex interpolation property, so the same holds for the corresponding left-hand spaces. We conclude from the isomorphisms property of $K_{\beta}^{+}$ (Lemma 3.3) in the allowed range of $\gamma$'s that the scales $\mathcal{F}_{\beta}^{\gamma}$ and $\mathcal{G}_{\beta}^{\gamma}$ of kinetic spaces have the complex interpolation property, respectively. $\square$

**Remark 3.4.** We also have the isomorphism

$$ L_t^2 \dot{X}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{X}_{\beta}^{\gamma-\beta} \to L_t^2 L_{\varphi,\xi}^2 (w^{2\gamma-4\beta} d\varphi d\xi) + L_t^1 L_{\varphi,\xi}^2 (w^{2\gamma-2\beta} d\varphi d\xi) $$

but it is not clear whether the scale of spaces on the right-hand side is an interpolation family, whence we cannot deduce the complex interpolation property for $\dot{\mathcal{L}}_{\beta}^{\gamma}$.

**Remark 3.5.** When $\gamma = \beta$, we can obtain multiplicative inequalities by scale invariance. Applying (i) to $f_{\lambda}(t, x, v) = f(\lambda t, x, \lambda v)$ we obtain the multiplicative inequality

$$ \|D_x^{\frac{\beta}{2\beta+1}} f\|_{L_{t,x,v}^2} \lesssim_{d,\beta} \|D_v^{\beta} f\|_{L_{t,x,v}^2}^{\frac{\beta+1}{2\beta+1}} \|(\partial_t + v \cdot \nabla_x) f\|_{L_{t,x}^2 \dot{H}_v^{-\beta}}^{\frac{\beta}{2\beta+1}} $$

after optimizing the choice of $\lambda$. Similarly, applying (ii) to $f_{\lambda}(t, x, v) = f(t, \lambda x, \lambda v)$ and optimizing we obtain the multiplicative inequality

$$ \sup_{t \in \mathbb{R}} \|f(t)\|_{L_{x,v}^2} \le (2 \|D_v^{\beta} f\|_{L_{t,x,v}^2} \|(\partial_t + v \cdot \nabla_x) f\|_{L_{t,x}^2 \dot{H}_v^{-\beta}})^{\frac{1}{2}}. $$
The following proposition will use elementary estimates on the kernel $K$. By abuse of notation, we also denote by $W^\epsilon$ the operator of multiplication by $W^\epsilon(t)$.

**Proposition 2.19 (Weighted temporal estimates).** Let $\beta > 0$ and $\gamma \in \mathbb{R}$. Then with bounds for the norms that only depend on $\beta, \gamma, d$,

$T: L_t^2(W^{2\gamma-4\beta}) \to L_t^2(W^{2\gamma})$ is bounded.

$T: L_t^1(W^{\gamma-\beta}) \to L_t^2(W^{2\gamma})$ is bounded.

$W^{\gamma-\beta}T: L_t^2(W^{2\gamma-4\beta}) \to C_0(\mathbb{R}_t)$ is bounded.

$W^{\gamma-\beta}T: L_t^1(W^{\gamma-\beta}) \to C_0(\mathbb{R}_t)$ is bounded.

The same estimates are valid for $T^*$.

**2.3.7. Elementary estimates on the function K.** To prove Proposition 2.19, we begin with two elementary lemmas about the function $K$.

**Lemma 2.20.** Let $\beta > 0$. There are constants $0 < c_\beta < C_\beta < \infty$ such that for all $s, t \in \mathbb{R}$ with $s < t$ and $(\varphi, \xi) \neq (0,0)$,

$$c_\beta \le \frac{\int_s^t |\xi - \tau\varphi|^{2\beta} d\tau}{(t-s)(|\xi - s\varphi|^{2\beta} + ((t-s)|\varphi|)^{2\beta})} \le C_\beta$$

and

$$c_\beta \le \frac{\int_s^t |\xi - \tau\varphi|^{2\beta} d\tau}{(t-s)(|\xi - t\varphi|^{2\beta} + ((t-s)|\varphi|)^{2\beta})} \le C_\beta.$$

*Proof.* Set $\xi' = \xi - s\varphi$ and $\varphi' = (t-s)\varphi$, then a change of variable yields

$$\int_s^t |\xi - \tau\varphi|^{2\beta} d\tau = (t-s) \int_0^1 |\xi' - \tau\varphi'|^{2\beta} d\tau.$$

Now the function

$$(\xi', \varphi') \mapsto \int_0^1 |\xi' - \tau\varphi'|^{2\beta} d\tau$$

is continuous on $\mathbb{R}^{2d}$, hence bounded above and below by positive numbers $C_\beta, c_\beta$ on the compact set defined by $|\xi'|^{2\beta} + |\varphi'|^{2\beta} = 1$. One can conclude the first inequality by using that it is also homogeneous of order $2\beta$. One can do similarly with $\xi' = \xi - t\varphi$ writing

$$\int_s^t |\xi - \tau\varphi|^{2\beta} d\tau = (t-s) \int_0^1 |\xi' + \tau\varphi'|^{2\beta} d\tau$$

and conclude as before.

□
### 3.2.2. *Proof of Theorem 1.1.* We turn to the proof of the first main theorem of this work.

**Proof of Theorem 1.1.** For $S = S_1 + (-\Delta_v)^\beta f + S_2 + S_3 \in L_{t,x}^2 \dot{H}_v^{-\beta} + L_{t,x}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$, we have $(\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^\beta f = S$. As $K_\beta^+ S \in L_{t,x}^2 \dot{H}_v^\beta$ is a distributional solution of this equation, it is the only one by uniqueness in Lemma 3.1. It follows that $f = K_\beta^+ S$, which also belongs to $L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2)$ by Proposition 2.11 and the estimate follows.

Let us prove the absolute continuity on $\mathbb{R}$. As above, we write $f = K_\beta^+ S$ with $S = S_1 + (-\Delta_v)^\beta f + S_2 + S_3$. Define $f_1 = K_\beta^+(S_1 + (-\Delta_v)^\beta f)$, $f_2 = K_\beta^+ S_2$, $f_3 = K_\beta^+ S_3$, so that $f = f_1 + f_2 + f_3$. By Lemma 2.10, we can find approximations $(S_1 + (-\Delta_v)^\beta f)_k$, $(S_2)_k$ and $(S_3)_k$ of $S_1 + (-\Delta_v)^\beta f$, $S_2$ and $S_3$ in the respective topologies so that $(f_1)_k = K_\beta^+(S_1 + (-\Delta_v)^\beta f)_k$, $(f_2)_k = K_\beta^+(S_2)_k$ and $(f_3)_k = K_\beta^+(S_3)_k$ belong to $S(\Omega)$. Altogether, we have, when $k \to \infty$, and $i = 1, 2, 3$,

$$(S_1 + (-\Delta_v)^\beta f)_k \to S_1 + (-\Delta_v)^\beta f, \quad \text{in } L_{t,x}^2 \dot{H}_v^{-\beta},$$

$$(S_2)_k \to S_2, \quad \text{in } L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}},$$

$$(S_3)_k \to S_3, \quad \text{in } L_t^1 L_{x,v}^2,$$

$$(f_i)_k \to f_i, \quad \text{in } L_{t,x}^2 \dot{H}_v^\beta \cap L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2).$$

We now derive the absolute continuity using the approximations. Let $g_k = (f_1)_k + (f_2)_k + (f_3)_k$. As $\Gamma((\partial_t + v \cdot \nabla_x)g_k) = \partial_t \Gamma g_k$, for all $s < t$,

$$ \begin{aligned} \|g_k(t)\|_{L_{x,v}^2}^2 - \|g_k(s)\|_{L_{x,v}^2}^2 &= \iint_{\mathbb{R}^{2d}} |(\Gamma g_k)(t, x, v)|^2 dx dv - \iint_{\mathbb{R}^{2d}} |(\Gamma g_k)(s, x, v)|^2 dx dv \\ &= 2 \operatorname{Re} \iint_{\mathbb{R}^{2d}} \int_s^t \partial_t (\Gamma g_k)(\tau, x, v) \overline{\Gamma g_k}(\tau, x, v) d\tau dx dv \\ &= 2 \operatorname{Re} \int_s^t \iint_{\mathbb{R}^{2d}} \partial_t (\Gamma g_k)(\tau, x, v) \overline{\Gamma g_k}(\tau, x, v) dx dv d\tau \\ &= 2 \operatorname{Re} \int_s^t \iint_{\mathbb{R}^{2d}} ((\partial_t + v \cdot \nabla_x)g_k)(\tau, x, v) \overline{g_k}(\tau, x, v) dx dv d\tau \end{aligned} $$

where we used Fubini's theorem, and change of variables $\Gamma(t), \Gamma(s)$ and $\Gamma(\tau)$ in the $(x, v)$-integrals. The left hand-side converges to $\|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2$. For the right-hand side, we split the integral using the decomposition of $(\partial_t + v \cdot \nabla_x)g_k$ into three terms and argue differently for each. Realizing the $v$-integral as the duality bracket between $\dot{H}_v^{-\beta}$ and $\dot{H}_v^{\beta}$, we have, using the dominated convergence theorem for the $(\tau, x)$-integral,

$$ \begin{aligned} & \int_s^t \iint_{\mathbb{R}^{2d}} ((S_1 + (-\Delta_v)^\beta f)_k - (-\Delta_v)^\beta g_k) \overline{g_k} dx dv d\tau \\ &= \int_s^t \int_{\mathbb{R}^d} \langle (S_1 + (-\Delta_v)^\beta f)_k - (-\Delta_v)^\beta g_k, g_k \rangle dx d\tau \to \int_s^t \int_{\mathbb{R}^d} \langle S_1, f \rangle dx d\tau. \end{aligned} $$
**Lemma 2.21.** Let $\beta > 0$, $\varepsilon \in \mathbb{R}$, $\alpha > 0$ and $(\varphi, \xi) \neq (0, 0)$. One has the following estimates with implicit constants depending on $\beta, \varepsilon, \alpha$ and independent of $s, t, \varphi, \xi$.

$$ (2.14) \qquad \int_{-\infty}^{t} K(t, s)^{\alpha} ds \approx_{\beta, \alpha} W^{-2\beta}(t), $$

$$ (2.15) \qquad \int_{s}^{\infty} K(t, s)^{\alpha} dt \approx_{\beta, \alpha} W^{-2\beta}(s), $$

$$ (2.16) \qquad \sup(W^{\varepsilon}(t), W^{\varepsilon}(s)) K(t, s) \lesssim_{\beta, \varepsilon} \inf(W^{\varepsilon}(t), W^{\varepsilon}(s)), \quad s \le t. $$

$$ (2.17) \qquad \int_{-\infty}^{t} K(t, s)W^{2\beta-\varepsilon}(s) ds \lesssim_{\beta, \varepsilon} W^{-\varepsilon}(t), $$

$$ (2.18) \qquad \int_{s}^{\infty} K(t, s)W^{2\beta-\varepsilon}(t) dt \lesssim_{\beta, \varepsilon} W^{-\varepsilon}(s). $$

*Proof.* Proof of (2.14) and (2.15). It is easily checked for all $A, B \ge 0$, and $c > 0$,

$$ (2.19) \qquad \int_{0}^{\infty} e^{-cu^{1+2\beta}A^{2\beta}} e^{-cuB^{2\beta}} du \approx_{c, \beta} \inf(B^{-2\beta}, A^{-\frac{2\beta}{1+2\beta}}) = (\sup(B, A^{\frac{1}{1+2\beta}}))^{-2\beta}. $$

Thus, use the comparisons in Lemma 2.20, the change of variable $t-s = u$ and (2.19).

*Proof of (2.16).* First, there is nothing to do if $s = t$, and we assume $s < t$. Also for $\varepsilon = 0$, we see that $0 < K(t, s) \le 1$, so we are done. Assume next $\varepsilon \ne 0$. Set $A = (t-s)^{\frac{1}{2\beta}}|\xi - t\varphi|$, $B = (t-s)^{\frac{1}{2\beta}}|\xi - s\varphi|$ and $C = (t-s)^{\frac{2\beta+1}{2\beta}}|\varphi|$. We have $C > 0$, $A, B \ge 0$ and $|A - B| \le C$. Observe that using both inequalities in Lemma 2.20 yields

$$ K(t, s) \le \exp \left( - \frac{c_{\beta}}{2} \left( A^{2\beta} + B^{2\beta} + 2C^{2\beta} \right) \right) $$

so that after multiplication by $(t-s)^{\frac{1}{2\beta}}$, the desired inequality follows from

$$ \sup(\rho^{\varepsilon}, \rho^{-\varepsilon}) \exp \left( - \frac{c_{\beta}}{2} \left( A^{2\beta} + B^{2\beta} + 2C^{2\beta} \right) \right) \le C_{\beta, \varepsilon}, $$

where

$$ \rho = \frac{\sup(A, C^{\frac{1}{2\beta+1}})}{\sup(B, C^{\frac{1}{2\beta+1}})}. $$

The case $\varepsilon < 0$ reduces to the case $\varepsilon > 0$ by symmetry. Thus, we assume now $\varepsilon > 0$. If $\sup(A, C^{\frac{1}{2\beta+1}}) = C^{\frac{1}{2\beta+1}}$, then $\rho \le 1$. If $\sup(A, C^{\frac{1}{2\beta+1}}) = A$, then, using $A \le B+C$,

$$ \rho \le 1 + \frac{C}{\sup(B, C^{\frac{1}{2\beta+1}})} \le 1 + \left( \sup(B, C^{\frac{1}{2\beta+1}}) \right)^{2\beta}. $$

Similarly, we have $\rho^{-1} \le 1 + \left( \sup(A, C^{\frac{1}{2\beta+1}}) \right)^{2\beta}$. Hence, the term $\sup(\rho^{\varepsilon}, \rho^{-\varepsilon})$ can be absorbed by the exponential, and this finishes the proof of (2.16).
Similarly, realizing the x-integral as the duality bracket between $\dot{\mathbf{H}}_x^{-\frac{\beta}{2\beta+1}}$ and $\dot{\mathbf{H}}_x^{\frac{\beta}{2\beta+1}}$ and using dominated convergence in the $(t, v)$-integral

$$ \int_s^t \iint_{\mathbb{R}^{2d}} (S_2)_k \overline{g_k} \, dx \, dv \, d\tau = \int_s^t \int_{\mathbb{R}^d} \langle (S_2)_k, g_k \rangle \, dv \, d\tau \to \int_s^t \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv \, d\tau. $$

Eventually, using merely dominated convergence for the $(t, x, v)$-integral

$$ \int_s^t \iint_{\mathbb{R}^{2d}} (S_3)_k \overline{g_k} \, dx \, dv \, d\tau \to \int_s^t \iint_{\mathbb{R}^d} S_3 \overline{f} \, dx \, dv \, d\tau. \quad \square $$

The following result is a general version of the Theorem 1.1.

**Theorem 3.6** (Kinetic embedding and transfer of regularity - polarized version). Let $-\beta \le \varepsilon \le \beta$ such that $\beta \pm \varepsilon < d/2$. Let $f, \tilde{f} \in \mathcal{D}'(\Omega)$ be such that $f \in \dot{\mathcal{L}}_{\beta}^{\beta+\varepsilon}$ and $\tilde{f} \in \dot{\mathcal{L}}_{\beta}^{\beta-\varepsilon}$. Then

$$ \text{(i) } f \in L_{t,v}^2 \dot{\mathbf{H}}_x^{\frac{\beta+\varepsilon}{2\beta+1}} \cap C_0(\mathbb{R}_t; \dot{\mathbf{X}}_{\beta}^{\varepsilon}) \text{ with} \\ \qquad \| D_x^{\frac{\beta+\varepsilon}{2\beta+1}} f \|_{L_{t,x,v}^2} + \sup_{t \in \mathbb{R}} \| f(t) \|_{\dot{\mathbf{X}}_{\beta}^{\varepsilon}} \lesssim_{d,\beta,\varepsilon} \| f \|_{\dot{\mathcal{L}}_{\beta}^{\beta+\varepsilon}} $$

$$ \text{(ii) } \tilde{f} \in L_{t,v}^2 \dot{\mathbf{H}}_x^{\frac{\beta-\varepsilon}{2\beta+1}} \cap C_0(\mathbb{R}_t; \dot{\mathbf{X}}_{\beta}^{-\varepsilon}) \text{ with} \\ \qquad \| D_x^{\frac{\beta-\varepsilon}{2\beta+1}} \tilde{f} \|_{L_{t,x,v}^2} + \sup_{t \in \mathbb{R}} \| \tilde{f}(t) \|_{\dot{\mathbf{X}}_{\beta}^{-\varepsilon}} \lesssim_{d,\beta,\varepsilon} \| \tilde{f} \|_{\dot{\mathcal{L}}_{\beta}^{\beta-\varepsilon}}. $$

(iii) The map $t \mapsto \langle f(t), \tilde{f}(t) \rangle$, where the bracket is for the duality between $\dot{\mathbf{X}}_{\beta}^{\varepsilon}, \dot{\mathbf{X}}_{\beta}^{-\varepsilon}$ is absolutely continuous on $\mathbb{R}$ and its almost everywhere derivative can be computed as follows. Write

$$ (\partial_t + v \cdot \nabla_x) f = S_1 + S_2 + S_3 $$

$$ \text{with } S_1 \in L_{t,x}^2 \dot{\mathbf{H}}_v^{-\beta+\varepsilon}, S_2 \in L_{t,v}^2 \dot{\mathbf{H}}_x^{-\frac{\beta+\varepsilon}{2\beta+1}} \text{ and } S_3 \in L_t^1 \dot{\mathbf{X}}_{\beta}^{-\beta+\varepsilon}, \text{ and} $$

$$ (\partial_t + v \cdot \nabla_x) \tilde{f} = \tilde{S}_1 + \tilde{S}_2 + \tilde{S}_3 $$

$$ \text{with } \tilde{S}_1 \in L_{t,x}^2 \dot{\mathbf{H}}_v^{-\beta-\varepsilon}, \tilde{S}_2 \in L_{t,v}^2 \dot{\mathbf{H}}_x^{-\frac{\beta-\varepsilon}{2\beta+1}} \text{ and } \tilde{S}_3 \in L_t^1 \dot{\mathbf{X}}_{\beta}^{-\beta-\varepsilon}. \text{ Then} $$

$$ \frac{d}{dt} \langle f(t), \tilde{f}(t) \rangle = \int_{\mathbb{R}^d} \langle (S_1, \tilde{f}) + \langle f, \tilde{S}_1 \rangle \rangle dx + \int_{\mathbb{R}^d} \langle (S_2, \tilde{f}) + \langle f, \tilde{S}_2 \rangle \rangle dv + \langle S_3, \tilde{f} \rangle + \langle f, \tilde{S}_3 \rangle $$

where each bracket denotes a different sesquilinear duality extending the $L^2$ inner product by tracing the spaces involved. Putting absolute values around each bracket gives an integrable function in the missing variables.

*Proof.* Items (i) and (ii) are already in Theorem 2.2. Item (iii) is the polarized version (1.6) of Theorem 1.1 where the exponents are symmetric with respect to $\beta$ for duality reasons. Details are left to the reader. $\square$
Proof of (2.17). Using the pointwise inequality (2.16) and the integral one (2.15),

$$
\begin{align*}
\int_{-\infty}^{t} K(t, s)W^{2\beta-\varepsilon}(s) \, ds &= \int_{-\infty}^{t} K(t, s)^{\frac{1}{2}}(W^{4\beta-2\varepsilon}(s)K(t, s))^{\frac{1}{2}} \, ds \\
&\lesssim_{\beta, 2\beta-\varepsilon} W^{2\beta-\varepsilon}(t) \int_{-\infty}^{t} K(t, s)^{\frac{1}{2}} \, ds \\
&\lesssim_{\beta, \varepsilon} W^{-\varepsilon}(t).
\end{align*}
$$

The proof of (2.18) is the same.

<figure><img src="image_2.png" /></figure>

2.3.8. An intermediate result. Before proving Proposition 2.19, we prove estimates for $T$ in $L^1$ and $L^\infty$ spaces.

**Proposition 2.22** ($L^1$ and $L^\infty$ estimates for $T$). Let $\beta > 0$ and $\varepsilon \in \mathbb{R}$. Then with bounds for the norms that only depend on $\beta$ and $\varepsilon$,

$W^\varepsilon T W^{-\varepsilon} : L_t^1 \to C_0(\mathbb{R}_t)$ is bounded.

$W^\varepsilon T W^{2\beta-\varepsilon} : L_t^1 \to L_t^1$ is bounded.

$W^\varepsilon T W^{2\beta-\varepsilon} : L_t^\infty \to L_t^\infty$ is bounded.

The same estimates are valid for $T^*$.

*Proof.* We only treat $T$ as the proofs for $T^*$ are similar. We start with the $L_t^\infty$ bound for $W^\varepsilon T(W^{-\varepsilon}f)$ when $f \in L_t^1$. We have by (2.16),

$$
W^{\varepsilon}(t)|T(W^{-\varepsilon}f)(t)| \leq \int_{-\infty}^{t} W^{\varepsilon}(t)K(t,s)W^{-\varepsilon}(s)|f(s)| \, ds \lesssim_{\beta, \varepsilon} \int_{-\infty}^{t} |f(s)| \, ds,
$$

which is controlled by $\|f\|_{L_t^1}$. This also proves the fact $W^\varepsilon(t)|T(W^{-\varepsilon}f)(t)| \to 0$ when $t \to -\infty$.

By a density argument and closedness of $C_0(\mathbb{R}_t)$ in $L_t^\infty$, it is now enough to prove the continuity of $W^\varepsilon T(W^{-\varepsilon}f)$ and limit $0$ at $+\infty$ assuming $f$ to be compactly supported in $[-R, R]$ for some fixed $R > 0$ and continuous. For the limit, the above estimate rewrites for $t \ge R$,

$$
W^{\epsilon}(t)|T(W^{-\epsilon}f)(t)| \leq \int_{-R}^{R} W^{\epsilon}(t)K(t,s)W^{-\epsilon}(s)|f(s)| \, ds
$$

and clearly the integral converges to 0 by dominated convergence as $t \to +\infty$.

For the continuity, we use that $W^\epsilon$ and $W^{-\epsilon}$ are continuous and also that $T(W^{-\epsilon}f)$ is locally Lipschitz in time since, as $f$ is continuous with compact support, it is easy to see that $g = T(W^{-\epsilon}f)$ solves the ordinary differential equation $g'(t) + |\xi - t\varphi|^{2\beta}g(t) = (W^{-\epsilon}f)(t)$ in the classical sense.

Next, the $L_t^1 - L_t^1$ and $L_t^\infty - L_t^\infty$ bounds of $W^\epsilon T W^{2\beta-\epsilon}$ follow from (2.17) and (2.18) by Fubini's theorem. $\square$

<figure><img src="image_3.png" /></figure>

2.3.9. *Proof of Proposition 2.19.* We are now ready to prove the temporal weighted estimates for the operator $T$.

*Proof of Proposition 2.19.* The last boundedness statement is already contained in Proposition 2.22. We prove the other three properties in the order they are stated.
## 4. CONSTRUCTION OF WEAK SOLUTIONS TO ROUGH EQUATIONS

This section is devoted to the construction of weak solutions. It relies on a theorem due to J.-L. Lions [24, Chap. II, Thm. 1.1]. Since this article is written in French, we recall the statement before proving Theorem 1.12.

**4.1. Existence and uniqueness of weak solutions.** The proof of the existence of weak solutions to Kolmogorov-Fokker-Planck equations with rough coefficients uses the following theorem.

**Theorem 4.1 (Lions – [24]).** Let $F$ be a Hilbert space equipped with a scalar product $\langle \cdot, \cdot \rangle_F$ and the associated norm $\| \cdot \|_F$. Let $\mathcal{H} \subset F$ be a prehibertian space equipped with a scalar product $\langle \cdot, \cdot \rangle_{\mathcal{H}}$ and the associated norm $\| \cdot \|_{\mathcal{H}}$. Assume that there exists $c_1 > 0$ such that for all $f \in \mathcal{H}$, $\|f\|_F \le c_1 \|f\|_{\mathcal{H}}$. Let $E$ be a sesquilinear form on $F \times \mathcal{H}$ such that

(H1) for all $h \in \mathcal{H}$, the linear form $f \mapsto E(f, h)$ is continuous on $F$,

(H2) there exists $\alpha > 0$ such that $\mathrm{Re} E(h, h) \ge \alpha \|h\|_{\mathcal{H}}^2$ for all $h \in \mathcal{H}$.

Then for any semi-linear continuous form $L$ on $\mathcal{H}$, there exists $f \in F$ such that $E(f, h) = L(h)$ for all $h \in \mathcal{H}$.

The following lemma is a consequence of the transfer of regularity obtained in Theorem 1.1.

**Lemma 4.2 (Energy equality).** Let $\beta \in (0, d/2)$ and $S = S_1 + S_2 + S_3$ with $S_1 \in L_{t,x}^2 \dot{H}_v^{-\beta}$ and $S_2 \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}}$ and $S_3 \in L_t^1 L_{x,v}^2$. Any weak solution to $(\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S$ in the sense of Definition 1.9 satisfies the energy equality: for any $s, t \in \mathbb{R}$ with $s < t$,

$$ \|f(t)\|_{L_{x,v}^2}^2 + 2 \mathrm{Re} \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau = \\ \qquad (4.1) \quad \|f(s)\|_{L_{x,v}^2}^2 + 2 \mathrm{Re} \int_s^t \left( \int_{\mathbb{R}^d} \langle S_1, f \rangle \, dx + \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv + \iint_{\mathbb{R}^{2d}} S_3 \bar{f} \, dx \, dv \right) \, d\tau. $$

*Proof.* As $(\partial_t + v \cdot \nabla_x) f = S - \mathcal{A} f$, we know from Theorem 1.1 that $t \mapsto \|f(t)\|_{L_{x,v}^2}^2$ is absolutely continuous, with limit 0 at $\pm\infty$ and, for $s < t$, $\|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2$ is equal to

$$ 2 \mathrm{Re} \int_s^t \left( \int_{\mathbb{R}^d} \langle S_1 - \mathcal{A} f, f \rangle \, dx + \int_{\mathbb{R}^d} \langle S_2, f \rangle \, dv + \iint_{\mathbb{R}^{2d}} S_3 \bar{f} \, dx \, dv \right) \, d\tau. $$

The bracket inside the first integral denotes the one expressing the $\dot{H}_v^{-\beta}, \dot{H}_v^{\beta}$ duality as a function of $(\tau, x)$, so by definition and writing the variables for clarity

$$ \langle \mathcal{A} f, f \rangle(\tau, x) = a_{\tau,x}(f(\tau, x), f(\tau, x)) = a_{\tau,x}(f, f) $$

by convention in the last equality.

<figure><img src="image_5.png" /></figure>

Before proving Theorem 1.12, we recall that $\dot{F}^{\beta} = L_{t,x}^2 \dot{H}_v^{\beta}$ with norm $\|D_v^{\beta} u\|_{L_{t,x,v}^2}$ for $D_v = (-\Delta_v)^{1/2}$ is a Hilbert space as $0 < \beta < d/2$.
First, interpolating $L_t^1 - L_t^1$ and $L_t^\infty - L_t^\infty$ boundedness of $W^\gamma TW^{2\beta-\gamma}$, we obtain $L_t^2 - L_t^2$ boundedness which is equivalent to the first item.

Secondly, we have $W^\gamma T : L_t^1(W^{\gamma-2\beta}) \to L_t^1$ and $W^\gamma T : L_t^1(W^\gamma) \to L_t^\infty$. Using Stein-Weiss interpolation with change of measure (see [9, Theorem 5.4.1]), we obtain $W^\gamma T : L_t^1(W^{\gamma-\beta}) \to L_t^2$ is bounded, which is equivalent to the second item.

Thirdly, the second case applied to $T^*$ rewrites $W^\gamma T^* W^{\beta-\gamma} : L_t^1 \to L_t^2$ is bounded, so by duality and changing $\beta - \gamma$ to $\gamma - \beta$, we have $W^{\gamma-\beta}TW^{2\beta-\gamma}$ maps $L_t^2$ into $L_t^\infty$. Note that if we apply this to continuous and compactly supported functions $f$ then we also have $W^{\gamma-\beta}TW^{2\beta-\gamma}f = W^{\gamma-\beta}TW^{\beta-\gamma}(W^\beta f) \in C_0(\mathbb{R}_t)$ as $W^\beta f \in L_t^1$. By density of such functions $f$ in $L_t^2$ and closedness of $C_0(\mathbb{R}_t)$ in $L_t^\infty$, we conclude that $W^{\gamma-\beta}TW^{2\beta-\gamma}$ maps $L_t^2$ into $C_0(\mathbb{R}_t)$ which is equivalent to the third item. $\square$

2.3.10. *Proof of Proposition 2.15.* We can now obtain the desired a priori estimates for the Kolmogorov operators.

*Proof of Proposition 2.15.* Let us concentrate on $K_\beta^+$, the proof for $K_\beta^-$ being similar. We come back to the definition of $T$ depending on the three variables and recall that the weight $W$ also depends on the three variables. By the Fourier transform correspondence $S \mapsto \hat{\Gamma}S$, Lemma 2.10 and the construction in Lemma 2.12, the space $\hat{\Gamma}S_K$ is dense in $L_t^p L_{\varphi, \xi}^2(W^\alpha)$ for all $\alpha \in \mathbb{R}$ and $1 \le p < \infty$, where $L_{\varphi, \xi}^2(W^\alpha)$ is here the weighted $L^2$ space with respect to $W^\alpha(t, \varphi, \xi) d\varphi d\xi$, for fixed $t$. Using additionally Fubini's theorem, the first assertion for $K_\beta^+$ is equivalent to the weighted boundedness

$$T : L_{\varphi, \xi}^2 L_t^2(W^{2\gamma-4\beta}) \to L_{\varphi, \xi}^2 L_t^2(W^{2\gamma}).$$

Using the conjugate $\hat{\Gamma}$ of the Galilean change of variables through the Fourier transform, and using the fact that this map preserves $C_0(\mathbb{R}_t; L_{\varphi, \xi}^2)$, the second is equivalent to

$$W^{\gamma-\beta} T : L_{\varphi, \xi}^2 L_t^2(W^{2\gamma-4\beta}) \to C_0(\mathbb{R}_t; L_{\varphi, \xi}^2).$$

The third is equivalent to

$$T : L_t^1 L_{\varphi, \xi}^2(W^{2\gamma-2\beta}) \to L_{\varphi, \xi}^2 L_t^2(W^{2\gamma}).$$

With the same idea as for the second assertion, the fourth one is equivalent to

$$W^{\gamma-\beta} T : L_t^1 L_{\varphi, \xi}^2(W^{2\gamma-2\beta}) \to C_0(\mathbb{R}_t; L_{\varphi, \xi}^2).$$

The first two assertions on $T$ are direct consequences of Proposition 2.19.

For the third one, we have by Proposition 2.19 and by Minkowski integral inequality

$$\begin{aligned} \|W^\gamma Th\|_{L_t^2}\|_{L_{\varphi, \xi}^2} &\lesssim_{\beta, \gamma} \|W^{\gamma-\beta}h\|_{L_t^1}\|_{L_{\varphi, \xi}^2} \\ &\le \|W^{\gamma-\beta}h\|_{L_{\varphi, \xi}^2}\|_{L_t^1}. \end{aligned}$$

For the $L^\infty$ bound in the fourth one, write

$$\begin{aligned} \|W^{\gamma-\beta}(t) Th(t)\|_{L_{\varphi, \xi}^2} &\le \int_{-\infty}^t \|W^{\gamma-\beta}(t) K(t, s) h(s)\|_{L_{\varphi, \xi}^2} \, ds \\ &\lesssim_{\beta, \gamma} \int_{-\infty}^t \|W^{\gamma-\beta}(s) h(s)\|_{L_{\varphi, \xi}^2} \, ds \end{aligned}$$
*Proof of Theorem 1.12.* Lemma 4.2 already proves that any weak solution satisfies the energy equality.

In the second step, we prove uniqueness. Assume that $f \in \dot{\mathbb{F}}^{\beta}$ with $(\partial_t + v \cdot \nabla_x)f + Af = 0$ (in the sense of Definition 1.9). The energy equality reads when $s < t$,

$$ \|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2 = -2 \operatorname{Re} \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau. $$

Letting now $s \to -\infty$ and $t \to +\infty$ and using that limits of $L_{x,v}^2$ norms are zero, we obtain $\operatorname{Re} \int_{-\infty}^{+\infty} \int_{\mathbb{R}^d} a_{\tau,x}(f, f) \, dx \, d\tau = 0$. Ellipticity implies $f = 0$.

In a third step, we prove the existence in the case where $S \in L_{t,x}^2 \dot{\mathbb{H}}_v^{-\beta} = (\dot{\mathbb{F}}^{\beta})'$ by use of Theorem 4.1. We verify the hypotheses there.

We let $\mathcal{H} = \mathcal{D}(\Omega)$ equipped with the prehilbertian norm $\|h\|_{\mathcal{H}} = \|h\|_{\dot{\mathbb{F}}^{\beta}}$. In particular, the inclusion $\mathcal{H} \subset \dot{\mathbb{F}}^{\beta}$ is continuous.

Next, consider the sesquilinear form $E$ on $\dot{\mathbb{F}}^{\beta} \times \mathcal{H}$ defined by

$$ E(f, h) = - \iiint_{\Omega} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + a(f, h), $$

where $a$ is defined in (1.9).

We need to check (H1), that is: for all $h \in \mathcal{H}$, $f \mapsto E(f, h)$ is continuous on $\dot{\mathbb{F}}^{\beta}$. To this end, we show that $|E(f, h)| \lesssim \|f\|_{\dot{\mathbb{F}}^{\beta}} \|h\|_{\dot{\mathbb{F}}^{\beta}}$, where the $\dot{\mathbb{F}}^{\beta}$ norm is defined in (2.5). This implies (H1) as $\|h\|_{\dot{\mathbb{F}}^{\beta}} < \infty$ for $h \in \mathcal{H}$. Let $h \in \mathcal{H}$ and $f \in \dot{\mathbb{F}}^{\beta}$. Then, by Hölder inequality and Sobolev embedding, we have

$$ \left| \iiint_{\Omega} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv \right| \le \|f\|_{L_{t,x}^2 L_v^q} \|(\partial_t + v \cdot \nabla_x)h\|_{L_{t,x}^2 L_v^{q'}} \\ \lesssim \|f\|_{L_{t,x}^2 \dot{\mathbb{H}}_v^{\beta}} \|(\partial_t + v \cdot \nabla_x)h\|_{L_{t,x}^2 \dot{\mathbb{H}}_v^{-\beta}}. $$

For the other term, we use the assumption on $a$ in order to get,

$$ |a(f, h)| \le \Lambda \|f\|_{\dot{\mathbb{F}}^{\beta}} \|h\|_{\dot{\mathbb{F}}^{\beta}}. $$

Gathering the two terms using the definition of the $\dot{\mathbb{F}}^{\beta}$ norm in (2.5) gives us the conclusion.

We next check (H2), namely that $\operatorname{Re} E(h, h) \ge \lambda \|h\|_{\mathcal{H}}^2$ for all $h \in \mathcal{H}$. Indeed,

$$ (4.2) \qquad - \operatorname{Re} \iiint_{\Omega} h(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv = 0. $$

$$ \text{Thus, } \operatorname{Re} E(h, h) = \operatorname{Re} a(h, h) \ge \lambda \|h\|_{\dot{\mathbb{F}}^{\beta}}^2 = \lambda \|h\|_{\mathcal{H}}^2. $$

As $S \in (\dot{\mathbb{F}}^{\beta})'$, the map $h \mapsto \iint_{\mathbb{R} \times \mathbb{R}^d} \langle S, h \rangle \, dx \, dt$, where the brackets inside the integral are the $\dot{\mathbb{H}}_v^{-\beta}$, $\dot{\mathbb{H}}_v^{\beta}$ duality as usual, is a continuous semi-linear functional on $\mathcal{H}$. Applying the Theorem 4.1, there exists $f \in \dot{\mathbb{F}}^{\beta}$ such that

$$ E(f, h) = \iint_{\mathbb{R} \times \mathbb{R}^d} \langle S, h \rangle \, dx \, dt \quad \text{for all } h \in \mathcal{H}. $$
using Minkowski integral inequality and (2.16) directly, rather than Proposition 2.19. Continuity and limits follow the argument in Proposition 2.22. We leave such details to the reader.

We turn to the specific case $0 \le \gamma \le 2\beta$. The first weigthed boundedness above reformulates as the $L_{t,\varphi,\xi}^2$ boundedness of $W^{\gamma-2\beta}TW^{\gamma}$. Using the definition of $W$ in (2.8) and $\gamma, 2\beta - \gamma \ge 0$, we can split this into the $L_{t,\varphi,\xi}^2$ boundedness of four operators. Coming back to the Kolmogorov operators and noting that the operators $D_v^\alpha$ and $D_x^\alpha$ preserve $S_K$, we obtain the $L_{t,x,v}^2$ control of

$$D_v^\gamma K_\beta^+ D_v^{2\beta-\gamma} S, D_x^{\frac{\gamma}{2\beta+1}} K_\beta^+ D_v^{2\beta-\gamma} S, D_v^\gamma K_\beta^+ D_x^{\frac{2\beta-\gamma}{2\beta+1}} S, D_x^{\frac{\gamma}{2\beta+1}} K_\beta^+ D_x^{\frac{2\beta-\gamma}{2\beta+1}} S$$

by $\|S\|_{L_{t,x,v}^2}$ when $S \in S_K$. We argue similarly for the final inequality when $\beta \le \gamma \le 2\beta$. $\square$

<figure><img src="image_4.png" /></figure>

**Remark 2.23.** Note that $D_x$ commutes with $K_\beta^+$, but not $D_v$.

### 3. UNIQUENESS AND KINETIC EMBEDDINGS

**3.1. Uniqueness.** The kinetic embeddings rely on a crucial uniqueness result.

**Lemma 3.1 (Uniqueness).** Let $\beta > 0$ and $\gamma > 0$. If $f \in \dot{U}^\gamma$ satisfies $\pm(\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^\beta f = 0$ in $D'(\Omega)$, then $f = 0$.

Recall that $\dot{U}^\gamma$ is defined just before Definition 1.2.

*Proof.* We only consider $(\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^\beta f = 0$, while the backward equation can be treated by a similar argument. When $f \in \dot{U}^\gamma$, $f$ and $(-\Delta_v)^\beta f$ belong to $L_{\text{loc},t}^1 S'_{x,v}$, with partial Fourier transform in $L_{\text{loc}}^1(\Omega)$, and the equation above can be interpreted in $D'_t S'_{x,v}$. Setting $G = \tilde{\Gamma}f$, we have $G \in L_{\text{loc}}^1(\Omega)$ and $\partial_t G + |\xi - t\varphi|^{2\beta}G = 0$ in $D'_t S'_{x,v} \subset D'(\Omega)$. From $\partial_t G = -|\xi - t\varphi|^{2\beta}G \in L_{\text{loc}}^1(\Omega)$, we deduce that $G$ has a representative in $C(\mathbb{R}_t; L_{\text{loc},\varphi,\xi}^1)$ and then in $C^1(\mathbb{R}_t; L_{\text{loc},\varphi,\xi}^1)$. Identifying $G$ with this representative, the equation can be interpreted as a first-order differential equation valued in $L_{\text{loc},\varphi,\xi}^1$. Thus, there exists $G_0 \in L_{\text{loc},\varphi,\xi}^1$ such that $G = K(t,0)G_0$ for all $t \in \mathbb{R}$, where $K(t,0)$ is the kernel defined in (2.11). As $(-\Delta_v)^{\gamma/2}f \in L_{t,x,v}^2$, we have that $|\xi - t\varphi|^\gamma G \in L_{t,\varphi,\xi}^2$, that is

$$(3.1) \quad \iiint_{\Omega} |\xi - t\varphi|^{2\gamma} \exp \left( -2 \int_0^t |\xi - \tau\varphi|^{2\beta} d\tau \right) |G_0(\varphi, \xi)|^2 dt d\varphi d\xi < \infty.$$

We can use Fubini's theorem and notice that the $t$-integral on $\mathbb{R}$ is infinite when $(\varphi, \xi) \ne (0,0)$ because of the exponential growth when $t \to -\infty$. This proves that $G_0 = 0$ almost everywhere. Hence, $G = 0$ and so is $f$. $\square$

<figure><img src="image_5.png" /></figure>

**Remark 3.2.** The argument works for a larger class of $f$. From the argument, we see that it suffices that $f, (-\Delta_v)^\beta f \in L_{\text{loc},t}^1 S'_{x,v}$ with $G = \tilde{\Gamma}f \in L_{\text{loc}}^1(\Omega)$ such that there is some measurable function $m$ for which $mG \in L_t^1 L_{\text{loc},\varphi,\xi}^1$ and

$$\int_{\mathbb{R}} |m(t, \varphi, \xi)| \exp \left( - \int_0^t |\xi - \tau\varphi|^{2\beta} d\tau \right) dt = \infty$$
In our language, this means that $f$ is a weak solution to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ in the sense of Definition 1.9. And since $(\partial_t + v \cdot \nabla_x)f = S - \mathcal{A}f \in (\dot{\mathcal{F}}^\beta)'$, we have that $f \in \dot{\mathcal{F}}_\beta^\beta$.

The second and third steps show the isomorphism property of $(\partial_t + v \cdot \nabla_x) + \mathcal{A}$ from $\dot{\mathcal{F}}_\beta^\beta$ onto $(\dot{\mathcal{F}}^\beta)'$.

In the fourth step, we assume $S \in L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$. We prove the existence of a weak solution by a duality scheme and the use of kinetic embedding and transfer of regularity in Theorem 1.1. The same argument as above applies to $((\partial_t + v \cdot \nabla_x) + \mathcal{A})^* = -(\partial_t + v \cdot \nabla_x) + \mathcal{A}^*$, because (4.2) is not changed in changing the sign of the field and $\mathcal{A}^*$ is also elliptic. It is also an isomorphism from $\dot{\mathcal{F}}_\beta^\beta$ onto $(\dot{\mathcal{F}}^\beta)'$. In particular, Theorem 1.1 implies that

$$ ((\partial_t + v \cdot \nabla_x) + \mathcal{A}^*)^{-1} : (\dot{\mathcal{F}}^\beta)' \to \dot{\mathcal{F}}_\beta^\beta \hookrightarrow \dot{\mathcal{L}}_\beta^\beta \hookrightarrow L_{t,v}^2 \dot{\mathcal{H}}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2). $$

Hence, the operator $\mathcal{T} : L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2 \to \dot{\mathcal{F}}^\beta$ defined by

$$ \langle \mathcal{T}S, h \rangle = \langle S, ((\partial_t + v \cdot \nabla_x) + \mathcal{A}^*)^{-1}h \rangle, \quad \text{for } h \in (\dot{\mathcal{F}}^\beta)' \text{ and } S \in L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2 $$
is continuous.

If $S \in L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ satisfies the extra assumption $S \in (\dot{\mathcal{F}}^\beta)'$, then $\mathcal{T}S$ agrees with $((\partial_t + v \cdot \nabla_x) + \mathcal{A})^{-1}S$ as previously defined, so the function $f = \mathcal{T}S$ satisfies $E(f, h) = \langle S, h \rangle$ for all $h \in \mathcal{H} = \mathcal{D}(\Omega)$.

We now remove this extra assumption. Approximate $S \in L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ by a sequence $S_k \in \mathcal{S}_K \subset \mathcal{S}(\Omega)$ as in Lemma 2.10, and thus, letting $f_k = \mathcal{T}S_k$, we have $E(f_k, h) = \langle S_k, h \rangle$ when $h \in \mathcal{D}(\Omega)$. For fixed $h$, and using the proper meaning of $\langle S, h \rangle$ explained in Theorem 1.1, the second term converges to $\langle S, h \rangle$ since $S_k \to S$ in $L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$. As $f_k \to \mathcal{T}S$ in $\dot{\mathcal{F}}^\beta$, we can pass to the limit in the first one using the estimate proved for (H1) above. We have shown that $\mathcal{T}S$ is a weak solution to (1.3). Eventually, the fact that $\mathcal{T}S \in L_{t,v}^2 \dot{\mathcal{H}}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2)$ follows from Theorem 1.1. $\square$

**4.2. Operators of Kolmogorov type.** Let us mention a corollary of Theorem 1.12. We introduce the operator

$$ (4.3) \qquad \mathcal{K}_A = ((\partial_t + v \cdot \nabla_x) + \mathcal{A})^{-1} $$

defined as follows: For $S \in \mathcal{S}'(\Omega)$, $\mathcal{K}_A S$ is the weak solution in $\dot{\mathcal{F}}^\beta$, whenever it exists, to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ in the sense of Definition 1.9.

**Corollary 4.3 (Boundedness and isomorphism).** Assume $0 < \beta < d/2$ and the assumptions of Theorem 1.12 on $\mathcal{A}$.

(i) We have the continuous map

$$ (4.4) \qquad \mathcal{K}_A : L_{t,x}^2 \dot{\mathcal{H}}_v^{-\beta} + L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2 \to L_{t,x}^2 \dot{\mathcal{H}}_v^\beta \cap L_{t,v}^2 \dot{\mathcal{H}}_x^{\frac{\beta}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2). $$
almost everywhere. For example this works for $f \in L_{t,v}^2 \dot{H}_x^{\frac{\gamma}{2\beta+1}}$ if $\frac{\gamma}{2\beta+1} < d/2$, with additionally $f \in L_{\text{loc},t}^1 L_{x,v}^2$ otherwise. This also works for sums of spaces for which these conditions hold.

**Remark 3.3.** The argument shows that it is only the behaviour at $t \to -\infty$ that forces $f$ to be 0. For the backward equation, it is the behaviour at $t \to +\infty$ that counts.

**3.2. Isomorphism properties.** A consequence of uniqueness is the isomorphism properties of the Kolmogorov operators for our scales of spaces in an appropriate range of exponents.

**Lemma 3.4.** Assume $\beta > 0$, $0 \le \gamma \le 2\beta$ with $\gamma < d/2$. If $S \in L_t^2 \dot{X}_\beta^{\gamma-2\beta} + L_t^1 \dot{X}_\beta^{\gamma-\beta}$, then $\pm(\partial_t + v \cdot \nabla_x)K_\beta^\pm S + (-\Delta_v)^\beta K_\beta^\pm S = S$ in $S'(\Omega)$, hence in $D'(\Omega)$. If $\gamma = \beta$, one can drop the condition $\gamma < d/2$.

*Proof.* We do the argument for $K_\beta^+$ and the one for $K_\beta^-$ is similar. For $S$ in the density class $S_K$ of Lemma 2.12, the proof of that lemma shows that for $g = Th$ with $h = \Gamma_S$, we have $g, \partial_t g, |\xi - t\varphi|^{2\beta} g \in L_{t,\varphi,\xi}^2$ and $\partial_t g + |\xi - t\varphi|^{2\beta} g = h$. It easily follows that the equation holds in $S'(\Omega)$, hence the formula holds by applying the correspondence via the Fourier transform. Using the bounds for the operators $K_\beta^+$ in Proposition 2.15, one can deduce the formula in $S'(\Omega)$ for general $S$ by density, hence it holds also in $D'(\Omega)$. If $\gamma = \beta$, then we already know the bounds of $K_\beta^\pm$ by Corollary 2.18 for all $\beta > 0$, and we conclude by the same density argument. $\square$

We have already established that Kolmogorov solutions are the unique weak solutions.

*Proof of Corollary 1.14.* Lemma 3.4 and Corollary 2.18 prove that the Kolmogorov solutions $K_\beta^\pm S$ are weak solutions to $\pm(\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^\beta f = S$ in $D'(\Omega)$ respectively when $S \in \dot{Z}^\beta$. Moreover, they are the only ones by Lemma 3.1. $\square$

In fact, we have obtained more, and this is the key ingredient for developing our theory of weak solutions.

**Lemma 3.5 (Isomorphisms).** Assume $0 \le \gamma \le 2\beta$ with $\gamma < d/2$. The Kolmogorov operators $K_\beta^\pm$ are isomorphisms from $L_{t,x}^2 \dot{H}_v^{\gamma-2\beta}$ onto $\dot{F}_\beta^\gamma$, from $L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ onto $\dot{G}_\beta^\gamma$ and from $L_t^2 \dot{X}_\beta^{\gamma-2\beta} + L_t^1 \dot{X}_\beta^{\gamma-\beta}$ onto $\dot{L}_\beta^\gamma$, with inverses given by $\pm(\partial_t + v \cdot \nabla_x) + (-\Delta_v)^\beta$. If $\gamma = \beta \ge d/2$, then the statement holds provided we define the kinetic spaces with $f \in \dot{U}^\beta$ replacing $f \in L_{t,x}^2 \dot{H}_v^\beta$.

*Proof.* We do the proof of the last statement for $K_\beta^+$ as the other cases are all the same.

First we assume $0 \le \gamma \le 2\beta$ with $\gamma < d/2$. It follows from Proposition 2.16 that $K_\beta^+$ maps continuously $S \in L_t^2 \dot{X}_\beta^{\gamma-2\beta} + L_t^1 \dot{X}_\beta^{\gamma-\beta}$ into $\dot{L}_\beta^\gamma$. Indeed, $K_\beta^+ S \in L_{t,x}^2 \dot{H}_v^\gamma$ as $\gamma \ge 0$, and by the equation in Lemma 3.4, $(\partial_t + v \cdot \nabla_x)K_\beta^+ S = S - (-\Delta_v)^\beta K_\beta^+ S \in L_t^2 \dot{X}_\beta^{\gamma-2\beta} + L_t^1 \dot{X}_\beta^{\gamma-\beta}$ since $(-\Delta_v)^\beta K_\beta^+ S \in L_{t,x}^2 \dot{H}_v^{\gamma-2\beta} \subset L_t^2 \dot{X}_\beta^{\gamma-2\beta}$ when $\gamma - 2\beta \le 0$.

Lemma 3.4 shows that $K_\beta^+$ is injective.
(ii) The operator $(\partial_t + v \cdot \nabla_x) + \mathcal{A}$ is an isomorphism from $\dot{\mathcal{F}}_\beta^\beta$ onto $L_{t,x}^2 \dot{\mathcal{H}}_v^{-\beta}$, from $\dot{\mathcal{K}}_\beta^\beta$ onto $L_{t,x}^2 \dot{\mathcal{X}}_\beta^{-\beta}$, and from $\dot{\mathcal{L}}_\beta^\beta$ onto $L_t^2 \dot{\mathcal{X}}_\beta^{-\beta} + L_t^1 L_{x,v}^2$.

*Proof.* The first item is a consequence of the proof of Theorem 1.12. The first isomorphism was also established in the course of the argument. The other two follow by combining existence in steps 3 and 4, uniqueness in step 2 and the open mapping theorem. □

**Remark 4.4.** The formal adjoint of $\mathcal{K}_A$, namely $(\mathcal{K}_A)^* = -(\partial_t + v \cdot \nabla_x) + \mathcal{A}^*)^{-1}$, has the same properties. When $\mathcal{A} = (-\Delta_v)^\beta$, strictly speaking we should write $\mathcal{A} = I_{t,x} \otimes (-\Delta_v)^\beta$ with $I_{t,x}$ being the identity on $L_{t,x}^2$, then we recover the Kolmogorov operators as $\mathcal{K}_A = \mathcal{K}_\beta^+$ and $(\mathcal{K}_A)^* = \mathcal{K}_\beta^-$.

## 5. KINETIC CAUCHY PROBLEMS

This section is devoted to the construction of weak solutions to kinetic Cauchy problems on strips for time intervals of the form $[0, \infty)$ or $[0, T]$ with $T$ finite. In order to construct solutions on strips, we will introduce inhomogeneous kinetic spaces and prove an embedding theorem. We begin with working on infinite intervals and obtain homogeneous estimates.

**5.1. The kinetic Cauchy problem in homogeneous spaces.** We turn our attention to the kinetic Cauchy problem

$$ (5.1) \qquad \begin{cases} (\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S, \\ f(0) = \psi, \end{cases} $$

For the kinetic Cauchy problem on infinite time interval, we replace $\Omega$ by $\Omega_+ = (0, \infty) \times \mathbb{R}^d \times \mathbb{R}^d$ and one can again use the homogeneous spaces on $\Omega_+$ with same notation, see Section 2.2.2. We use the same notation as well for the operator $\mathcal{A}$.

**Definition 5.1** (Weak solutions to the kinetic Cauchy problem). Let $\beta \in (0, d/2)$. Let $S \in \mathcal{D}'(\Omega_+)$ and $\psi \in L_{x,v}^2$. The distribution $S$ is assumed to extend to a continuous linear functional on $\mathcal{D}(\bar{\Omega}_+)$. A distribution $f \in \mathcal{D}'(\Omega_+)$ is said to be a weak solution to (5.1) if $f \in L_{t,x}^2 \dot{\mathcal{H}}_v^\beta$ and for all $h \in \mathcal{D}(\bar{\Omega}_+)$,

$$ - \iint\limits_{\Omega_+} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + a(f, h) = \langle S, h \rangle + \iint_{\mathbb{R}^{2d}} \psi(x, v) \bar{h}(0, x, v) \, dx \, dv. $$

Again, $f \in L_{t,x}^2 \dot{\mathcal{H}}_v^\beta$ implies $f \in L_{t,x}^2 S'_v \cap L_{t,x}^2 L_{\text{loc},v}^2$ with our definition. When $h$ is smooth, the expression $(\partial_t + v \cdot \nabla_x) \bar{h}$ is nothing but the pointwise partial derivative calculation. Thus the left-hand side is well-defined; in the right-hand side, $\langle S, h \rangle$ expresses the ad-hoc duality for the extension of $S$ (it will be clear in the following). The function $\psi$ could be also a distribution in $\mathbb{R}^{2d}$, in which case the corresponding term should be a sesquilinear duality bracket $\langle \psi, h(0) \rangle$. When $h \in \mathcal{D}(\Omega_+)$, the last term disappears and this means in particular that $f$ is a weak solution to the first equation $(\partial_t + v \cdot \nabla_x) f + \mathcal{A} f = S$ in the sense of Definition 1.9 on $\Omega_+$. The second equation in (5.1) is encoded by allowing test functions that do not vanish at $t = 0$.
Next, we show the ontoness of $K_β^+$. Consider $f \in \dot{\mathcal{L}}_β^\gamma$ and set $S = (\partial_t + v \cdot \nabla_x)f + (-\Delta_v)^\beta f$. By definitions of the spaces $\dot{\mathcal{L}}_β^\gamma$ and $\dot{X}_β^{\gamma-2\beta}$ with $\gamma - 2\beta \le 0$, we have $S \in L_t^2 \dot{X}_β^{\gamma-2\beta} + L_t^1 \dot{X}_β^{\gamma-\beta}$ with norm controlled by that of $f$ in $\dot{\mathcal{L}}_β^\gamma$. We observe that $K_β^+ S \in L_t^2 \dot{X}_β^\gamma \subset L_{t,x}^2 \dot{H}_v^\gamma$ by Proposition 2.16 and $\gamma \ge 0$. As $K_β^+ S$ is also a distributional solution of the equation satisfied by $f$ from Lemma 3.4, it agrees with $f$ by uniqueness from Lemma 3.1.

This establishes continuity and bijectivity of $K_β^+$ and that its inverse is $(\partial_t + v \cdot \nabla_x) + (-\Delta_v)^\beta$. That the latter is continuous from $\dot{\mathcal{L}}_β^\gamma$ into $L_t^2 \dot{X}_β^{\gamma-2\beta} + L_t^1 \dot{X}_β^{\gamma-\beta}$ has been proven above. Hence, the isomorphism property is proved.

Now we consider the case $\gamma = \beta \ge d/2$. The definition of the kinetic space replacing (2.7), seen already in Corollary 1.11, is

$$ (3.2) \qquad \dot{\mathcal{L}}_\beta^\beta = \{f \in \mathcal{D}'(\Omega); f \in \dot{U}^\beta \ \& \ (\partial_t + v \cdot \nabla_x)f \in L_t^2 \dot{X}_\beta^{-\beta} + L_t^1 L_{x,v}^2\} $$

with

$$ \|f\|_{\dot{\mathcal{L}}_{\beta}^{\beta}} = \|D_{v}^{\beta} f\|_{L_{t,x,v}^{2}} + \|(\partial_{t} + v \cdot \nabla_{x}) f\|_{L_{t}^{2} \dot{X}_{\beta}^{-\beta} + L_{t}^{1} L_{x,v}^{2}}. $$

Then the same argument as above applies replacing $\gamma$ by $\beta$ and $L_{t,x}^2 \dot{H}_v^\gamma$ by $\dot{U}^\beta$ when they appear. This proves the isomorphism property. $\square$

We mention an immediate consequence, which is not direct from the definition of kinetic spaces.

**Corollary 3.6.** In the range of exponents prescribed in Lemma 3.5, the kinetic spaces $\dot{F}_\beta^\gamma$, $\dot{G}_\beta^\gamma$ and $\dot{\mathcal{L}}_\beta^\gamma$ (with the modification of definition when $\gamma = \beta \ge d/2$) are complete normed spaces.

### 3.2.1. *Proof of the embeddings and transfers of regularity.*

*Proof of Theorem 2.2.* The first two items (i) and (ii) follow from Lemma 3.5 and Proposition 2.16, or Corollary 2.18 if $\gamma = \beta$.

*Proof of (iii):* By Lemma 2.12, the space $S_K$ which is dense in any $L_t^2 \dot{X}_\beta^{\gamma-2\beta} + L_t^1 \dot{X}_\beta^{\gamma-\beta}$ and has range under $K_\beta^+$ contained in $L_{t,x,v}^2$ (and, if we replace it by $S_K$ of Lemma 2.14 if $d \ge 2$, contained in $S(\Omega)$). We conclude using the isomorphism property in Lemma 3.5 that this range is dense in each of our kinetic spaces. The same holds with the range under $K_\beta^-$ of this dense subspace.

*Proof of (iv):* When $\gamma - 2\beta < d/2$, the Fourier transform in the $(x, v)$ variables provides us with isomorphisms

$$ L_{t,x}^2 \dot{H}_v^{\gamma-2\beta} \to L_{t,\varphi,\xi}^2 (|\xi|^{2\gamma-4\beta} dt d\varphi d\xi), $$

$$ L_t^2 \dot{X}_\beta^{\gamma-2\beta} \to L_{t,\varphi,\xi}^2 (w^{2\gamma-4\beta} dt d\varphi d\xi), $$

where we recall that $w(\varphi, \xi) = \sup(|\xi|, |\varphi|^{\frac{1}{1+2\beta}})$. Hence, complex interpolation for weighted $L^2$ spaces, see [9], implies that the right-hand spaces in the two lines have the complex interpolation property, so the same holds for the corresponding left-hand spaces. We conclude from the isomorphism properties of $K_\beta^+$ (Lemma 3.5) in
**Theorem 5.2** (Weak solutions to the kinetic Cauchy problem). Let $S \in L_{t,x}^2 \dot{H}_v^{-\beta} + L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ and $\psi \in L_{x,v}^2$. There exists a unique weak solution $f \in L_{t,x}^2 \dot{H}_v^{\beta}$ to the kinetic Cauchy problem in the sense of Definition 5.1.

Moreover, $f \in L_{t,v}^2 \dot{H}_x^{\frac{\beta}{2\beta+1}} \cap C_0([0, \infty); L_{x,v}^2)$, hence there is $L_{x,v}^2$ convergence of $f(t)$ to $\psi$ as $t \to 0$, and

$$ \|D_v^\beta f\|_{L_{t,x,v}^2} + \|D_x^{\frac{\beta}{2\beta+1}} f\|_{L_{t,x,v}^2} + \sup_{t \in [0,\infty)} \|f(t)\|_{L_{x,v}^2} \\ \lesssim \|S\|_{L_{t,x}^2 \dot{H}_v^{-\beta} + L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2} + \|\psi\|_{L_{x,v}^2} $$

for an implicit constant depending only on $d, \beta, \lambda, \Lambda$. In addition, $f$ satisfies the energy equality (4.1) for non-negative times.

*Proof.* The strategy has several steps.

Step 1: The estimates of the backward Kolmogorov operator in Proposition 2.11 when $\gamma = \beta$ hold on $\Omega_+$, by restriction.

Step 2: The uniqueness statement in Lemma 3.1 holds for the backward Kolmogorov operator on $\Omega_+$, in particular when $\gamma = \beta$, with the same proof.

Step 3: Combining Steps 1 and 2, the space $\dot{\mathcal{L}}_\beta^\beta$ on $\Omega_+$ embeds into $C_0([0, \infty); L_{x,v}^2)$. In particular, for all $f \in \dot{\mathcal{L}}_\beta^\beta$,

$$ \|f\|_{C_0([0,\infty); L_{x,v}^2)} \lesssim_{d,\beta} \|f\|_{\dot{\mathcal{L}}_\beta^\beta} $$

Step 4: More generally, the statement of Theorem 1.1 holds on $\Omega_+$ with $C_0(\mathbb{R}_t; L_{x,v}^2)$ replaced by $C_0([0, \infty); L_{x,v}^2)$. The absolute continuity on $[0, \infty)$ is proved similarly using the backward kinetic Kolmogorov operator to construct approximations for the proof. In particular, any weak solution to the kinetic Cauchy problem as in the statement satisfies the energy equality (4.1) in Lemma 4.2 for non-negative times and converges in the sense of $L_{x,v}^2$ limit at $t = 0$.

Step 5: The uniqueness in the kinetic Cauchy problem proceeds with the energy equality as in the proof of Theorem 1.12.

Step 6: Existence when $S \in L_{t,x}^2 \dot{H}_v^{-\beta} = (\dot{F}^\beta)'$ and $\psi \in L_{x,v}^2$ also proceeds as in the proof of Theorem 1.12 with modifications as follows. We change here $\mathcal{H}$ to $\mathcal{H}_+ = \mathcal{D}(\overline{\Omega}_+)$ equipped with prehilbertian norm $\|h\|_{\mathcal{H}_+}^2 = \|h\|_{\dot{F}^\beta}^2 + \|h(0)\|_{L_{x,v}^2}^2$. Clearly $\mathcal{H}_+$ is continuously included in the Hilbert space $\dot{F}^\beta$ on $\Omega_+$.

Setting

$$ E_+(f, h) := \iiint_{\Omega_+} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + a(f, h), $$

the proof of condition (H1) with $|E_+(f, h)| \lesssim \|f\|_{\dot{F}^\beta} \|h\|_{\dot{F}_\beta^\beta}$ for $f \in \dot{F}^\beta$ and $h \in \mathcal{H}_+$ is unchanged. Now, for condition (H2), if $h \in \mathcal{H}_+$ then

$$ - \mathrm{Re} \iiint_{\Omega_+} h(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv = \frac{1}{2} \|h(0)\|_{L_{x,v}^2}^2. $$
the allowed range of $\gamma$'s that the scales $\dot{\mathcal{F}}_{\beta}^{\gamma}$ and $\dot{\mathcal{G}}_{\beta}^{\gamma}$ of kinetic spaces have the complex interpolation property, respectively. $\square$

**Remark 3.7.** We also have the isomorphism

$$L_t^2 \dot{\mathcal{X}}_{\beta}^{\gamma-2\beta} + L_t^1 \dot{\mathcal{X}}_{\beta}^{\gamma-\beta} \to L_t^2 L_{\varphi,\xi}^2 (w^{2\gamma-4\beta} d\varphi d\xi) + L_t^1 L_{\varphi,\xi}^2 (w^{2\gamma-2\beta} d\varphi d\xi)$$

but it is not clear whether the scale of spaces on the right-hand side is an interpolation family, whence we cannot deduce the complex interpolation property for $\dot{\mathcal{L}}_{\beta}^{\gamma}$.

**Remark 3.8.** When $\gamma = \beta$, we can obtain multiplicative inequalities by scale invariance for $f \in \dot{\mathcal{F}}_{\beta}^{\gamma}$. Applying (i) to $f_{\delta}(t, x, v) = f(\delta t, x, \delta v)$ and optimizing the choice of $\delta > 0$ we obtain the multiplicative inequality

$$\|D_x^{\frac{\beta}{2\beta+1}} f\|_{L_{t,x,v}^2} \lesssim_{d,\beta} \|D_v^{\beta} f\|_{L_{t,x,v}^2}^{\frac{\beta+1}{2\beta+1}} \|(\partial_t + v \cdot \nabla_x) f\|_{L_{t,x}^2 \dot{\mathcal{H}}_v^{-\beta}}^{\frac{\beta}{2\beta+1}}.$$

Similarly, applying (ii) to $f_{\delta}(t, x, v) = f(t, \delta x, \delta v)$ and optimizing in $\delta > 0$, we obtain the multiplicative inequality

$$\sup_{t \in \mathbb{R}} \|f(t)\|_{L_{x,v}^2} \le (2 \|D_v^{\beta} f\|_{L_{t,x,v}^2} \|(\partial_t + v \cdot \nabla_x) f\|_{L_{t,x}^2 \dot{\mathcal{H}}_v^{-\beta}})^{\frac{1}{2}}.$$

The space $\dot{\mathcal{F}}_{\beta}^{\gamma}$ has been defined for $\beta < d/2$. For $\beta \ge d/2$, we set $\dot{\mathcal{F}}_{\beta}^{\gamma} = \{f \in \dot{\mathcal{U}}^{\beta} ; (\partial_t + v \cdot \nabla_x) f \in L_{t,x}^2 \dot{\mathcal{H}}_v^{-\beta}\}$.

### 3.2.2. *Proof of Theorem 1.16.* We turn to the proof of our main tool in this work.

**Proof of Theorem 1.16.** We assume $f \in \dot{\mathcal{U}}^{\beta}$ and $(\partial_t + v \cdot \nabla_x) f = S_1 + S_2 + S_3$ which belongs to $L_{t,x}^2 \dot{\mathcal{H}}_v^{-\beta} + L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$. Thus $S := S_1 + (-\Delta_v)^{\beta} f + S_2 + S_3$ is in the same space and $(\partial_t + v \cdot \nabla_x) f + (-\Delta_v)^{\beta} f = S$. By Corollary 2.18, $\mathcal{K}_{\beta}^{+} S \in L_{t,x}^2 \dot{\mathcal{H}}_v^{\beta} \cap L_{t,v}^2 \dot{\mathcal{H}}_x^{\frac{2\beta+1}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2)$. Hence, $\mathcal{K}_{\beta}^{+} S \in \dot{\mathcal{U}}^{\beta}$ and it is a distributional solution of the same equation as $f$ (Lemma 3.4 in the case $\gamma = \beta$). It is the only one by uniqueness in Lemma 3.1. It follows that $f = \mathcal{K}_{\beta}^{+} S$, and the estimate on $f$ follows from the ones on $\mathcal{K}_{\beta}^{+} S$.

Let us prove the absolute continuity on $\mathbb{R}$. We just showed that $f = \mathcal{K}_{\beta}^{+} S$ with $S = S_1 + (-\Delta_v)^{\beta} f + S_2 + S_3$. Define $f_1 = \mathcal{K}_{\beta}^{+} (S_1 + (-\Delta_v)^{\beta} f)$, $f_2 = \mathcal{K}_{\beta}^{+} S_2$, $f_3 = \mathcal{K}_{\beta}^{+} S_3$, so that $f = f_1 + f_2 + f_3$. By Lemma 2.12, consider approximations $(S_1 + (-\Delta_v)^{\beta} f)_k$, $(S_2)_k$ and $(S_3)_k$ of $S_1 + (-\Delta_v)^{\beta} f$, $S_2$ and $S_3$ in $L_{t,x}^2 \dot{\mathcal{H}}_v^{-\beta}$, $L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}}$, $L_t^1 L_{x,v}^2$ respectively. Set $(f_1)_k = \mathcal{K}_{\beta}^{+} (S_1 + (-\Delta_v)^{\beta} f)_k$, $(f_2)_k = \mathcal{K}_{\beta}^{+} (S_2)_k$ and $(f_3)_k = \mathcal{K}_{\beta}^{+} (S_3)_k$. We have, when $k \to \infty$, and $i = 1, 2, 3$,

$$(S_1 + (-\Delta_v)^{\beta} f)_k \to S_1 + (-\Delta_v)^{\beta} f, \quad \text{in } L_{t,x}^2 \dot{\mathcal{H}}_v^{-\beta},$$

$$(S_2)_k \to S_2, \quad \text{in } L_{t,v}^2 \dot{\mathcal{H}}_x^{-\frac{\beta}{2\beta+1}},$$

$$(S_3)_k \to S_3, \quad \text{in } L_t^1 L_{x,v}^2,$$

$$(f_i)_k \to f_i, \quad \text{in } L_{t,x}^2 \dot{\mathcal{H}}_v^{\beta} \cap L_{t,v}^2 \dot{\mathcal{H}}_x^{\frac{2\beta+1}{2\beta+1}} \cap C_0(\mathbb{R}_t; L_{x,v}^2).$$
Thus,

$$ \mathrm{Re} E_+(h, h) = \frac{1}{2} \|h(0)\|_{L_{x,v}^2}^2 + \mathrm{Re} a(h, h) \geq \min(\lambda, \frac{1}{2}) \|h\|_{\mathcal{H}_+}^2. $$

Finally, the map $\mathcal{H}_+ \ni h \mapsto L(h) = \iint_{(0,\infty) \times \mathbb{R}^d} \langle S, h \rangle dx dt + \langle \psi, h(0) \rangle$, where the brackets inside the integral are the $\dot{H}_v^{-\beta}$, $\dot{H}_v^{\beta}$ duality as usual, is a continuous semi-linear functional on $\mathcal{H}_+$. For the first term, this is as before, and for the second, this is simply Cauchy-Schwarz inequality,

$$ |\langle \psi, h(0) \rangle| \leq \|\psi\|_{L_{x,v}^2} \|h(0)\|_{L_{x,v}^2} \leq \|\psi\|_{L_{x,v}^2} \|h\|_{\mathcal{H}_+}. $$

Applying Theorem 4.1 again, there exists $f \in \dot{\mathbb{F}}^{\beta}$ such that for all $h \in \mathcal{D}(\overline{\Omega}_+)$, $E_+(f, h) = L(h)$, that is explicitly

$$ \begin{aligned} & (5.2) \quad - \iiint_{\Omega_+} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dt \, dx \, dv + \iint_{(0,\infty) \times \mathbb{R}^d} a_{t,x}(f,h) \, dx \, dt \\ & = \iint_{(0,\infty) \times \mathbb{R}^d} \langle S, h \rangle \, dx \, dt + \langle \psi, h(0) \rangle. \end{aligned} $$

In particular, $f$ is a weak solution on $\Omega_+$ to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$. It remains to show $f(0) = \psi$ as we already know that $t \mapsto f(t)$ is continuous on $[0, \infty)$ in $L_{x,v}^2$. We remark again that $t \mapsto \langle f(t), h(t) \rangle$ is absolutely continuous and

$$ \begin{aligned} \langle f(t), h(t) \rangle - \langle f(0), h(0) \rangle = & - \int_s^t \int_{\mathbb{R}^d} a_{\tau,x}(f,h) \, dx \, d\tau \\ & + \int_0^t \int_{\mathbb{R}^d} \langle S, h \rangle \, dx \, d\tau + \int_0^t \iint_{\mathbb{R}^{2d}} f(\partial_t + v \cdot \nabla_x) \bar{h} \, dx \, dv \, d\tau. \end{aligned} $$

Letting $t \to +\infty$, the right hand side tends to $-\langle \psi, h(0) \rangle$ by (5.2) and the left hand side to $-\langle f(0), h(0) \rangle$. As $h(0)$ can be arbitrary in $\mathcal{D}(\mathbb{R}^{2d})$, we obtain $u(0) = \psi$.

**Step 7:** It remains to prove existence when $S \in L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ and $\psi = 0$. In order to do so, we extend $S$ by $0$ for $t < 0$ and take the weak solution $f$ to $(\partial_t + v \cdot \nabla_x)f + \mathcal{A}f = S$ on $\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d$ of Theorem 1.12. Using the energy equality, we have when $s < t \leq 0$,

$$ \|f(t)\|_{L_{x,v}^2}^2 - \|f(s)\|_{L_{x,v}^2}^2 = -2 \mathrm{Re} \int_s^t \int_{\mathbb{R}^d} a_{t,x}(f,f) \, dx \, dt, $$

and letting $s \to -\infty$, we obtain $\|f(t)\|_{L_{x,v}^2}^2 \leq 0$. Therefore, $f$ restricted to $\Omega_+$ is a weak solution on $\Omega_+$ with $f(0) = 0$. $\square$

**Corollary 5.3 (Causality principle).** Let $S \in L_{t,x}^2 \dot{H}_v^{-\beta} + L_{t,v}^2 \dot{H}_x^{-\frac{\beta}{2\beta+1}} + L_t^1 L_{x,v}^2$ on $\Omega_+$. If $S_0$ is the zero extension of $S$ for $t < 0$, then $K_A S_0$, with $K_A$ as in (4.3), vanishes for $t \leq 0$ and its restriction to $\Omega_+$ is the solution to the kinetic Cauchy problem (5.1) with zero initial value.

*Proof.* This is the argument in Step 7. $\square$