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# On homogeneous Sobolev and Besov spaces on the whole and the half space

Anatole Gaudin

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Anatole Gaudin. On homogeneous Sobolev and Besov spaces on the whole and the half space. 2023. [hal-03850461v5](https://hal HAL Id: hal-03850461)

https://amu.hal.science/hal-03850461v5

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# On homogeneous Sobolev and Besov spaces on the whole and the half space

Anatole Gaudin

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

## To cite this version:

Anatole Gaudin. On homogeneous Sobolev and Besov spaces on the whole and the half space. 2024. hal-03850461v6

HAL Id: hal-03850461

https://amu.hal.science/hal-03850461v6

Preprint submitted on 14 Jul 2024

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_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# On homogeneous Sobolev and Besov spaces on the whole and the half space.‡

Anatole GAUDIN‡

July 25, 2023

## Abstract

In this paper, we propose an elementary construction of homogeneous Sobolev spaces of fractional order on $\mathbb{R}^n$ and $\mathbb{R}_+^n$ in the scope of the treatment of non-linear partial differential equations. This construction extends the construction of homogeneous Besov spaces on $S_h'(\mathbb{R}^n)$ started by Bahouri, Chemin and Danchin on $\mathbb{R}^n$. We will also extend the treatment done by Danchin and Mucha on $\mathbb{R}_+^n$, and the construction of homogeneous Sobolev spaces of integer orders started by Danchin, Hieber, Mucha and Tolksdorf on $\mathbb{R}^n$ and $\mathbb{R}_+^n$.

Properties of real and complex interpolation, duality, and density are discussed. Trace results are also reviewed. Our approach relies mostly on interpolation theory and yields simpler proofs of some already known results in the case of Besov spaces.

The lack of completeness on the whole scale will lead to consideration of the intersection spaces with decoupled estimates to circumvent this issue.

As standard and simple applications, we treat the problems of Dirichlet and Neumann Laplacians in these homogeneous functions spaces.

## Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>1.1</td><td>Motivations and interests</td><td>2</td></tr><tr><td>1.2</td><td>Notation, definition and usual concepts</td><td>4</td></tr><tr><td>2</td><td>Homogeneous function spaces on the whole space</td><td>6</td></tr><tr><td>2.1</td><td>Definition, usual properties</td><td>6</td></tr><tr><td>2.2</td><td>Interpolation, duality and the fundamental Sobolev multiplier</td><td>10</td></tr><tr><td>3</td><td>Function spaces on the upper half-space</td><td>14</td></tr><tr><td>3.1</td><td>Quick overview of inhomogeneous function spaces on $\mathbb{R}_+^n$</td><td>15</td></tr><tr><td>3.2</td><td>Homogeneous function spaces on $\mathbb{R}_+^n$</td><td>15</td></tr><tr><td>3.2.1</td><td>Homogeneous Sobolev spaces</td><td>16</td></tr><tr><td>3.2.2</td><td>Homogeneous Besov spaces, Interpolation</td><td>22</td></tr><tr><td>4</td><td>On traces of functions</td><td>28</td></tr><tr><td>4.1</td><td>On inhomogeneous function spaces.</td><td>28</td></tr><tr><td>4.2</td><td>On homogeneous function spaces.</td><td>29</td></tr><tr><td>5</td><td>Additional notations and some remarks</td><td>33</td></tr><tr><td>5.1</td><td>Non-exhaustivity of the construction</td><td>33</td></tr><tr><td>5.2</td><td>Operators on Sobolev and Besov spaces</td><td>33</td></tr></table>

<footnote>*MSC 2020: 35J05, 42B37, 46B70, 46E35</footnote>

<footnote>†Key words: Homogeneous Sobolev spaces, Homogeneous Besov spaces, Homogeneous tempered distributions, Traces, Interpolation of non-complete spaces, Half-space, Dirichlet and Neumann Laplacians.</footnote>

<footnote>‡Aix-Marseille Université, CNRS, I2M, Marseille, France - email: anatole.gaudin@univ-amu.fr</footnote>
# On homogeneous Sobolev and Besov spaces on the whole and the half space.‡

Anatole GAUDIN‡

## Abstract

In this paper, we propose an elementary construction of homogeneous Sobolev spaces of fractional order on $\mathbb{R}^n$ and $\mathbb{R}_+^n$ in the scope of the treatment of non-linear partial differential equations. This construction extends the construction of homogeneous Besov spaces on $S'_h(\mathbb{R}^n)$ started by Bahouri, Chemin and Danchin on $\mathbb{R}^n$. We will also extend the treatment done by Danchin and Mucha on $\mathbb{R}_+^n$, and the construction of homogeneous Sobolev spaces of integer orders started by Danchin, Hieber, Mucha and Tolksdorf on $\mathbb{R}^n$ and $\mathbb{R}_+^n$.

Properties of real and complex interpolation, duality, and density are discussed. Trace results are also reviewed. Our approach relies mostly on interpolation theory and yields simpler proofs of some already known results in the case of Besov spaces.

The lack of completeness for our function spaces with high regularities will lead to the consideration of the intersection with a complete space to enforce the behavior of low frequencies. From this point one performs decoupled estimates, in order to obtain results similar to the one obtained in the case of low regularities.

As standard and simple applications, we treat the problems of Dirichlet and Neumann Laplacians in these homogeneous function spaces.

THIS MANUSCRIPT HAS BEEN ACCEPTED FOR PUBLICATION IN *Tunisian Journal of Mathematics*, Vol. 6 (2024), No. 2, 343–404, DOI: 0.2140/tunis.2024.6.343

## Contents

<table><tbody><tr><td><b>1 Introduction</b></td><td><b>2</b></td></tr><tr><td>1.1 Motivations and interests</td><td>2</td></tr><tr><td>1.2 Notation, definition, and usual concepts</td><td>4</td></tr><tr><td><b>2 Homogeneous function spaces on the whole space</b></td><td><b>6</b></td></tr><tr><td>2.1 Definition, usual properties</td><td>6</td></tr><tr><td>2.2 Interpolation, duality and the fundamental Sobolev multiplier result</td><td>10</td></tr><tr><td><b>3 Function spaces on the upper half-space</b></td><td><b>14</b></td></tr><tr><td>3.1 Quick overview of inhomogeneous function spaces on $\mathbb{R}_+^n$</td><td>15</td></tr><tr><td>3.2 Homogeneous function spaces on $\mathbb{R}_+^n$</td><td>16</td></tr><tr><td>3.2.1 Homogeneous Sobolev spaces</td><td>16</td></tr><tr><td>3.2.2 Homogeneous Besov spaces, Interpolation</td><td>23</td></tr><tr><td><b>4 On traces of functions</b></td><td><b>28</b></td></tr><tr><td>4.1 On inhomogeneous function spaces.</td><td>29</td></tr><tr><td>4.2 On homogeneous function spaces.</td><td>30</td></tr><tr><td>4.3 Several remarks</td><td>34</td></tr></tbody></table>

<footnote>*MSC 2020: 35J05, 42B37, 46B70, 46E35</footnote>

<footnote>†Key words: Homogeneous Sobolev spaces, Homogeneous Besov spaces, Homogeneous tempered distributions, Traces, Interpolation of non-complete spaces, Half-space, Dirichlet and Neumann Laplacians.</footnote>

<footnote>‡Aix-Marseille Université, CNRS, I2M, Marseille, France - email: anatole.gaudin@univ-amu.fr</footnote>
6 Applications: the Dirichlet and Neumann Laplacians on the half-space 34

A Complex interpolation for intersection of homogeneous Besov spaces 40

B Estimates for the Poisson semigroup 40

# 1 Introduction

## 1.1 Motivations and interests

We want to give an appropriate construction of homogeneous Sobolev spaces as subspaces of tempered distributions instead of a quotient space of distributions by polynomials. This construction is motivated by the fact that one would make sense of (para-)products laws, stability under global diffeomorphism, or to look at boundary conditions, and therefore traces, when one restrict those spaces on a domain. This could be somewhat difficult if we work with tempered distributions up to polynomials. Indeed, it is not clear that one can perform previous operations in a way that does not depend on a choice of a representative $u + P \in S'(\mathbb{R}^n)$ of $[u] \in S'(\mathbb{R}^n) / \mathbb{C}[x]$. This is inconvenient when it comes to study non-linear partial differential equations, or partial differential equations on a domain with boundary conditions (moreover, applying an extension operator to a polynomial does not give back a polynomial). However, the interested reader could consult, for instance [BL76, Chapter 6, Section 6.3], [Tri83, Chapter 5], or [Saw18, Chapter 2, Section 2.4] for such a construction on the whole space.

Bourdaud and Triebel have highlighted such problems in the choice of such constructions, see their respective work [Bou88, Bou13] and [Tri15, Chapter 2, Section 2.4].

In the case of the common realization in the quotient structure, for $[u], [v] \in S'(\mathbb{R}^n) / \mathbb{C}[x]$, and $u + P, v + Q, u + \tilde{P}, v + \tilde{Q} \in S'(\mathbb{R}^n)$ two representatives of $[u]$ and $[v]$, we have

$$ (u + P)(v + Q) - (u + \tilde{P})(v + \tilde{Q}) = (P - \tilde{P})v + (Q - \tilde{Q})u + PQ - \tilde{P}\tilde{Q}. $$

Therefore, subject to a (para-)product law that makes sense (in terms of suitable bilinear estimates), although $PQ - \tilde{P}\tilde{Q}$ is a polynomial, this is not the case for $(P - \tilde{P})v + (Q - \tilde{Q})u$ so that the product depends on the choice of representatives!

Another possibility to build homogeneous function spaces would be to naively complete the Schwartz class with respect to the homogeneous norm we want to consider. This construction has the disadvantage of producing elements that may no longer be distributions. For example, one can check that

$$ C_c^\infty(\mathbb{R}^n) \not\subset \dot{H}^{-\frac{n}{2}}(\mathbb{R}^n). $$

This prevents us, a priori, from identifying elements of $\dot{H}^{\frac{n}{2}}(\mathbb{R}^n)$ (as a completion) as distributions. This phenomenon is known as *infrared divergence* and relates to convergence problems for the sum of low frequencies in the sense of Fourier.

Thus, for a consistent realization of homogeneous function spaces, we can only choose two out of the following three properties:

(i) functions spaces whose elements are distributions, in a reasonable sense;

(ii) well-defined product laws;

(iii) all spaces are complete.

The idea of Bahouri, Chemin and Danchin in [BCD11, Chapter 2] was to accept the loss of point (iii) and to introduce a subspace of $S'(\mathbb{R}^n)$ such that we get rid of polynomials, see [BCD11, Examples, p.23]. The aforementioned subspace of $S'(\mathbb{R}^n)$ is

$$ S'_{h}(\mathbb{R}^{n}) := \left\{ u \in S'(\mathbb{R}^{n}) \mid \forall \Theta \in C_{c}^{\infty}(\mathbb{R}^{n}), \|\Theta(\lambda \mathfrak{D})u\|_{L^{\infty}(\mathbb{R}^{n})} \xrightarrow{\lambda \to +\infty} 0 \right\}, \quad (1.1) $$
<table>
   <tr>
    <td>5</td>
    <td>Applications: the Dirichlet and Neumann Laplacians on the half-space</td>
    <td>34</td>
   </tr>
   <tr>
    <td>A</td>
    <td>Complex interpolation for intersection of homogeneous Besov spaces</td>
    <td>41</td>
   </tr>
   <tr>
    <td>B</td>
    <td>Estimates for the Poisson semigroup</td>
    <td>41</td>
   </tr>
   <tr>
    <td>C</td>
    <td>Weak* continuity of translations for endpoint Besov spaces</td>
    <td>43</td>
   </tr>
  </table>

# 1 Introduction

## 1.1 Motivations and interests

We want to give an appropriate construction of homogeneous Sobolev spaces as subspaces of tempered distributions instead of a quotient space of distributions by polynomials. This construction is motivated by the fact that we would like to: give meaning to (para)product laws, be stable under the action of a global diffeomorphism, or even to look at boundary values, and thus verify the precise meaning of traces, when these spaces are restricted to a domain. This could be somewhat difficult if we work with tempered distributions up to polynomials. Indeed, it is not clear that one can perform previous operations in a way that does not depend on a choice of a representative $u+P \in S'(\mathbb{R}^n)$ of $[u] \in S'(\mathbb{R}^n) / C[x]$. This is inconvenient when it comes to study non-linear partial differential equations, or partial differential equations on a domain with boundary conditions (moreover, applying an extension operator to a polynomial does not give back a polynomial). However, the interested reader could consult, for instance [BL76, Chapter 6, Section 6.3], [Tri83, Chapter 5], or [Saw18, Chapter 2, Section 2.4] for such a construction on the whole space.

Bourdaud and Triebel have highlighted such problems in the choice of such constructions, see their respective work [Bou88, Bou13] and [Tri15, Chapter 2, Section 2.4].

In the case of the common realization in the quotient structure, for $[u], [v] \in S'(\mathbb{R}^n) / C[x]$, and $u+P, v+Q, u+\tilde{P}, v+\tilde{Q} \in S'(\mathbb{R}^n)$ two representatives of $[u]$ and $[v]$, we have

$$ (u+P)(v+Q) - (u+\tilde{P})(v+\tilde{Q}) = (P-\tilde{P})v + (Q-\tilde{Q})u + PQ - \tilde{P}\tilde{Q}. $$

Therefore, under the existence of meaningful bilinear (para)product estimates, although $PQ - \tilde{P}\tilde{Q}$ is a polynomial, this is however not the case for $(P-\tilde{P})v + (Q-\tilde{Q})u$ so that the product **depends on the choice of representatives!**

Another possibility to build homogeneous function spaces would be to naively complete the Schwartz class with respect to the homogeneous norm we want to consider. This construction has the disadvantage of producing elements that may no longer be distributions. For example, one can check that

$$ C_c^\infty(\mathbb{R}^n) \not\subset \dot{H}^{-\frac{n}{2}}(\mathbb{R}^n). $$

This prevents us, a priori, from identifying elements of $\dot{H}^{-\frac{n}{2}}(\mathbb{R}^n)$ (as a completion) as distributions. This phenomenon is known as *infrared divergence* and relates to convergence problems for the sum of low frequencies in the sense of Fourier.

Thus, for a consistent realization of homogeneous function spaces, we can only choose two out of the following three properties:

(i) functions spaces whose elements are distributions, in a reasonable sense;

(ii) well-defined product laws;

(iii) all spaces are complete.

The idea of Bahouri, Chemin and Danchin in [BCD11, Chapter 2] was to accept the loss of point (iii) and to introduce a subspace of $S'(\mathbb{R}^n)$ such that we get rid of polynomials, see [BCD11,
where for $\lambda > 0$, $\Theta(\lambda\mathfrak{D})u := \mathcal{F}^{-1}\Theta(\lambda)\mathcal{F}u$, $\mathcal{F}$ being the Fourier transform.

The condition of uniform convergence for low frequencies in the above definition ensures that for $u \in S'_h(\mathbb{R}^n)$, the series

$$ \sum_{j \le 0} \dot{\Delta}_j u $$

converges in $L^\infty(\mathbb{R}^n)$, and then, by [BCD11, Proposition 2.14], the following equality holds in $S'(\mathbb{R}^n)$

$$ u = \sum_{j \in \mathbb{Z}} \dot{\Delta}_j u, $$

where $(\dot{\Delta}_j)_{j \in \mathbb{Z}}$ is the homogeneous Littlewood-Paley decomposition on $\mathbb{R}^n$. With $S'_h(\mathbb{R}^n)$ as an ambient space, Bahouri, Chemin and Danchin gave a construction of homogeneous Besov spaces $\dot{B}_{p,q}^s(\mathbb{R}^n)$ which are complete whenever $(s, p, q) \in \mathbb{R} \times (1, +\infty) \times [1, +\infty]$ satisfies

$$ \left[ s < \frac{n}{p} \right] \text{ or } \left[ q = 1 \text{ and } s \le \frac{n}{p} \right]. \qquad (1.2) $$

Later, this has also led Danchin and Mucha to consider homogeneous Besov spaces on $\mathbb{R}_+^n$ and on exterior domains, see [DM09, DM15], and Danchin, Hieber, Mucha and Tolksdorf [DHMT21] to consider homogeneous Sobolev spaces $\dot{H}^{m,p}$ on $\mathbb{R}^n$ and $\mathbb{R}_+^n$, for $m \in \mathbb{N}$, $p \in (1, +\infty)$. Each iteration led to various important applications in fluid dynamics, such as Navier-Stokes equations with variable density in [DM09, DM15], or free boundary problems as in [DHMT21]. This highlights the needs of stability under global diffeomorphism, and (para)product laws that do not rely on a choice of a representative up to a polynomial.

The main purpose of this paper is TO NOT use the canonical embedding of $S'_h(\mathbb{R}^n)$ in tempered distributions modulo polynomials $S'(\mathbb{R}^n) / C[x]$. We want to preserve consistency in the realization of our scales of homogeneous function spaces.

We mention that several attempts to define (homogeneous) Besov spaces on have been made over the last years. See for instance the work Auscher and Amenta [AA18] for complete operator-adapted realizations over Tent and $Z$-spaces on $\mathbb{R}^n$, and the work of Iwabuchi, Matsuyama and Taniguchi [IMT19] for an operator-adapted-construction based on the spectral theory of Schrödinger operators on a general openset $\Omega \subset \mathbb{R}^n$. See also the references therein for more historical background about Besov spaces and their operator-adapted counterparts. However, such kind of construction is either constrained to the pure linear theory, and is not compatible with standard distribution theory.

We want to summarize, complete and extend the given construction of homogeneous Besov spaces in [BCD11, Chapter 2] and the one of homogeneous Sobolev spaces started in [DHMT21, Chapter 3]. We are going to discuss in Section 2 their construction and usual and expected properties, and especially their behavior through complex and real interpolation. The whole space case is treated first, then the case of the half-space will follow.

To be clearer, firstly, in the case of the whole space $\mathbb{R}^n$, we want to check that the real interpolation identity like

$$ (\dot{H}^{s_0,p}(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n))_{\theta,q} = \dot{B}_{p,q}^{(1-\theta)s_0+\theta s_1}(\mathbb{R}^n) \qquad (1.3) $$

still makes sense for our $S'_h$-realization of homogeneous function spaces, for $s_0, s_1 \in \mathbb{R}$, $\theta \in (0, 1)$, $p \in (1, +\infty)$, $q \in [1, +\infty]$. This is done in Theorem 2.6. This is the **first key result** of this paper. We will also check that expected duality results still hold in this framework.

Secondly, we show via some extension-restriction operators, few duality arguments, and interpolation theory, that we still have:

* **Expected density results:** Propositions 3.9, Lemma 3.16 and Corollaries 3.12 and 3.18. For $p \in (1, +\infty)$, $q \in [1, +\infty)$, $s > -1 + 1/p$, when (1.2) is satisfied,

$$ \dot{H}_0^{s,p}(\mathbb{R}_+^n) = \overline{C_c^\infty(\mathbb{R}_+^n)}^{\|\cdot\|_{\dot{H}^{s,p}(\mathbb{R}^n)}}, \quad \text{and} \quad \dot{B}_{p,q,0}^s(\mathbb{R}_+^n) = \overline{C_c^\infty(\mathbb{R}_+^n)}^{\|\cdot\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)}}; \qquad (1.4) $$

We need to make clear now that this is not a definition but a **consequence** from the definition written at the beginning of Section 3.
Examples, p.23]. The aforementioned subspace of $S'(\mathbb{R}^n)$ is

$$ S'_{h}(\mathbb{R}^n) := \left\{ u \in S'(\mathbb{R}^n) \mid \forall \Theta \in C_c^\infty(\mathbb{R}^n), \|\Theta(\lambda\mathfrak{D})u\|_{L^\infty(\mathbb{R}^n)} \xrightarrow{\lambda \to +\infty} 0 \right\}, \quad (1.1) $$

where for $\lambda > 0$, $\Theta(\lambda\mathfrak{D})u := \mathcal{F}^{-1}\Theta(\lambda\cdot)Fu$, $\mathcal{F}$ being the Fourier transform.

The condition of uniform convergence for low frequencies in the above definition ensures that for $u \in S'_h(\mathbb{R}^n)$, the series

$$ \sum_{j \le 0} \dot{\Delta}_j u $$

converges in $L^\infty(\mathbb{R}^n)$, and then, by [BCD11, Proposition 2.14], the following equality holds in $S'(\mathbb{R}^n)$

$$ u = \sum_{j \in \mathbb{Z}} \dot{\Delta}_j u, $$

where $(\dot{\Delta}_j)_{j \in \mathbb{Z}}$ is the homogeneous Littlewood-Paley decomposition on $\mathbb{R}^n$. With $S'_h(\mathbb{R}^n)$ as an ambient space, Bahouri, Chemin and Danchin gave a construction of homogeneous Besov spaces $\dot{B}_{p,q}^s(\mathbb{R}^n)$ which are complete whenever $(s, p, q) \in \mathbb{R} \times (1, +\infty) \times [1, +\infty]$ satisfies

$$ \left[ s < \frac{n}{p} \right] \text{ or } \left[ q = 1 \text{ and } s \le \frac{n}{p} \right]. \qquad (C_{s,p,q}) $$

Later, this has also led Danchin and Mucha to consider homogeneous Besov spaces on $\mathbb{R}_+^n$ and on exterior domains, see [DM09, DM15], and Danchin, Hieber, Mucha and Tolksdorf [DHMT21] to consider homogeneous Sobolev spaces $\dot{H}^{m,p}$ on $\mathbb{R}^n$ and $\mathbb{R}_+^n$, for $m \in \mathbb{N}$, $p \in (1, +\infty)$. Each iteration led to various important applications in fluid dynamics, such as Navier-Stokes equations with variable density in [DM09, DM15], or free boundary problems as in [DHMT21]. This highlights the needs of stability under global diffeomorphism, and (para)product laws that do not rely on a choice of a representative up to a polynomial.

The main purpose of this paper is TO NOT use the canonical embedding of $S'_h(\mathbb{R}^n)$ in tempered distributions modulo polynomials $S'(\mathbb{R}^n) / C[x]$. We want to preserve consistency in the realization of our scales of homogeneous function spaces: hence, to have a unique common representative for each element of our function spaces, independent of the chosen regularity and integrability parameters. This is not possible in the quotient structure involving polynomials, for more details see [Saw18, Chapter 2, Section 2.4.3].

We mention that several attempts to define (homogeneous) Besov spaces have been made over the last years. See for instance the work by Auscher and Amenta [AA18] for complete operator-adapted realizations over Tent and $Z$-spaces on $\mathbb{R}^n$, and the work of Iwabuchi, Matsuyama and Taniguchi [IMT19] for an operator-adapted-construction based on the spectral theory of Schrödinger operators on a general openset $\Omega \subset \mathbb{R}^n$. See also the references therein for more historical background about Besov spaces and their operator-adapted counterparts. However, such kind of construction is either constrained to the pure linear theory, or is not compatible with the standard distribution theory.

We want to summarize, complete and extend the given construction of homogeneous Besov spaces in [BCD11, Chapter 2] and the one of homogeneous Sobolev spaces started in [DHMT21, Chapter 3]. We are going to discuss in Section 2 their construction and usual and expected properties, and especially their behavior through complex and real interpolation. The whole space case is treated first, then the case of the half-space will follow.

To be clearer, firstly, in the case of the whole space $\mathbb{R}^n$, we want to check that the real interpolation identity like

$$ (\dot{H}^{s_0,p}(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n))_{\theta,q} = \dot{B}_{p,q}^{(1-\theta)s_0+\theta s_1}(\mathbb{R}^n) \quad (1.2) $$

still makes sense for our $S'_h$-realization of homogeneous function spaces, for $s_0, s_1 \in \mathbb{R}$, $\theta \in (0, 1)$, $p \in (1, +\infty)$, $q \in [1, +\infty]$. This is done in Theorem 2.6. This is the **first key result** of this paper. We will also check that expected duality results still hold in this framework.

Secondly, we show via some extension-restriction operators, few duality arguments, and interpo-
* **Expected duality results:** Propositions 3.11 and 3.23.

For all $p \in (1, +\infty)$, $q \in (1, +\infty]$, $s > -1 + \frac{1}{p}$, when (1.2) is satisfied, it holds

$$ (\dot{H}^{s,p}(\mathbb{R}_+^n))' = \dot{H}_0^{-s,p'}(\mathbb{R}_+^n), \quad (\dot{B}_{p',q'}^{-s}(\mathbb{R}_+^n))' = \dot{B}_{p,q,0}^s(\mathbb{R}_+^n), \quad (1.5) $$

$$ (\dot{H}_0^{s,p}(\mathbb{R}_+^n))' = \dot{H}^{-s,p'}(\mathbb{R}_+^n), \quad (\dot{B}_{p',q',0}^{-s}(\mathbb{R}_+^n))' = \dot{B}_{p,q}^s(\mathbb{R}_+^n). \quad (1.6) $$

* **Expected interpolation results:** Propositions 3.22 and 3.17.

If $(\dot{h}, \dot{b}) \in \{(\dot{H}, \dot{B}), (\dot{H}_0, \dot{B}_{\cdot,0})\}$, with $(p_0, q_0)$, $(p_1, q_1)$, $(p, q) \in (1, +\infty) \times [1, +\infty]$, $\theta \in (0, 1)$, $s_j, s > -1 + 1/p_j$, $j \in \{0, 1\}$, with $s > -1 + 1/p$, where $s_0, s_1, s$ are three real numbers, so that one can set

$$ \left( s, \frac{1}{p_{\theta}}, \frac{1}{q_{\theta}} \right) := (1 - \theta) \left( s_0, \frac{1}{p_0}, \frac{1}{q_0} \right) + \theta \left( s_1, \frac{1}{p_1}, \frac{1}{q_1} \right), $$

such that (1.2) is satisfied. Then, one has

$$ [\dot{h}^{s_0,p_0}(\mathbb{R}_+^n), \dot{h}^{s_1,p_1}(\mathbb{R}_+^n)]_{\theta} = \dot{h}^{s,p_{\theta}}(\mathbb{R}_+^n), \quad (\dot{h}_{p,q_0}^{s_0}(\mathbb{R}_+^n), \dot{h}_{p,q_1}^{s_1}(\mathbb{R}_+^n))_{\theta,q} = \dot{h}_{p,q}^s(\mathbb{R}_+^n), \quad (1.7) $$

$$ (\dot{h}^{s_0,p}(\mathbb{R}_+^n), \dot{h}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} = \dot{b}_{p,q}^s(\mathbb{R}_+^n), \quad [\dot{b}_{p_0,q_0}^{s_0}(\mathbb{R}_+^n), \dot{b}_{p_1,q_1}^{s_1}(\mathbb{R}_+^n)]_{\theta} = \dot{b}_{p_{\theta},q_{\theta}}^s(\mathbb{R}_+^n). \quad (1.8) $$

Note that Proposition 2.9, telling that $1_{\mathbb{R}_+^n}$ yields a bounded multiplication operator on $\dot{H}^{s,p}(\mathbb{R}^n)$ for all $p \in (1, +\infty)$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$, is the **second key point** of this paper in order to obtain the expected results. In particular, this will imply the following equalities of homogeneous Sobolev and Besov spaces with equivalent norms

$$ \dot{H}^{s,p}(\mathbb{R}_+^n) = \dot{H}_0^{s,p}(\mathbb{R}_+^n), \quad \dot{B}_{p,q}^s(\mathbb{R}_+^n) = \dot{B}_{p,q,0}^s(\mathbb{R}_+^n). \quad (1.9) $$

Some already existing density and boundedness results in Besov spaces presented here are already known, but redone here in a different manner giving some minor improvements with regard to [DHMT21, Chapter 3], allowing sometimes to deal sometimes with $s > -1 + \frac{1}{p}$ or $q = +\infty$. Some other results, despite being well known in the construction of usual Sobolev and Besov spaces, are quite new due to the ambient framework. This leads to some new proofs in a different spirit than the ones already available in the literature.

Due to the lack of completeness for homogeneous Sobolev (and Besov) spaces with high regularity exponents, one will need to consider intersection spaces $\dot{H}^{s_0,p_0} \cap \dot{H}^{s_1,p_1}$, with either, $\dot{H}^{s_0,p_0}$ or $\dot{H}^{s_1,p_1}$ known to be complete (i.e. $s_j < n/p_j$). Therefore, one will have to check boundedness of operators with decoupled estimates.

In Section 4, we will review the meaning of traces on the boundary. As an application, in Section 6, we treat the well-posedness of Neumann and Dirichlet Laplacians on the half-space with fine enough behavior of solutions. The "fine enough behavior" have to be understood in the sense that the decay to 0 at infinity is given a very precise sense.

## 1.2 Notation, definition and usual concepts

Throughout this paper the dimension will be $n \ge 2$, and $\mathbb{N}$ will be the set of non-negative integers. For $a, b \in \mathbb{R}$ with $a \le b$, we write $[a, b] := [a, b] \cap \mathbb{Z}$.

For $x \in \mathbb{R}^n$, the (open) ball centered in $x$ of radius $r > 0$ is given by

$$ B(x, r) := \{ y \in \mathbb{R}^n \mid |x - y| < r \}. $$

For two real numbers $A, B \in \mathbb{R}$, $A \lesssim_{a,b,c} B$ means that there exists a constant $C > 0$ depending on $a, b, c$ such that $A \le CB$. When both $A \lesssim_{a,b,c} B$ and $B \lesssim_{a,b,c} A$ are true, we simply write $A \sim_{a,b,c} B$. When the number of indices is overloaded, we allow ourselves to write $A \lesssim_{d,e,f} B \sim_{a,b,c} A$ instead of $A \lesssim_{a,b,c,d,e,f} B$.

### Spaces of measurable or smooth functions.

Denote by $S(\mathbb{R}^n, \mathbb{C})$ the space of complex valued Schwartz functions, and $S'(\mathbb{R}^n, \mathbb{C})$ its dual called the space of tempered distributions. The Fourier transform on $S'(\mathbb{R}^n, \mathbb{C})$ is written $\mathcal{F}$, and is
lation theory, that we still have:

* **Expected density results:** Proposition 3.9, Lemma 3.16 and Corollaries 3.12 and 3.18.
For $p \in (1, +\infty)$, $q \in [1, +\infty)$, $s > -1 + \frac{1}{p}$, when $(C_{s,p,q})$ is satisfied,

$$ \dot{H}_0^{s,p}(\mathbb{R}_+^n) = \overline{C_c^\infty(\mathbb{R}_+^n)}^{\|\cdot\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}}, \quad \text{and} \quad \dot{B}_{p,q,0}^s(\mathbb{R}_+^n) = \overline{C_c^\infty(\mathbb{R}_+^n)}^{\|\cdot\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}}; \quad (1.3) $$

We need to make clear now that this is not a definition but a **consequence** of the definition written at the beginning of Section 3.

* **Expected duality results:** Propositions 3.11 and 3.23.
For all $p \in (1, +\infty)$, $q \in (1, +\infty]$, $s > -1 + \frac{1}{p}$, when $(C_{s,p,q})$ is satisfied, it holds

$$ (\dot{H}^{s,p}(\mathbb{R}_+^n))' = \dot{H}_0^{-s,p'}(\mathbb{R}_+^n), \quad (\dot{B}_{p',q'}^{-s}(\mathbb{R}_+^n))' = \dot{B}_{p,q,0}^s(\mathbb{R}_+^n), \quad (1.4) $$

$$ (\dot{H}_0^{s,p}(\mathbb{R}_+^n))' = \dot{H}^{-s,p'}(\mathbb{R}_+^n), \quad (\dot{B}_{p',q',0}^{-s}(\mathbb{R}_+^n))' = \dot{B}_{p,q}^s(\mathbb{R}_+^n). \quad (1.5) $$

* **Expected interpolation results:** Propositions 3.22 and 3.17.
If $(\dot{h}, \dot{b}) \in \{(\dot{H}, \dot{B}), (\dot{H}_0, \dot{B}_{\cdot, \cdot, 0})\}$, with $(p_0, q_0)$, $(p_1, q_1)$, $(p, q) \in (1, +\infty) \times [1, +\infty]$, $\theta \in (0, 1)$, $s_j, s > -1 + 1/p_j$, $j \in \{0, 1\}$, with $s > -1 + 1/p$, where $s_0, s_1, s$ are three real numbers, so that one can set

$$ \left(s, \frac{1}{p_{\theta}}, \frac{1}{q_{\theta}}\right) := (1 - \theta) \left(s_0, \frac{1}{p_0}, \frac{1}{q_0}\right) + \theta \left(s_1, \frac{1}{p_1}, \frac{1}{q_1}\right), $$

such that $(C_{s,p,q})$ is satisfied. Then, one has

$$ [\dot{h}^{s_0, p_0}(\mathbb{R}_+^n), \dot{h}^{s_1, p_1}(\mathbb{R}_+^n)]_{\theta} = \dot{h}^{s, p_{\theta}}(\mathbb{R}_+^n), \quad (\dot{h}_{p_0, q_0}^{s_0}(\mathbb{R}_+^n), \dot{h}_{p_1, q_1}^{s_1}(\mathbb{R}_+^n))_{\theta, q} = \dot{h}_{p, q}^s(\mathbb{R}_+^n), \quad (1.6) $$

$$ (\dot{h}^{s_0, p}(\mathbb{R}_+^n), \dot{h}^{s_1, p}(\mathbb{R}_+^n))_{\theta, q} = \dot{h}_{p, q}^s(\mathbb{R}_+^n), \quad [\dot{h}_{p_0, q_0}^{s_0}(\mathbb{R}_+^n), \dot{h}_{p_1, q_1}^{s_1}(\mathbb{R}_+^n)]_{\theta} = \dot{h}_{p_{\theta}, q_{\theta}}^s(\mathbb{R}_+^n). \quad (1.7) $$

Note that Proposition 2.9, telling that $1_{\mathbb{R}_+^n}$ yields a bounded multiplication operator on $\dot{H}^{s,p}(\mathbb{R}^n)$ for all $p \in (1, +\infty)$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$, is the **second key point** of this paper in order to obtain the expected results. In particular, this will imply the following equalities of homogeneous Sobolev and Besov spaces with equivalent norms

$$ \dot{H}^{s,p}(\mathbb{R}_+^n) = \dot{H}_0^{s,p}(\mathbb{R}_+^n), \quad \dot{B}_{p,q}^s(\mathbb{R}_+^n) = \dot{B}_{p,q,0}^s(\mathbb{R}_+^n). \quad (1.8) $$

Some already existing density and boundedness results in Besov spaces presented here are commonly known, but redone here differently giving some minor improvements with regard to [DHMT21, Chapter 3], allowing to deal in several cases with $s > -1 + \frac{1}{p}$ or $q = +\infty$. Some other results, despite being well known in the construction of usual Sobolev and Besov spaces, are quite new due to the ambient framework. This leads to some new proofs in a different spirit than the ones already available in the literature.

Due to the lack of completeness for homogeneous Sobolev (and Besov) spaces with high regularity exponents, one will need to consider intersection spaces $\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}$, with $\dot{H}^{s_0, p_0}$ known to be complete (i.e., $s_0 < n/p_0$). Therefore, one will have to check the boundedness of operators with decoupled estimates.

In Section 4, we will review the meaning of traces at the boundary. As an application, in Section 5, we treat the well-posedness of Neumann and Dirichlet Laplacians on the half-space with fine enough behavior of solutions. The “fine enough behavior” have to be understood in the sense that the decay to 0 at infinity is given in a very precise sense.

## 1.2 Notation, definition, and usual concepts

Throughout this paper the dimension will be $n \ge 1$, and $\mathbb{N}$ will be the set of non-negative integers. For $a, b \in \mathbb{R}$ with $a \le b$, we write $[a, b] := [a, b] \cap \mathbb{Z}$.

For $x \in \mathbb{R}^n$, the (open) ball centered in $x$ of radius $r > 0$ is given by

$$ B(x, r) := \{ y \in \mathbb{R}^n \mid |x - y| < r \}. $$
pointwise defined for any $f \in L^1(\mathbb{R}^n, \mathbb{C})$ by

$$ \mathcal{F}f(\xi) := \int_{\mathbb{R}^n} f(x) e^{-ix \cdot \xi} dx, \xi \in \mathbb{R}^n. $$

Additionnally, for $p \in [1 + \infty]$, we write $p' = \frac{p}{p-1}$ its **Hölder conjugate**.

For any $m \in \mathbb{N}$, the map $\nabla^m : S'(\mathbb{R}^n, \mathbb{C}) \to S'(\mathbb{R}^n, \mathbb{C}^{n^m})$ is defined as $\nabla^m u := (\partial^\alpha u)|_{|\alpha|=m}$. We denote by $(e^{-t(-\Delta)^{\frac{1}{2}}})_{t \ge 0}$ the Poisson semigroup on $\mathbb{R}^n$. We also introduce operators $\nabla'$ and $\Delta'$ which are respectively the gradient and the Laplacian on $\mathbb{R}^{n-1}$ identified with the $n-1$ first variables of $\mathbb{R}^n$, i.e. $\nabla' = (\partial_{x_1}, \dots, \partial_{x_{n-1}})$ and $\Delta' = \partial_{x_1}^2 + \dots + \partial_{x_{n-1}}^2$.

When $\Omega$ is an open set of $\mathbb{R}^n$, $C_c^\infty(\Omega, \mathbb{C})$ is the set of smooth compactly supported functions in $\Omega$, and $D'(\Omega, \mathbb{C})$ is its topological dual. For $p \in [1, +\infty)$, $L^p(\Omega, \mathbb{C})$ is the normed vector space of complex valued (Lebesgue-) measurable functions whose $p$-th power is integrable with respect to the Lebesgue measure, $S(\bar{\Omega}, \mathbb{C})$ (resp. $C_c^\infty(\bar{\Omega}, \mathbb{C})$) stands for functions which are restrictions on $\Omega$ of elements of $S(\mathbb{R}^n, \mathbb{C})$ (resp. $C_c^\infty(\mathbb{R}^n, \mathbb{C})$). Unless the contrary is explicitly stated, we will always identify $L^p(\Omega, \mathbb{C})$ (resp. $C_c^\infty(\Omega, \mathbb{C})$) as the subspace of functions in $L^p(\mathbb{R}^n, \mathbb{C})$ (resp. $C_c^\infty(\mathbb{R}^n, \mathbb{C})$) supported in $\bar{\Omega}$ through the extension by 0 outside $\Omega$. $L^\infty(\Omega, \mathbb{C})$ stands for the space of essentially bounded (Lebesgue-) measurable functions.

For $s \in \mathbb{R}$, $p \in [1, +\infty)$, $\ell_s^p(\mathbb{Z}, \mathbb{C})$, stands for the normed vector space of $p$-summable sequences of complexes numbers with respect to the counting measure $2^{ksp}dk$; $\ell_s^\infty(\mathbb{Z}, \mathbb{C})$ stands for sequences $(x_k)_{k \in \mathbb{Z}}$ such that $(2^{ks}x_k)_{k \in \mathbb{Z}}$ is bounded. More generally, when $X$ is a Banach space, for $p \in [1, +\infty]$, one may also consider $L^p(\Omega, X)$ which stands for the space of (Bochner-)measurable functions $u : \Omega \to X$, such that $t \mapsto \|u(t)\|_X \in L^p(\Omega, \mathbb{R})$, similarly one may consider $\ell_s^p(\mathbb{Z}, X)$. Finally, $C^0(\Omega, X)$ stands for the space of continuous functions on $\Omega \subset \mathbb{R}^n$ with values in $X$. The subspace $C_b^0(\mathbb{R}, X)$ is made of uniformly bounded continuous functions and $C_b^0(\mathbb{R}, X)$ is the set of continuous functions that vanish at infinity. For $C \in \{C^0, C_b^0, C_0^0\}$, we set $C(\bar{\Omega}, X)$ to be the set of continuous functions on $\bar{\Omega}$ which are restrictions of elements that belongs to $C(\mathbb{R}^n, X)$.

## Interpolation of normed vector spaces

Let $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ be two normed vector spaces. We write $X \hookrightarrow Y$ to say that $X$ embeds continuously into $Y$. Now let us recall briefly basics of interpolation theory. If there exists a Hausdorff topological vector space $Z$, such that $X, Y \subset Z$, then $X \cap Y$ and $X + Y$ are normed vector spaces with their canonical norms, and one can define the $K$-functional of $z \in X + Y$, for any $t > 0$ by

$$ K(t, z, X, Y) := \inf_{\substack{(x,y) \in X \times Y, \\ z=x+y}} (\|x\|_X + t \|y\|_Y). $$

This allows us to construct, for any $\theta \in (0, 1)$, $q \in [1, +\infty]$, the real interpolation spaces between $X$ and $Y$ with indexes $\theta, q$ as

$$ (X, Y)_{\theta, q} := \left\{ x \in X + Y \mid t \mapsto t^{-\theta} K(t, x, X, Y) \in L_*^q(\mathbb{R}_+) \right\}, $$

where $L_*^q(\mathbb{R}_+) := L^q((0, +\infty), dt/t)$. The interested reader could check [Lun18, Chapter 1], [BL76, Chapter 3] for more informations about real interpolation and its applications.

If moreover we assume that $X$ and $Y$ are complex Banach spaces, one can consider $F(X, Y)$ the set of all continuous functions $f : \bar{S} \to X + Y$, $S$ being the strip of complex numbers whose real part is between 0 and 1, with $f$ holomorphic in $S$, and such that

$$ t \mapsto f(it) \in C_b^0(\mathbb{R}, X) \quad \text{and} \quad t \mapsto f(1 + it) \in C_b^0(\mathbb{R}, Y). $$

We can endow the space $F(X, Y)$ with the norm

$$ \|f\|_{F(X,Y)} := \max \left( \sup_{t \in \mathbb{R}} \|f(it)\|_X, \sup_{t \in \mathbb{R}} \|f(1+it)\|_Y \right), $$

which makes $F(X, Y)$ a Banach space since it is a closed subspace of $C^0(\bar{S}, X + Y)$. Hence for
For two real numbers $A, B \in \mathbb{R}$, $A \lesssim_{a,b,c} B$ means that there exists a constant $C > 0$ depending on $a, b, c$ such that $A \le CB$. When both $A \lesssim_{a,b,c} B$ and $B \lesssim_{a,b,c} A$ are true, we simply write $A \sim_{a,b,c} B$. When the number of indices is overloaded, we allow ourselves to write $A \lesssim_{a,b,c}^{d,e,f} B$ instead of $A \lesssim_{a,b,c,d,e,f} B$.

## Spaces of measurable or smooth functions.

Denote by $S(\mathbb{R}^n, \mathbb{C})$ the space of complex valued Schwartz functions, and $S'(\mathbb{R}^n, \mathbb{C})$ its dual called the space of tempered distributions. The Fourier transform on $S'(\mathbb{R}^n, \mathbb{C})$ is written $\mathcal{F}$, and is pointwise defined for any $f \in L^1(\mathbb{R}^n, \mathbb{C})$ by

$$ \mathcal{F}f(\xi) := \int_{\mathbb{R}^n} f(x) e^{-ix \cdot \xi} dx, \xi \in \mathbb{R}^n. $$

Additionally, for $p \in [1 + \infty]$, we write $p' = \frac{p}{p-1}$ its Hölder conjugate.

For any $m \in \mathbb{N}$, the map $\nabla^m : S'(\mathbb{R}^n, \mathbb{C}) \to S'(\mathbb{R}^n, \mathbb{C}^{n^m})$ is defined as $\nabla^m u := (\partial^\alpha u)|_{\alpha|=m}$. We denote by $(e^{-t(-\Delta)^{\frac{1}{2}}})_{t \ge 0}$ the Poisson semigroup on $\mathbb{R}^n$. We also introduce operators $\nabla'$ and $\Delta'$ which are respectively the gradient and the Laplacian on $\mathbb{R}^{n-1}$ identified with the $n-1$ first variables of $\mathbb{R}^n$, i.e. $\nabla' = (\partial_{x_1}, \dots, \partial_{x_{n-1}})$ and $\Delta' = \partial_{x_1}^2 + \dots + \partial_{x_{n-1}}^2$.

When $\Omega$ is an open set of $\mathbb{R}^n$, $C_c^\infty(\Omega, \mathbb{C})$ is the set of smooth compactly supported functions in $\Omega$, and $D'(\Omega, \mathbb{C})$ is its topological dual. For $p \in [1, +\infty)$, $L^p(\Omega, \mathbb{C})$ is the normed vector space of complex valued (Lebesgue-) measurable functions whose $p$-th power is integrable with respect to the Lebesgue measure, $S(\bar{\Omega}, \mathbb{C})$ (resp. $C_c^\infty(\bar{\Omega}, \mathbb{C})$) stands for functions which are restrictions on $\Omega$ of elements of $S(\mathbb{R}^n, \mathbb{C})$ (resp. $C_c^\infty(\mathbb{R}^n, \mathbb{C})$). Unless the contrary is explicitly stated, we will always identify $L^p(\Omega, \mathbb{C})$ (resp. $C_c^\infty(\Omega, \mathbb{C})$) as the subspace of functions in $L^p(\mathbb{R}^n, \mathbb{C})$ (resp. $C_c^\infty(\mathbb{R}^n, \mathbb{C})$) supported in $\bar{\Omega}$ through the extension by 0 outside $\Omega$. $L^\infty(\Omega, \mathbb{C})$ stands for the space of essentially bounded (Lebesgue-) measurable functions.

For $s \in \mathbb{R}$, $p \in [1, +\infty)$, $\ell_s^p(\mathbb{Z}, \mathbb{C})$, stands for the normed vector space of $p$-summable sequences of complex numbers with respect to the counting measure $2^{ksp}dk$; $\ell_s^\infty(\mathbb{Z}, \mathbb{C})$ stands for sequences $(x_k)_{k \in \mathbb{Z}}$ such that $(2^{ks}x_k)_{k \in \mathbb{Z}}$ is bounded. More generally, when $X$ is a Banach space, for $p \in [1, +\infty]$, one may also consider $L^p(\Omega, X)$ which stands for the space of (Bochner-)measurable functions $u : \Omega \to X$, such that $t \mapsto \|u(t)\|_X \in L^p(\Omega, \mathbb{R})$, similarly one may consider $\ell_s^p(\mathbb{Z}, X)$. Finally, $C^0(\Omega, X)$ stands for the space of continuous functions on $\Omega \subset \mathbb{R}^n$ with values in $X$. The subspace $C_b^0(\mathbb{R}, X)$ is made of uniformly bounded continuous functions and $C_0^0(\mathbb{R}, X)$ is the set of continuous functions that vanish at infinity. For $C \in \{C^0, C_b^0, C_0^0\}$, we set $C(\bar{\Omega}, X)$ to be the set of continuous functions on $\bar{\Omega}$ which are restrictions of elements that belongs to $C(\mathbb{R}^n, X)$.

## Interpolation of normed vector spaces

Let $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ be two normed vector spaces. We write $X \hookrightarrow Y$ to say that $X$ embeds continuously into $Y$. Now let us recall briefly basics of interpolation theory. If there exists a Hausdorff topological vector space $Z$, such that $X, Y \subset Z$, then $X \cap Y$ and $X+Y$ are normed vector spaces with their canonical norms, and one can define the $K$-functional of $z \in X+Y$, for any $t > 0$ by

$$ K(t, z, X, Y) := \inf_{\substack{(x,y) \in X \times Y, \\ z=x+y}} (\|x\|_X + t \|y\|_Y). $$

This allows us to construct, for any $\theta \in (0, 1)$, $q \in [1, +\infty]$, the real interpolation spaces between $X$ and $Y$ with indexes $\theta, q$ as

$$ (X, Y)_{\theta, q} := \left\{ x \in X + Y \mid t \mapsto t^{-\theta} K(t, x, X, Y) \in L_q^*(\mathbb{R}_+) \right\}, $$

where $L_q^*(\mathbb{R}_+) := L^q((0, +\infty), dt/t)$. The interested reader could check [Lun18, Chapter 1], [BL76, Chapter 3] for more information about real interpolation and its applications.

If moreover we assume that $X$ and $Y$ are complex Banach spaces, one can consider $F(X, Y)$ the set of all continuous functions $f : \bar{S} \to X+Y, S$ being the strip of complex numbers whose real
$\theta \in (0, 1)$, the normed vector space given by

$$[X, Y]_{\theta} := \{ f(\theta) \mid f \in \mathbb{F}(X, Y) \}, \\ \|x\|_{[X, Y]_{\theta}} := \inf_{\substack{f \in \mathbb{F}(X, Y), \\ f(\theta) = x}} \|f\|_{\mathbb{F}(X, Y)},$$

is a Banach space called the complex interpolation space between X and Y associated with $\theta$. Again, the interested reader could check [Lun18, Chapter 2], [BL76, Chapter 4] for more informations about complex interpolation and its applications.

# 2 Homogeneous function spaces on the whole space

All the function spaces considered here are scalar complex valued. Hence, to alleviate the notations during this whole section we will write $L^p(\Omega)$ instead of $L^p(\Omega, \mathbb{C})$, and similarly for any other function spaces: we drop the arrival space $\mathbb{C}$.

## 2.1 Definition, usual properties

To deal with Besov spaces on the whole space, we need to introduce the Littlewood-Paley decomposition given by $\phi \in C_c^\infty(\mathbb{R}^n)$, radial, real-valued, non-negative, such that

* $\text{supp} \phi \subset B(0, 4/3)$;

* $\phi|_{B(0,3/4)} = 1$;

so we define the following functions for any $j \in \mathbb{Z}$ for all $\xi \in \mathbb{R}^n$,

$$\phi_j(\xi) := \phi(2^{-j}\xi), \quad \psi_j(\xi) := \phi_j(\xi/2) - \phi_j(\xi),$$

and the family $(\psi_j)_{j \in \mathbb{Z}}$ has the following properties

* $\text{supp}(\psi_j) \subset \{\xi \in \mathbb{R}^n | 3 \cdot 2^{j-2} \le |\xi| \le 2^{j+3}/3\}$;

* $\forall \xi \in \mathbb{R}^n \setminus \{0\}, \sum_{j=-M}^{N} \psi_j(\xi) \xrightarrow[N,M \to +\infty]{} 1$.

Such a family $(\phi_j)_{j \in \mathbb{Z}}$ is called a Littlewood-Paley family. Now, we consider the two following families of operators associated with their Fourier multipliers:

* The homogeneous family of Littlewood-Paley dyadic decomposition operators $(\hat{\Delta}_j)_{j \in \mathbb{Z}}$, where
$$\hat{\Delta}_j := \mathcal{F}^{-1}\psi_j\mathcal{F},$$

* The inhomogeneous family of Littlewood-Paley dyadic decomposition operators $(\Delta_k)_{k \in \mathbb{Z}}$, where

$$\Delta_{-1} := \mathcal{F}^{-1}\phi\mathcal{F},$$

$$\Delta_k := \hat{\Delta}_k \text{ for any } k \ge 0, \text{ and } \Delta_k := 0 \text{ for any } k \le -2.$$

* The j-th frequency cut-off operators given for all $j \in \mathbb{Z}$ by

$$\dot{S}_j := \mathcal{F}^{-1}\phi_j\mathcal{F}.$$

One may notice, as a direct application of Young's inequality for the convolution, that they are all uniformly bounded families of operators on $L^p(\mathbb{R}^n)$, $p \in [1, +\infty]$.

Both family of operators lead for $s \in \mathbb{R}$, $p, q \in [1, +\infty]$, $u \in S'(\mathbb{R}^n)$ to the following quantities,

$$\|u\|_{B_{p,q}^s(\mathbb{R}^n)} = \|(2^{ks} \|\Delta_k u\|_{L^p(\mathbb{R}^n)})_{k \in \mathbb{Z}}\|_{\ell^q(\mathbb{Z})} \quad \text{and} \quad \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} = \|(2^{js} \|\hat{\Delta}_j u\|_{L^p(\mathbb{R}^n)})_{j \in \mathbb{Z}}\|_{\ell^q(\mathbb{Z})},$$

respectively named the inhomogeneous and homogeneous Besov norms, but the homogeneous norm is not really a norm since $\|u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} = 0$ does not imply that $u = 0$. Thus, following [BCD11,
part is between 0 and 1, with $f$ holomorphic in $S$, and such that

$$t \mapsto f(it) \in C_b^0(\mathbb{R}, X) \quad \text{and} \quad t \mapsto f(1 + it) \in C_b^0(\mathbb{R}, Y).$$

We can endow the space $F(X, Y)$ with the norm

$$\|f\|_{F(X,Y)} := \max \left( \sup_{t \in \mathbb{R}} \|f(it)\|_X, \sup_{t \in \mathbb{R}} \|f(1+it)\|_Y \right),$$

which makes $F(X, Y)$ a Banach space since it is a closed subspace of $C^0(\bar{S}, X + Y)$. Hence, for $\theta \in (0, 1)$, the normed vector space given by

$$[X, Y]_{\theta} := \{ f(\theta) \mid f \in F(X, Y) \}, \\ \|x\|_{[X, Y]_{\theta}} := \inf_{\substack{f \in F(X, Y), \\ f(\theta) = x}} \|f\|_{F(X, Y)},$$

is a Banach space called the complex interpolation space between $X$ and $Y$ associated with $\theta$. Again, the interested reader could check [Lun18, Chapter 2], [BL76, Chapter 4] for more information about complex interpolation and its applications.

# 2 Homogeneous function spaces on the whole space

All the function spaces considered here are scalar complex valued. To alleviate the notations during this whole section, we will write $L^p(\Omega)$ instead of $L^p(\Omega, \mathbb{C})$, and similarly for any other function spaces: we drop the arrival space $\mathbb{C}$.

## 2.1 Definition, usual properties

To deal with Besov spaces on the whole space, we need to introduce the Littlewood-Paley decomposition given by $\phi \in C_c^\infty(\mathbb{R}^n)$, radial, real-valued, non-negative, such that

* $\text{supp } \phi \subset B(0, 4/3)$;

* $\phi|_{B(0,3/4)} = 1$;

so we define the following functions for any $j \in \mathbb{Z}$ for all $\xi \in \mathbb{R}^n$,

$$\phi_j(\xi) := \phi(2^{-j}\xi), \quad \psi_j(\xi) := \phi_j(\xi/2) - \phi_j(\xi),$$

and the family $(\psi_j)_{j \in \mathbb{Z}}$ has the following properties

* $\text{supp}(\psi_j) \subset \{\xi \in \mathbb{R}^n | 3 \cdot 2^{j-2} \le |\xi| \le 2^{j+3}/3\}$;

* $\forall \xi \in \mathbb{R}^n \setminus \{0\}, \sum_{j=-M}^{N} \psi_j(\xi) \xrightarrow[N,M \to +\infty]{} 1$.

Such a family $(\phi, (\psi_j)_{j \in \mathbb{Z}})$ is called a Littlewood-Paley family. Now, we consider the two following families of operators associated with their Fourier multipliers:

* The **homogeneous** family of Littlewood-Paley dyadic decomposition operators $(\hat{\Delta}_j)_{j \in \mathbb{Z}}$, where $\hat{\Delta}_j := \mathcal{F}^{-1}\psi_j\mathcal{F}$,

* The **inhomogeneous** family of Littlewood-Paley dyadic decomposition operators $(\Delta_k)_{k \in \mathbb{Z}}$, where

$$\Delta_{-1} := \mathcal{F}^{-1}\phi\mathcal{F},$$

$$\Delta_k := \hat{\Delta}_k \text{ for any } k \ge 0, \text{ and } \Delta_k := 0 \text{ for any } k \le -2.$$

* The $j$-th frequency cut-off operators given for all $j \in \mathbb{Z}$ by

$$\hat{S}_j := \mathcal{F}^{-1}\phi_j\mathcal{F}.$$
Chapter 2] and [DHMT21, Chapter 3], we introduce a subspace of tempered distributions such that $||\cdot||_{\dot{B}_{p,q}^s(\mathbb{R}^n)}$ is point-separating, say

$$ S'_{h}(\mathbb{R}^n) := \left\{ u \in S'(\mathbb{R}^n) \mid \forall \Theta \in C_c^\infty(\mathbb{R}^n), \|\Theta(\lambda\mathfrak{D})u\|_{L^\infty(\mathbb{R}^n)} \xrightarrow{\lambda \to +\infty} 0 \right\}, $$

where for $\lambda > 0$, $\Theta(\lambda\mathfrak{D})u = \mathcal{F}^{-1}\Theta(\lambda)\mathcal{F}u$. Notice that $S'_h(\mathbb{R}^n)$ does not contain any polynomials, and for any $p \in [1, +\infty)$, $L^p(\mathbb{R}^n) \subset S'_h(\mathbb{R}^n)$.

One can also define the following quantities called the inhomogeneous and homogeneous Sobolev spaces' potential norms

$$ \|u\|_{H^{s,p}(\mathbb{R}^n)} := \|(I - \Delta)^{\frac{s}{2}} u\|_{L^p(\mathbb{R}^n)} \quad \text{and} \quad \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} := \left\| \sum_{j \in \mathbb{Z}} (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u \right\|_{L^p(\mathbb{R}^n)}, $$

where $(-\Delta)^{\frac{s}{2}}$ is understood on $u \in S'_h(\mathbb{R}^n)$ by the action on its dyadic decomposition, i.e.

$$ (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u := \mathcal{F}^{-1}(|\cdot|^s \mathcal{F} \dot{\Delta}_j u), $$

which gives a family of $C^\infty$ functions with at most polynomial growth. Thanks to [DHMT21, Lemma 3.3, Definition 3.4],

$$ \sum_{j \in \mathbb{Z}} (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u \in S'_h(\mathbb{R}^n) $$

holds for all $u \in S'_h(\mathbb{R}^n)$, whenever $s \in [0, +\infty)$.

When $u \in S'_h(\mathbb{R}^n)$ and $\sum_{j \in \mathbb{Z}} (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u \in S'_h(\mathbb{R}^n)$, for $s \in \mathbb{R}$, one will simply write without distinction,

$$ (-\Delta)^{\frac{s}{2}} u = \sum_{j \in \mathbb{Z}} (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u \in S'_h(\mathbb{R}^n), $$

which is somewhat consistent in this case with the fact that $(-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u = \dot{\Delta}_j (-\Delta)^{\frac{s}{2}} u$, $j \in \mathbb{Z}$.

Hence for any $p, q \in [1, +\infty]$, $s \in \mathbb{R}$, we define

* the inhomogeneous and homogeneous Sobolev (Bessel and Riesz potential) spaces,

$$ H^{s,p}(\mathbb{R}^n) = \{ u \in S'(\mathbb{R}^n) \mid \|u\|_{H^{s,p}(\mathbb{R}^n)} < +\infty \}, \quad \dot{H}^{s,p}(\mathbb{R}^n) = \{ u \in S'_h(\mathbb{R}^n) \mid \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} < +\infty \}; $$

* and the inhomogeneous and homogeneous Besov spaces,

$$ B_{p,q}^{s}(\mathbb{R}^{n}) = \{ u \in S'(\mathbb{R}^{n}) \mid \|u\|_{B_{p,q}^{s}(\mathbb{R}^{n})} < +\infty \}, \quad \dot{B}_{p,q}^{s}(\mathbb{R}^{n}) = \{ u \in S'_{h}(\mathbb{R}^{n}) \mid \|u\|_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})} < +\infty \}, $$

which are all normed vector spaces. We also introduce the following closures

$$ B_{p,\infty}^{s}(\mathbb{R}^{n}) = \overline{S(\mathbb{R}^{n})}^{\|\cdot\|_{B_{p,\infty}^{s}(\mathbb{R}^{n})}} \quad \text{and} \quad \dot{B}_{p,\infty}^{s}(\mathbb{R}^{n}) = \overline{S_{0}(\mathbb{R}^{n})}^{\|\cdot\|_{B_{p,\infty}^{s}(\mathbb{R}^{n})}}. $$

Here $S_0(\mathbb{R}^n)$ is defined as

$$ S_0(\mathbb{R}^n) := \{ u \in S(\mathbb{R}^n) \mid 0 \notin \text{supp} (\mathcal{F}f) \}. $$

The treatment of homogeneous Besov spaces $\dot{B}_{p,q}^s(\mathbb{R}^n)$, $s \in \mathbb{R}$, $p, q \in [1, +\infty]$, defined on $S'_h(\mathbb{R}^n)$ has been done in an extensive manner in [BCD11, Chapter 2]. However, the corresponding construction for homogeneous Sobolev spaces $\dot{H}^{s,p}(\mathbb{R}^n)$, $s \in \mathbb{R}$, $p \in (1, +\infty)$ has only been done in the case $(p, s) \in \{(2), \mathbb{R}\} \cup ((1, +\infty), \mathbb{N})$. See [BCD11, Chapter 1] for the case $p = 2$, [DHMT21, Chapter 3] for the case $s \in \mathbb{N}$.

The inhomogeneous spaces $L^p(\mathbb{R}^n)$, $H^{s,p}(\mathbb{R}^n)$, and $B_{p,q}^s(\mathbb{R}^n)$ are all complete for all $p, q \in [1, +\infty]$, $s \in \mathbb{R}$, but in this setting homogeneous function spaces are no longer always complete (see [BCD11, Proposition 1.34, Remark 2.26]).

For homogeneous Besov spaces, we have the following properties:

**Proposition 2.1** Let $p, q \in [1, +\infty]$, $s \in \mathbb{R}$. The following assertions hold

(i) if $(s, p, q)$ satisfies the condition

$$ \left[ s < \frac{n}{p} \right] \quad \text{or} \quad \left[ q = 1 \text{ and } s \le \frac{n}{p} \right] \qquad (C_{s,p,q}) $$
One may notice, as a direct application of Young's inequality for the convolution, that they are all uniformly bounded families of operators on $L^p(\mathbb{R}^n)$, $p \in [1, +\infty]$.

Both family of operators lead for $s \in \mathbb{R}$, $p, q \in [1, +\infty]$, $u \in S'(\mathbb{R}^n)$ to the following quantities,

$$ \|u\|_{B_{p,q}^s(\mathbb{R}^n)} = \|(2^{ks} \|\Delta_k u\|_{L^p(\mathbb{R}^n)})_{k \in \mathbb{Z}}\|_{\ell^q(\mathbb{Z})} \quad \text{and} \quad \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} = \|(2^{js} \|\dot{\Delta}_j u\|_{L^p(\mathbb{R}^n)})_{j \in \mathbb{Z}}\|_{\ell^q(\mathbb{Z})}, $$

respectively named the inhomogeneous and homogeneous Besov norms, but the homogeneous norm is not really a norm since $\|u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} = 0$ does not imply that $u = 0$. Thus, following [BCD11, Chapter 2] and [DHMT21, Chapter 3], we introduce a subspace of tempered distributions such that $\|\cdot\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)}$ is point-separating, say

$$ S'_h(\mathbb{R}^n) := \left\{ u \in S'(\mathbb{R}^n) \mid \forall \Theta \in C_c^\infty(\mathbb{R}^n), \|\Theta(\lambda\mathfrak{D})u\|_{L^\infty(\mathbb{R}^n)} \xrightarrow{\lambda \to +\infty} 0 \right\}, $$

where for $\lambda > 0$, $\Theta(\lambda\mathfrak{D})u = \mathcal{F}^{-1}\Theta(\lambda\cdot)Fu$. Notice that $S'_h(\mathbb{R}^n)$ does not contain any nonzero polynomials, and for any $p \in [1, +\infty)$, $L^p(\mathbb{R}^n) \subset S'_h(\mathbb{R}^n)$.

One can also define the following quantities called the inhomogeneous and homogeneous Sobolev spaces' potential norms

$$ \|u\|_{H^{s,p}(\mathbb{R}^n)} := \|(I - \Delta)^{\frac{s}{2}} u\|_{L^p(\mathbb{R}^n)} \quad \text{and} \quad \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} := \left\| \sum_{j \in \mathbb{Z}} (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u \right\|_{L^p(\mathbb{R}^n)}, $$

where $(-\Delta)^{\frac{s}{2}}$ is understood on $u \in S'_h(\mathbb{R}^n)$ by the action on its dyadic decomposition, i.e.

$$ (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u := \mathcal{F}^{-1}(|\cdot|^s \mathcal{F} \dot{\Delta}_j u), $$

which gives, a priori, a family of $C^\infty$ functions with at most polynomial growth. Thanks to [DHMT21, Lemma 3.3, Definition 3.4],

$$ \sum_{j \in \mathbb{Z}} (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u \in S'_h(\mathbb{R}^n) $$

holds for all $u \in S'_h(\mathbb{R}^n)$, whenever $s \in [0, +\infty)$.

When $u \in S'_h(\mathbb{R}^n)$ and $\sum_{j \in \mathbb{Z}} (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u \in S'_h(\mathbb{R}^n)$, for $s \in \mathbb{R}$, one will simply write without distinction,

$$ (-\Delta)^{\frac{s}{2}} u = \sum_{j \in \mathbb{Z}} (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u \in S'_h(\mathbb{R}^n), $$

which is somewhat consistent in this case with the fact that $(-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u = \dot{\Delta}_j (-\Delta)^{\frac{s}{2}} u$, $j \in \mathbb{Z}$.

Hence, for any $p, q \in [1, +\infty]$, $s \in \mathbb{R}$, we define

* the inhomogeneous and homogeneous Sobolev (Bessel and Riesz potential) spaces,

$$ H^{s,p}(\mathbb{R}^n) = \{ u \in S'(\mathbb{R}^n) \mid \|u\|_{H^{s,p}(\mathbb{R}^n)} < +\infty \}, \quad \dot{H}^{s,p}(\mathbb{R}^n) = \{ u \in S'_h(\mathbb{R}^n) \mid \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} < +\infty \}; $$

* and the inhomogeneous and homogeneous Besov spaces,

$$ B_{p,q}^{s}(\mathbb{R}^{n}) = \{ u \in S'(\mathbb{R}^{n}) \mid \|u\|_{B_{p,q}^{s}(\mathbb{R}^{n})} < +\infty \}, \quad \dot{B}_{p,q}^{s}(\mathbb{R}^{n}) = \{ u \in S'_{h}(\mathbb{R}^{n}) \mid \|u\|_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})} < +\infty \}, $$

which are all normed vector spaces. We also introduce the following closures

$$ B_{p,\infty}^{s}(\mathbb{R}^{n}) = \overline{S(\mathbb{R}^{n})}^{\|\cdot\|_{B_{p,\infty}^{s}(\mathbb{R}^{n})}} \quad \text{and} \quad \dot{B}_{p,\infty}^{s}(\mathbb{R}^{n}) = \overline{S_{0}(\mathbb{R}^{n})}^{\|\cdot\|_{B_{p,\infty}^{s}(\mathbb{R}^{n})}}. $$

Here $S_0(\mathbb{R}^n)$ is defined as

$$ S_0(\mathbb{R}^n) := \{ u \in S(\mathbb{R}^n) \mid 0 \notin \text{supp} (\mathcal{F}f) \}. $$

The treatment of homogeneous Besov spaces $\dot{B}_{p,q}^s(\mathbb{R}^n)$, $s \in \mathbb{R}$, $p, q \in [1, +\infty]$, defined on $S'_h(\mathbb{R}^n)$ has been done in an extensive manner in [BCD11, Chapter 2]. However, the corresponding construction for homogeneous Sobolev spaces $\dot{H}^{s,p}(\mathbb{R}^n)$, $s \in \mathbb{R}$, $p \in (1, +\infty)$ has only been done in the case $(p, s) \in (\{2\}, \mathbb{R}) \cup ((1, +\infty), \mathbb{N})$. See [BCD11, Chapter 1] for the case $p = 2$, [DHMT21, Chapter 3] for the case $s \in \mathbb{N}$.
holds then $\dot{B}_{p,q}^s(\mathbb{R}^n)$ is a complete normed vector space,

(ii) for all $m \in \mathbb{N}$, all $u \in S'_h(\mathbb{R}^n)$,

$$ \sum_{j=1}^{n} \| \partial_{x_j}^m u \|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \sim_{s,m,p,n} \| \nabla^m u \|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \sim_{s,m,p,n} \| u \|_{\dot{B}_{p,q}^{s+m}(\mathbb{R}^n)} \quad (2.1) $$

(iii) if $p, q < +\infty$, the space $S_0(\mathbb{R}^n)$ is dense in $\dot{B}_{p,q}^s(\mathbb{R}^n)$,

(iv) if $1/q = 1/p - s/n \in (0, 1)$, we have dense Sobolev embeddings,

$$ \dot{B}_{p,r}^s(\mathbb{R}^n) \hookrightarrow L^q(\mathbb{R}^n), \quad (p, q < +\infty, r \in [1, q], s < n/p) $$

$$ \dot{B}_{p,1}^{n/p}(\mathbb{R}^n) \hookrightarrow C_0^0(\mathbb{R}^n), \quad (p < +\infty) $$

(v) when $s > 0$, one has $B_{p,q}^s(\mathbb{R}^n) = L^p(\mathbb{R}^n) \cap \dot{B}_{p,q}^s(\mathbb{R}^n)$ with equivalence of norms and a continuous embedding

$$ \dot{B}_{p,q}^{-s}(\mathbb{R}^n) \hookrightarrow B_{p,q}^{-s}(\mathbb{R}^n). $$

see [BCD11, Theorem 2.25, Lemma 2.1, Propositions 2.27 & 2.39, Theorems 2.40 & 2.41] and [BL76, Theorem 6.3.2] for more details.

In the case of $S'_h$-realizations of homogeneous Sobolev spaces only few properties are explicitly stated in the literature. We repair this injustice here, for which usual proofs can be adapted (almost) straightforwardly and are well known, and therefore omitted here. Several references and comments are given after the next statement.

**Proposition 2.2** Let $p \in (1, +\infty)$, $s \in \mathbb{R}$. The following assertions hold:

(i) if $s \in [0, n/p)$ and $1/q = 1/p - s/n \in (0, 1)$, we have the standard Sobolev embeddings,

$$ \dot{H}^{s,p}(\mathbb{R}^n) \hookrightarrow L^q(\mathbb{R}^n), $$

$$ L^p(\mathbb{R}^n) \hookrightarrow \dot{H}^{-s,q}(\mathbb{R}^n), $$

(ii) if $(s, p)$ is such that it satisfies

$$ s < \frac{n}{p} \qquad (C_{s,p}) $$

then $\dot{H}^{s,p}(\mathbb{R}^n)$ is a complete normed vector space and

$$ (-\Delta)^{\frac{s}{2}} : \dot{H}^{s,p}(\mathbb{R}^n) \longrightarrow L^p(\mathbb{R}^n) $$

is a bijective isometry of Banach spaces,

(iii) for all $m \in \mathbb{N}$, all $u \in S'_h(\mathbb{R}^n)$,

$$ \sum_{j=1}^{n} \| \partial_{x_j}^m u \|_{\dot{H}^{s,p}(\mathbb{R}^n)} \sim_{s,m,p,n} \| \nabla^m u \|_{\dot{H}^{s,p}(\mathbb{R}^n)} \sim_{s,m,p,n} \| u \|_{\dot{H}^{s+m,p}(\mathbb{R}^n)} \quad (2.2) $$

(iv) the space $S_0(\mathbb{R}^n)$ is dense in $\dot{H}^{s,p}(\mathbb{R}^n)$,

(v) for all $u \in S'_h(\mathbb{R}^n)$, one has the equivalence of norms

$$ \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \sim_{s,p,n} \|u\|_{\dot{B}_{p,2}^s(\mathbb{R}^n)} := \|(2^{js} \dot{\Delta}_j u)_{j \in \mathbb{Z}}\|_{L^p(\mathbb{R}^n, \ell^2(\mathbb{Z}))}, $$

(vi) if $s \ge 0$, one has $H^{s,p}(\mathbb{R}^n) = L^p(\mathbb{R}^n) \cap \dot{H}^{s,p}(\mathbb{R}^n)$ with equivalence of norms and a continuous embedding

$$ \dot{H}^{-s,p}(\mathbb{R}^n) \hookrightarrow H^{-s,p}(\mathbb{R}^n), $$

(vii) if for some $\theta \in (0, 1)$, one has $(s, 1/p) = (1-\theta)(s_0, 1/p_0) + \theta(s_1, 1/p_1)$ for $j \in \{0, 1\}$, $s_j \in \mathbb{R}$, $p_j \in (1, +\infty)$, then for all $u \in S'_h(\mathbb{R}^n)$, we have

$$ \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{p_0, p_1, s_0, s_1, n} \|u\|_{\dot{H}^{s_0, p_0}(\mathbb{R}^n)}^{1-\theta} \|u\|_{\dot{H}^{s_1, p_1}(\mathbb{R}^n)}^{\theta}. $$
The inhomogeneous spaces $L^p(\mathbb{R}^n)$, $H^{s,p}(\mathbb{R}^n)$, and $B_{p,q}^s(\mathbb{R}^n)$ are all complete for all $p, q \in [1, +\infty]$, $s \in \mathbb{R}$, but in this setting homogeneous function spaces are no longer always complete (see [BCD11, Proposition 1.34, Remark 2.26]).

For homogeneous Besov spaces, we have the following properties:

**Proposition 2.1** Let $p, q \in [1, +\infty]$, $s \in \mathbb{R}$. The following assertions hold

(i) if $(s, p, q)$ satisfies the condition

$$ \left[ s < \frac{n}{p} \right] \text{ or } \left[ q = 1 \text{ and } s \le \frac{n}{p} \right] \qquad (C_{s,p,q}) $$

then $\dot{B}_{p,q}^s(\mathbb{R}^n)$ is a complete normed vector space,

(ii) for all $m \in \mathbb{N}$, all $u \in S'_h(\mathbb{R}^n)$,

$$ \sum_{j=1}^{n} \| \partial_{x_j}^m u \|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \sim_{s,m,p,n} \| \nabla^m u \|_{\dot{B}_{p,q}^{s+m}(\mathbb{R}^n)} \sim_{s,m,p,n} \| u \|_{\dot{B}_{p,q}^{s+m}(\mathbb{R}^n)}, \qquad (2.1) $$

(iii) if $p, q < +\infty$, the space $S_0(\mathbb{R}^n)$ is dense in $\dot{B}_{p,q}^s(\mathbb{R}^n)$,

(iv) if $1/q = 1/p - s/n \in (0, 1)$, we have dense Sobolev embeddings,

$$ \dot{B}_{p,r}^s(\mathbb{R}^n) \hookrightarrow L^q(\mathbb{R}^n), \quad (p, q < +\infty, r \in [1, q], s < n/p) $$

$$ \dot{B}_{p,1}^{n/p}(\mathbb{R}^n) \hookrightarrow C_0^0(\mathbb{R}^n), \quad (p < +\infty) $$

(v) when $s > 0$, one has $B_{p,q}^s(\mathbb{R}^n) = L^p(\mathbb{R}^n) \cap \dot{B}_{p,q}^s(\mathbb{R}^n)$ with equivalence of norms and a continuous embedding

$$ \dot{B}_{p,q}^{-s}(\mathbb{R}^n) \hookrightarrow \dot{B}_{p,q}^{-s}(\mathbb{R}^n). $$

see [BCD11, Theorem 2.25, Lemma 2.1, Propositions 2.27 & 2.39, Theorems 2.40 & 2.41] and [BL76, Theorem 6.3.2] for more details.

In the case of $S'_h$-realizations of homogeneous Sobolev spaces, only few properties are explicitly stated in the literature. We repair this injustice here, for which usual proofs can be adapted (almost) straightforwardly and are well known, and therefore omitted here. Several references and comments are given after the next statement.

**Proposition 2.2** Let $p \in (1, +\infty)$, $s \in \mathbb{R}$. The following assertions hold:

(i) if $s \in [0, n/p)$ and $1/q = 1/p - s/n \in (0, 1)$, we have the standard Sobolev embeddings,

$$ \dot{H}^{s,p}(\mathbb{R}^n) \hookrightarrow L^q(\mathbb{R}^n), $$

$$ L^p(\mathbb{R}^n) \hookrightarrow \dot{H}^{-s,q}(\mathbb{R}^n), $$

(ii) if $(s, p)$ is such that it satisfies

$$ s < \frac{n}{p} \qquad (C_{s,p}) $$

then $\dot{H}^{s,p}(\mathbb{R}^n)$ is a complete normed vector space and

$$ (-\Delta)^{\frac{s}{2}} : \dot{H}^{s,p}(\mathbb{R}^n) \to L^p(\mathbb{R}^n) $$

is a bijective isometry of Banach spaces,

(iii) for all $m \in \mathbb{N}$, all $u \in S'_h(\mathbb{R}^n)$,

$$ \sum_{j=1}^{n} \| \partial_{x_j}^m u \|_{\dot{H}^{s,p}(\mathbb{R}^n)} \sim_{s,m,p,n} \| \nabla^m u \|_{\dot{H}^{s,p}(\mathbb{R}^n)} \sim_{s,m,p,n} \| u \|_{\dot{H}^{s+m,p}(\mathbb{R}^n)}, \qquad (2.2) $$

(iv) the space $S_0(\mathbb{R}^n)$ is dense in $\dot{H}^{s,p}(\mathbb{R}^n)$,
**Remark 2.3** One may check first point (iii) as a direct consequence of [DHMT21, Lemma 2.6]. One may deduce the point (i) from [Gra14b, Theorem 1.2.3] and a density argument achieved manually (at this stage homogeneous Sobolev spaces are not known to be complete or not, only Lebesgue spaces are). The point (ii) is then a direct consequence of the point (i). The point (iii) follows directly from the proof of [DHMT21, Proposition 3.7] for the case $s = 0$.

The point (v) is a very well known result, based on extensive use of Khintchine's inequality $(L^p(\mathbb{R}^n)$-square function estimates) and the Hörmander-Mikhlin Fourier multiplier theorem, see for instance [Tri92, Remark 3, p.25] and [Gra14a, Proposition 6.1.2] for the case of $S'(\mathbb{R}^n)$ when $s = 0$. One may adapt the proof taking care of possible convergence issues (no density argument is a priori allowed).

See [BL76, Theorem 6.3.2] for the point (vi). The point (vii) is just a direct consequence of point (v), applying Hölder's inequality twice.

All the details can be found in the dissertation of the author [Gau23, Chapter 2, Section 2.1].

**Remark 2.4** One can make the comparison with homogeneous function spaces defined by tempered distributions quotiented by polynomials:

The point (i) of Proposition 2.1, the point (ii) of Proposition 2.2 and [Saw18, Theorems 2.31 & 2.32] tell us that all realizations of homogeneous Sobolev and Besov spaces can be isometrically identified whenever $s < n/p$.

**Lemma 2.5** Let $p_j \in (1, +\infty)$, $s_j \in \mathbb{R}$, for $j \in \{0, 1\}$. If $(C_{s_0, p_0})$ is satisfied then the intersection space $\dot{H}^{s_0, p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}^n)$ is a Banach space for which $S_0(\mathbb{R}^n)$ is dense in it.

*Proof.* — The completeness is straightforward. Concerning the claim about density, we follow the proof of [BCD11, Proposition 2.27] with minor modifications, in order to adapt it to our setting.

For $u \in \dot{H}^{s_0, p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}^n)$, and fixed $\varepsilon > 0$, for $k \in \{0, 1\}$ there exists $N \in \mathbb{N}$ such that for all $\tilde{N} \ge N$

$$ \|u - u_{\tilde{N}}\|_{\dot{H}^{s_k, p_k}(\mathbb{R}^n)} < \varepsilon. $$

Here, for any $K \in \mathbb{N}$,

$$ u_K := \sum_{|j| \le K} \dot{\Delta}_j u. $$

For $M \in [\tilde{N} + 1, +\infty[$, $R > 0$, provided $\Theta \in C_c^\infty(\mathbb{R}^n)$, real valued, supported in $B(0, 2)$, such that $\Theta|_{B(0,1)} = 1$, and $\Theta_R := \Theta(\cdot/R)$, we introduce

$$ u_{\tilde{N},M}^R := (I - \dot{S}_{-M})[\Theta_R u_{\tilde{N}}]. $$

Since $\dot{\Delta}_k u_{\tilde{N}} = 0$, $k \le -M - 1$, we have $\dot{S}_{-M} u_{\tilde{N}} = 0$, then

$$ u_{\tilde{N},M}^R - u_{\tilde{N}} = (I - \dot{S}_{-M})[(\Theta_R - 1)u_{\tilde{N}}]. $$

If one sets $m_k := \max(0, \lfloor s_k \rfloor + 2)$, since $0 \notin \text{supp } \mathcal{F}(u_{\tilde{N},M}^R - u_{\tilde{N}})$ by construction, we apply [BL76, Theorem 6.3.2] and decreasing embedding of inhomogeneous Sobolev spaces to deduce

$$ \begin{aligned} \|u_{\tilde{N},M}^R - u_{\tilde{N}}\|_{\dot{H}^{s_k, p_k}(\mathbb{R}^n)} & \lesssim_{M, s_k, p_k} \|u_{\tilde{N},M}^R - u_{\tilde{N}}\|_{\dot{H}^{s_k, p_k}(\mathbb{R}^n)} \\ & \lesssim_{M, s_k, p_k} \|(I - \dot{S}_{-M})[(\Theta_R - 1)u_{\tilde{N}}]\|_{\dot{H}^{m_k, p_k}(\mathbb{R}^n)} \\ & \lesssim_{M, s_k, p_k} \|[\Theta_R - 1]u_{\tilde{N}}\|_{\dot{H}^{m_k, p_k}(\mathbb{R}^n)}. \end{aligned} $$

Since one may check that $u_{\tilde{N}} \in \dot{H}^{m_k, p_k}(\mathbb{R}^n)$ for $k \in \{0, 1\}$, by dominated convergence theorem it follows that

$$ \|u_{\tilde{N},M}^R - u_{\tilde{N}}\|_{\dot{H}^{s_k, p_k}(\mathbb{R}^n)} \xrightarrow{R \to +\infty} 0. $$

Thus, for $R > 0$ big enough, we have for $k \in \{0, 1\}$

$$ \|u - u_{\tilde{N},M}^R\|_{\dot{H}^{s_k, p_k}(\mathbb{R}^n)} < 2\varepsilon. $$

The proof ends here since $u_{\tilde{N},M}^R \in S_0(\mathbb{R}^n)$.

<figure><img src="image_4.png" /></figure>
(v) for all $u \in S'_h(\mathbb{R}^n)$, one has the equivalence of norms

$$ \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \sim_{s,p,n} \|u\|_{\dot{H}^{s,p_2}(\mathbb{R}^n)} := \|(2^{js} \dot{\Delta}_j u)_{j \in \mathbb{Z}}\|_{L^p(\mathbb{R}^n, \ell^2(\mathbb{Z}))}, $$

(vi) if $s \ge 0$, one has $H^{s,p}(\mathbb{R}^n) = L^p(\mathbb{R}^n) \cap \dot{H}^{s,p}(\mathbb{R}^n)$ with equivalence of norms and a continuous embedding

$$ \dot{H}^{-s,p}(\mathbb{R}^n) \hookrightarrow \dot{H}^{-s,p}(\mathbb{R}^n), $$

(vii) if for some $\theta \in (0,1)$, one has $(s, 1/p) = (1-\theta)(s_0, 1/p_0) + \theta(s_1, 1/p_1)$ for $j \in \{0,1\}$, $s_j \in \mathbb{R}$, $p_j \in (1, +\infty)$, then for all $u \in S'_h(\mathbb{R}^n)$, we have

$$ \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s_0, s_1, p_0, p_1, n} \|u\|_{\dot{H}^{s_0, p_0}(\mathbb{R}^n)}^{1-\theta} \|u\|_{\dot{H}^{s_1, p_1}(\mathbb{R}^n)}^{\theta}. $$

**Remark 2.3** One may check first point (iii) as a direct consequence of [DHMT21, Lemma 2.6]. One may deduce the point (i) from [Gra14b, Theorem 1.2.3] and a density argument achieved manually (at this stage homogeneous Sobolev spaces are not known to be complete or not, only Lebesgue spaces are). The point (ii) is then a direct consequence of the point (i). The point (iv) follows directly from the proof of [DHMT21, Proposition 3.7] for the case $s = 0$.

The point (v) is a very well known result, based on extensive use of Khintchine's inequality ($L^p(\mathbb{R}^n)$ square estimates) and the Hörmander-Mikhlin Fourier multiplier theorem, see for instance [Tri92, Remark 3, p.25] and [Gra14a, Proposition 6.1.2] for the case of $S'(\mathbb{R}^n)$ when $s = 0$. One may adapt the proof, taking care of possible convergence issues (no density argument is a priori allowed).

See [BL76, Theorem 6.3.2] for the point (vi). The point (vii) is just a direct consequence of point (v), applying Hölder's inequality twice.

All the details can be found in the dissertation of the author [Gau23, Chapter 2, Section 2.1].

**Remark 2.4** One can make the comparison with homogeneous function spaces defined by tempered distributions quotiented by polynomials:

The point (i) of Proposition 2.1, the point (ii) of Proposition 2.2 and [Saw18, Theorems 2.31 & 2.32] tell us that all realizations of homogeneous Sobolev and Besov spaces can be isometrically identified whenever $s < n/p$.

**Lemma 2.5** Let $p_j \in (1, +\infty)$, $s_j \in \mathbb{R}$, for $j \in \{0,1\}$. If $(C_{s_0, p_0})$ is satisfied, then the intersection space $\dot{H}^{s_0, p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}^n)$ is a Banach space for which $S_0(\mathbb{R}^n)$ is dense in it.

*Proof.* — The completeness is straightforward: this follows from the completeness of $\dot{H}^{s_0, p_0}(\mathbb{R}^n)$, so that a Cauchy sequence of $\dot{H}^{s_0, p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}^n)$ admits a limit in $\dot{H}^{s_0, p_0}(\mathbb{R}^n) \subset S'_h(\mathbb{R}^n)$. From this point, the equivalence of norms given by point (v) of Proposition 2.2 allows passing to the limit in the norms for free using the Fatou Lemma.

Concerning the claim about density, we follow the proof of [BCD11, Proposition 2.27] with minor modifications, in order to adapt it to our setting.

For $u \in \dot{H}^{s_0, p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}^n)$, and fixed $\varepsilon > 0$, for $k \in \{0,1\}$ there exists $N \in \mathbb{N}$ such that for all $\tilde{N} \ge N$

$$ \|u - u_{\tilde{N}}\|_{\dot{H}^{s_k, p_k}(\mathbb{R}^n)} < \varepsilon. $$

Here, for any $K \in \mathbb{N}$,

$$ u_K := \sum_{|j| \le K} \dot{\Delta}_j u. $$

For $M \in [\tilde{N} + 1, +\infty]$, $R > 0$, provided $\Theta \in C_c^\infty(\mathbb{R}^n)$, real valued, supported in $B(0, 2)$, such that $\Theta_{|B(0,1)} = 1$, and $\Theta_R := \Theta(\cdot/R)$, we introduce

$$ u_{\tilde{N}, M} := (I - \dot{S}_{-M})[\Theta_R u_{\tilde{N}}]. $$

Since $\dot{\Delta}_k u_{\tilde{N}} = 0$, $k \le -M - 1$, we have $\dot{S}_{-M} u_{\tilde{N}} = 0$, then

$$ u_{\tilde{N}, M}^{R} - u_{\tilde{N}} = (I - \dot{S}_{-M})[(\Theta_R - 1)u_{\tilde{N}}]. $$
## 2.2 Interpolation, duality and the fundamental Sobolev multiplier

We recall the usual interpolation properties for inhomogeneous function spaces,

$$ 
\begin{aligned}
[H^{s_0,p_0}(\mathbb{R}^n), H^{s_1,p_1}(\mathbb{R}^n)]_{\theta} &= H^{s,p_\theta}(\mathbb{R}^n), & (B_{p,q_0}^{s_0}(\mathbb{R}^n), B_{p,q_1}^{s_1}(\mathbb{R}^n))_{\theta,q} &= B_{p,q}^s(\mathbb{R}^n), \\
(H^{s_0,p}(\mathbb{R}^n), H^{s_1,p}(\mathbb{R}^n))_{\theta,q} &= B_{p,q}^s(\mathbb{R}^n), & [B_{p_0,q_0}^{s_0}(\mathbb{R}^n), B_{p_1,q_1}^{s_1}(\mathbb{R}^n)]_{\theta} &= B_{p_\theta,q_\theta}^s(\mathbb{R}^n),
\end{aligned}
$$

whenever $(p_0, q_0), (p_1, q_1), (p, q) \in [1, +\infty]^2 (p \neq 1, +\infty)$, when dealing with Sobolev (Riesz potential) spaces), $\theta \in (0, 1)$, $s_0 \neq s_1$ two real numbers, such that

$$ \left(s, \frac{1}{p_\theta}, \frac{1}{q_\theta}\right) := (1 - \theta) \left(s_0, \frac{1}{p_0}, \frac{1}{q_0}\right) + \theta \left(s_1, \frac{1}{p_1}, \frac{1}{q_1}\right), $$

see [BL76, Theorem 6.4.5]. A similar statement is available for our homogeneous function spaces.

**Theorem 2.6** Let $(p_0, p_1, p, q, q_0, q_1) \in (1, +\infty)^3 \times [1, +\infty]^3$, $s_0, s_1 \in \mathbb{R}$, such that $s_0 \neq s_1$, and for $\theta \in (0, 1)$, let

$$ \left(s, \frac{1}{p_\theta}, \frac{1}{q_\theta}\right) := (1 - \theta) \left(s_0, \frac{1}{p_0}, \frac{1}{q_0}\right) + \theta \left(s_1, \frac{1}{p_1}, \frac{1}{q_1}\right). $$

Assuming $(C_{s_0,p_0})$ (resp. $(C_{s_0,p,q_0})$), we get the following

$$ (\dot{H}^{s_0,p}(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n))_{\theta,q} = (\dot{B}_{p,q_0}^{s_0}(\mathbb{R}^n), \dot{B}_{p,q_1}^{s_1}(\mathbb{R}^n))_{\theta,q} = \dot{B}_{p,q}^s(\mathbb{R}^n). \quad (2.3) $$

If moreover $(C_{s_0,p_0})$ and $(C_{s_1,p_1})$ are true then also is $(C_{s,p_\theta})$ and

$$ [\dot{H}^{s_0,p_0}(\mathbb{R}^n), \dot{H}^{s_1,p_1}(\mathbb{R}^n)]_{\theta} = \dot{H}^{s,p_\theta}(\mathbb{R}^n), \quad (2.4) $$

and similarly if $(C_{s_0,p_0,q_0})$ and $(C_{s_1,p_1,q_1})$ are satisfied then $(C_{s,p_\theta,q_\theta})$ is also satisfied and

$$ [\dot{B}_{p_0,q_0}^{s_0}(\mathbb{R}^n), \dot{B}_{p_1,q_1}^{s_1}(\mathbb{R}^n)]_{\theta} = \dot{B}_{p_\theta,q_\theta}^s(\mathbb{R}^n). \quad (2.5) $$

**Proof.** — **Step 1:** Let us deal with the real interpolation identity (2.3). Let us consider first the case of Sobolev spaces, with $u \in \dot{H}^{s_0,p}(\mathbb{R}^n) + \dot{H}^{s_1,p}(\mathbb{R}^n)$. For $(a, b) \in \dot{H}^{s_0,p}(\mathbb{R}^n) \times \dot{H}^{s_1,p}(\mathbb{R}^n)$, such that $u = a + b$, by point (v) in Proposition 2.2 we have

$$ (\dot{A}_j u)_{j \in \mathbb{Z}} = (\dot{A}_j a)_{j \in \mathbb{Z}} + (\dot{A}_j b)_{j \in \mathbb{Z}} \in L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})) + L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z})). $$

Therefore, by the definition of the K-functional and point (v) in Proposition 2.2, for $t > 0$,

$$ K(t, (\dot{A}_j u)_{j \in \mathbb{Z}}, L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))) \le \|a\|_{\dot{B}_{p,2}^{s_0}(\mathbb{R}^n)} + t\|b\|_{\dot{B}_{p,2}^{s_1}(\mathbb{R}^n)} \\ \lesssim_{p,s_0,s_1,n} \|a\|_{\dot{H}^{s_0,p}(\mathbb{R}^n)} + t\|b\|_{\dot{H}^{s_1,p}(\mathbb{R}^n)} $$

We then take the infimum on all such pairs $(a, b)$,

$$ K(t, (\dot{A}_j u)_{j \in \mathbb{Z}}, L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))) \lesssim_{p,s_0,s_1,n} K(t, u, \dot{H}^{s_0,p}(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n)). \quad (2.6) $$

Now, we want to prove the reverse estimate. Since $(\dot{A}_j u)_{j \in \mathbb{Z}} \in L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})) + L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))$, let $(A, B) \in L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})) \times L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))$ such that

$$ (\dot{A}_j u)_{j \in \mathbb{Z}} = A + B. \quad (2.7) $$

For $(w_j)_{j \in \mathbb{Z}} \subset S'(\mathbb{R}^n)$, say, for simplicity, with finite support in the discrete variable, we define the map

$$ \tilde{\Sigma}((w_j)_{j \in \mathbb{Z}}) := \sum_{j=-\infty}^{+\infty} \dot{A}_j [w_{j-1} + w_j + w_{j+1}], \quad (2.8) $$

and it satisfies for $v \in S'_h(\mathbb{R}^n)$

$$ \tilde{\Sigma}((\dot{A}_j v)_{j \in \mathbb{Z}}) = v. $$

By point (v) in Proposition 2.2 and [Gra14a, Proposition 6.1.4], one can check that

$$ \tilde{\Sigma} : L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})) \to \dot{H}^{s_0,p}(\mathbb{R}^n) \quad (2.9) $$

is well defined and bounded since $(C_{s_0,p_0})$ is satisfied. Now, we apply $\tilde{\Sigma}$ to (2.7) to deduce from
If one sets $m_k := \max(0, |s_k| + 2)$, since $0 \notin \text{supp } \mathcal{F}(u_{\tilde{N},M}^R - u_{\tilde{N}})$ by construction, we apply [BL76, Theorem 6.3.2] and decreasing embedding of inhomogeneous Sobolev spaces to deduce

$$
\begin{align*}
\| u_{\tilde{N},M}^R - u_{\tilde{N}} \|_{\dot{H}^{s_k,p_k}(\mathbb{R}^n)} &\lesssim_{M,s_k,p_k} \| u_{\tilde{N},M}^R - u_{\tilde{N}} \|_{H^{s_k,p_k}(\mathbb{R}^n)} \\
&\lesssim_{M,s_k,p_k} \|(I - \dot{S}_{-M})[(\Theta_R - 1)u_{\tilde{N}}]\|_{H^{m_k,p_k}(\mathbb{R}^n)} \\
&\lesssim_{M,s_k,p_k} \|[(\Theta_R - 1)u_{\tilde{N}}]\|_{H^{m_k,p_k}(\mathbb{R}^n)}.
\end{align*}
$$

Since one may check that $u_{\tilde{N}} \in H^{m_k, p_k}(\mathbb{R}^n)$ for $k \in \{0, 1\}$, by dominated convergence theorem it
follows that

$$
\| u_{\tilde{N},M}^R - u_{\tilde{N}} \|_{\dot{H}^{s_k,p_k}(\mathbb{R}^n)} \xrightarrow{R \to +\infty} 0.
$$

Thus, for $R > 0$ big enough, we have for $k \in \{0, 1\}$

$$
\|u - u_{\tilde{N},M}^R\|_{\dot{H}^{s_k,p_k}(\mathbb{R}^n)} < 2\epsilon.
$$

The proof ends here since $u_{\tilde{N},M}^R \in S_0(\mathbb{R}^n)$.

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

2.2 Interpolation, duality and the fundamental Sobolev multiplier result

We recall the usual interpolation properties for inhomogeneous function spaces,

$$
\begin{align*}
[\mathbf{H}^{s_0, p_0}(\mathbb{R}^n), \mathbf{H}^{s_1, p_1}(\mathbb{R}^n)]_{\theta} &= \mathbf{H}^{s, p_{\theta}}(\mathbb{R}^n), & (\mathbf{B}_{p,q_0}^{s_0}(\mathbb{R}^n), \mathbf{B}_{p,q_1}^{s_1}(\mathbb{R}^n))_{\theta,q} &= \mathbf{B}_{p,q}^{s}(\mathbb{R}^n), \\
(\mathbf{H}^{s_0, p}(\mathbb{R}^n), \mathbf{H}^{s_1, p}(\mathbb{R}^n))_{\theta,q} &= \mathbf{B}_{p,q}^{s}(\mathbb{R}^n), & [\mathbf{B}_{p_0,q_0}^{s_0}(\mathbb{R}^n), \mathbf{B}_{p_1,q_1}^{s_1}(\mathbb{R}^n)]_{\theta} &= \mathbf{B}_{p_{\theta},q_{\theta}}^{s}(\mathbb{R}^n),
\end{align*}
$$

whenever $(p_0, q_0)$, $(p_1, q_1)$, $(p, q) \in [1, +\infty]^2 (p \neq 1, +\infty)$, when dealing with Sobolev (Riesz potential) spaces), $\theta \in (0, 1)$, $s_0 \neq s_1$ two real numbers, such that

$$
\left(s, \frac{1}{p_{\theta}}, \frac{1}{q_{\theta}}\right) := (1 - \theta) \left(s_0, \frac{1}{p_0}, \frac{1}{q_0}\right) + \theta \left(s_1, \frac{1}{p_1}, \frac{1}{q_1}\right),
$$

see [BL76, Theorem 6.4.5]. A similar statement is available for our homogeneous function spaces.

**Theorem 2.6** Let $(p_0, p_1, p, q, q_0, q_1) \in (1, +\infty)^3 \times [1, +\infty]^3$, $s_0, s_1 \in \mathbb{R}$, such that $s_0 \neq s_1$, and for $\theta \in (0, 1)$, let

$$
\left(s, \frac{1}{p_{\theta}}, \frac{1}{q_{\theta}}\right) := (1 - \theta) \left(s_0, \frac{1}{p_0}, \frac{1}{q_0}\right) + \theta \left(s_1, \frac{1}{p_1}, \frac{1}{q_1}\right).
$$

Assuming $(C_{s_0,p_0})$ (resp. $(C_{s_0,p_0,q_0})$), we get the following

$$
(\dot{H}^{s_0,p}(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n))_{\theta,q} = (\dot{B}_{p,q_0}^{s_0}(\mathbb{R}^n), \dot{B}_{p,q_1}^{s_1}(\mathbb{R}^n))_{\theta,q} = \dot{B}_{p,q}^{s}(\mathbb{R}^n). \quad (2.3)
$$

If moreover $(C_{s_0,p_0})$ and $(C_{s_1,p_1})$ are true then also is $(C_{s,p_{\theta}})$ and

$$
[\dot{H}^{s_0, p_0}(\mathbb{R}^n), \dot{H}^{s_1, p_1}(\mathbb{R}^n)]_{\theta} = \dot{H}^{s, p_{\theta}}(\mathbb{R}^n), \quad (2.4)
$$

and similarly if $(C_{s_0,p_0,q_0})$ and $(C_{s_1,p_1,q_1})$ are satisfied then $(C_{s,p_{\theta},q_{\theta}})$ is also satisfied and when $q_{\theta} < +\infty$,

$$
[\dot{B}_{p_0, q_0}^{s_0}(\mathbb{R}^n), \dot{B}_{p_1, q_1}^{s_1}(\mathbb{R}^n)]_{\theta} = \dot{B}_{p_{\theta}, q_{\theta}}^{s}(\mathbb{R}^n). \quad (2.5)
$$

**Proof.** — **Step 1:** Let us deal with the real interpolation identity (2.3). Let us consider first the case of Sobolev spaces, with $u \in \dot{H}^{s_0,p}(\mathbb{R}^n) + \dot{H}^{s_1,p}(\mathbb{R}^n)$. For $(a,b) \in \dot{H}^{s_0,p}(\mathbb{R}^n) \times \dot{H}^{s_1,p}(\mathbb{R}^n)$, such that $u = a + b$, by point (v) in Proposition 2.2 we have

$$
(\dot{\Delta}_j u)_{j \in \mathbb{Z}} = (\dot{\Delta}_j a)_{j \in \mathbb{Z}} + (\dot{\Delta}_j b)_{j \in \mathbb{Z}} \in L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})) + L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z})).
$$

Therefore, by the definition of the $K$-functional and point (v) in Proposition 2.2, for $t > 0$,

$$
\begin{align*}
K(t, (\dot{\Delta}_j u)_{j \in \mathbb{Z}}, L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))) &\le \|a\|_{\dot{F}_{p,2}^{s_0}(\mathbb{R}^n)} + t\|b\|_{\dot{F}_{p,2}^{s_1}(\mathbb{R}^n)} \\
&\lesssim_{p,s_0,s_1,n} \|a\|_{\dot{H}^{s_0,p}(\mathbb{R}^n)} + t\|b\|_{\dot{H}^{s_1,p}(\mathbb{R}^n)}.
\end{align*}
$$
$\tilde{\Sigma}(\dot{\Delta}_j u)_{j \in \mathbb{Z}} = u \in S'_h(\mathbb{R}^n)$, and $\tilde{\Sigma}A \in \dot{\mathbb{H}}^{s_0,p}(\mathbb{R}^n) \subset S'_h(\mathbb{R}^n)$, that
$$ \tilde{\Sigma}B = u - \tilde{\Sigma}A \in S'_h(\mathbb{R}^n). $$

By mean of [Gra14a, Proposition 6.1.4], we obtain

$$ \|\tilde{\Sigma}B\|_{\dot{\mathbb{H}}_{p,2}^{s_1}(\mathbb{R}^n)} = \|(\dot{\Delta}_j \tilde{\Sigma} B)_{j \in \mathbb{Z}}\|_{L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))} \lesssim_{p, s_1, n} \|B\|_{L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))}. \quad (2.10) $$

Hence, by point (v) in Proposition 2.2, $\tilde{\Sigma}B$ is an element of $\dot{\mathbb{H}}^{s_1,p}(\mathbb{R}^n)$. Therefore by the definition of the K-functional, the boundedness properties of $\tilde{\Sigma}$, point (v) in Proposition 2.2, for $t > 0$,

$$ K(t, u, \dot{\mathbb{H}}^{s_0,p}(\mathbb{R}^n), \dot{\mathbb{H}}^{s_1,p}(\mathbb{R}^n)) \le \|\tilde{\Sigma}A\|_{\dot{\mathbb{H}}^{s_0,p}(\mathbb{R}^n)} + t\|\tilde{\Sigma}B\|_{\dot{\mathbb{H}}^{s_1,p}(\mathbb{R}^n)} \\ \lesssim_{p, s_0, s_1, n} \|A\|_{L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z}))} + t\|B\|_{L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))}. $$

Thus, let us take the infimum on all such pairs $(A, B)$, and invoke (2.6) to obtain for all $t > 0$, and all $u \in \dot{\mathbb{H}}^{s_0,p}(\mathbb{R}^n) + \dot{\mathbb{H}}^{s_1,p}(\mathbb{R}^n)$,

$$ K(t, (\dot{\Delta}_j u)_{j \in \mathbb{Z}}, L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))) \sim_{p, s_0, s_1, n} K(t, u, \dot{\mathbb{H}}^{s_0,p}(\mathbb{R}^n), \dot{\mathbb{H}}^{s_1,p}(\mathbb{R}^n)). \quad (2.11) $$

We recall that [BL76, Theorems 5.6.1 & 3.5.3] and [Tri78, Theorem, Section 1.18.4] give, all together, the well known real interpolation identity

$$ (L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z})))_{\theta,q} = \ell_{s_0}^q(\mathbb{Z}, L^p(\mathbb{R}^n)). \quad (2.12) $$

Thus, up to multiply the estimate (2.11) by $t^{-\theta}$ and taking its $L^q$-norm, it can be turned into

$$ \|u\|_{\dot{\mathbb{B}}_{p,q}^s(\mathbb{R}^n)} = \|(\dot{\Delta}_j u)_{j \in \mathbb{Z}}\|_{\ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n))} \sim_{p, s_0, s_1, \theta, n} \|(\dot{\Delta}_j u)_{j \in \mathbb{Z}}\|_{(L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z})))_{\theta,q}} \\ \sim_{p, s_0, s_1, \theta, n} \|u\|_{(\dot{\mathbb{H}}^{s_0,p}(\mathbb{R}^n), \dot{\mathbb{H}}^{s_1,p}(\mathbb{R}^n))_{\theta,q}}. $$

Therefore (2.3) is proved.

**Step 2:** For $p \in (1, +\infty)$, $q \in [1, +\infty]$ and $s \in \mathbb{R}$ such that $(C_{s,p,q})$ is satisfied, for $\tilde{\Sigma}$ introduced in (2.8), we want to show the boundedness of

$$ \tilde{\Sigma} : \ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n)) \to \dot{\mathbb{B}}_{p,q}^s(\mathbb{R}^n). \quad (2.13) $$

The idea is to consider instead the operator $(\dot{\Delta}_j \tilde{\Sigma})_{j \in \mathbb{Z}}$ and to show that it is a bounded operator seen as

$$ (\dot{\Delta}_j \tilde{\Sigma})_{j \in \mathbb{Z}} : \ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n)) \to \ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n)), \quad p \in (1, +\infty), \quad q \in [1, +\infty], \quad s \in \mathbb{R}. \quad (2.14) $$

In fact, as for (2.9) and (2.10), by [Gra14a, Proposition 6.1.4], for given $p \in (1, +\infty)$ and $s, s_0, s_1 \in \mathbb{R}$ with $s_0 < s < s_1$, one has the following boundedness property:

$$ (\dot{\Delta}_j \tilde{\Sigma})_{j \in \mathbb{Z}} : L^p(\mathbb{R}^n, \ell_{s_k}^2(\mathbb{Z})) \to L^p(\mathbb{R}^n, \ell_{s_k}^2(\mathbb{Z})), \quad k \in \{0, 1\}. $$

Therefore, (2.14) follows from real interpolation, thanks to the previous identity (2.12).

Now, in order to obtain (2.13) when $(C_{s,p,q})$ is satisfied, it suffices to consider an element $(u_j)_{j \in \mathbb{Z}}$ in the space $\ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n))$, then to approximate it by truncation with respect to the discrete variable, so that

$$ (\tilde{\Sigma}(u_j)_{j \in [-N, N]})_{N \in \mathbb{N}} \subset L^p(\mathbb{R}^n) \cap \dot{\mathbb{B}}_{p,q}^s(\mathbb{R}^n). $$

Then, the map $\tilde{\Sigma}$ extends uniquely to a bounded map with values in $\dot{\mathbb{B}}_{p,q}^s(\mathbb{R}^n)$ whenever $(C_{s,p,q})$ is satisfied and $q < +\infty$.

For the case $q = +\infty$ when $(C_{s,p,q})$ is satisfied, i.e. when $s < n/p$ is satisfied, the result follows in fact directly from **Step 1**.

In fact, the above manual real interpolation procedure was mainly performed to reach the endpoint couple $(\dot{\mathbb{B}}_{p,1}^{n/p}(\mathbb{R}^n), \ell_{n/p}^1(\mathbb{Z}, L^p(\mathbb{R}^n)))$.

**Step 3:** For the real interpolation identity (2.3) in the case of Besov spaces, by the previous **Step 2**, the proof presented in **Step 1** is still valid if we replace $(\dot{\mathbb{H}}^{s_0,p}, \dot{\mathbb{H}}^{s_1,p})$ and the condition $(C_{s_0,p})$ by $(\dot{\mathbb{B}}_{p,q_0}^{s_0}, \dot{\mathbb{B}}_{p,q_1}^{s_1})$ with the condition $(C_{s_0,p,q_0})$.

**Step 4:** As in the proof of [BL76, Theorem 6.4.5], being aware of [BL76, Definition 6.4.1], we can claim, thanks to previous steps, that
We then take the infimum on all such pairs $(a, b)$,

$$ K(t, (\dot{\Delta}_j u)_{j \in \mathbb{Z}}, L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))) \lesssim_{p, s_0, s_1, n} K(t, u, \dot{H}^{s_0, p}(\mathbb{R}^n), \dot{H}^{s_1, p}(\mathbb{R}^n)). \quad (2.6) $$

Now, we want to prove the reverse estimate. Since $(\dot{\Delta}_j u)_{j \in \mathbb{Z}} \in L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})) + L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))$, let $(A, B) \in L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})) \times L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))$ such that

$$ (\dot{\Delta}_j u)_{j \in \mathbb{Z}} = A + B. \quad (2.7) $$

For $(w_j)_{j \in \mathbb{Z}} \subset S'(\mathbb{R}^n)$, say, for simplicity, with finite support in the discrete variable, we define the map

$$ \tilde{\Sigma}((w_j)_{j \in \mathbb{Z}}) := \sum_{j=-\infty}^{+\infty} \dot{\Delta}_j [w_{j-1} + w_j + w_{j+1}], \quad (2.8) $$

and it satisfies for $v \in S'_h(\mathbb{R}^n)$

$$ \tilde{\Sigma}((\dot{\Delta}_j v)_{j \in \mathbb{Z}}) = v. $$

By point (v) in Proposition 2.2 and [Gra14a, Proposition 6.1.4], one can check that

$$ \tilde{\Sigma} : L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})) \to \dot{H}^{s_0, p}(\mathbb{R}^n) \quad (2.9) $$

is well-defined and bounded since $(C_{s_0, p})$ is satisfied. Now, we apply $\tilde{\Sigma}$ to (2.7) to deduce from $\tilde{\Sigma}(\dot{\Delta}_j u)_{j \in \mathbb{Z}} = u \in S'_h(\mathbb{R}^n)$, and $\tilde{\Sigma}A \in \dot{H}^{s_0, p}(\mathbb{R}^n) \subset S'_h(\mathbb{R}^n)$, that

$$ \tilde{\Sigma}B = u - \tilde{\Sigma}A \in S'_h(\mathbb{R}^n). $$

By mean of [Gra14a, Proposition 6.1.4], we obtain

$$ \|\tilde{\Sigma}B\|_{\dot{H}_{p,q}^{s_1}(\mathbb{R}^n)} = \|(\dot{\Delta}_j \tilde{\Sigma}B)_{j \in \mathbb{Z}}\|_{L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))} \lesssim_{p, s_1, n} \|B\|_{L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))}. \quad (2.10) $$

Hence, by point (v) in Proposition 2.2, $\tilde{\Sigma}B$ is an element of $\dot{H}^{s_1, p}(\mathbb{R}^n)$. Therefore, by the definition of the K-functional, the boundedness properties of $\tilde{\Sigma}$, point (v) in Proposition 2.2, for $t > 0$,

$$ K(t, u, \dot{H}^{s_0, p}(\mathbb{R}^n), \dot{H}^{s_1, p}(\mathbb{R}^n)) \le \|\tilde{\Sigma}A\|_{\dot{H}^{s_0, p}(\mathbb{R}^n)} + t\|\tilde{\Sigma}B\|_{\dot{H}^{s_1, p}(\mathbb{R}^n)} \\ \lesssim_{p, s_0, s_1, n} \|A\|_{L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z}))} + t\|B\|_{L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))}. $$

Thus, let us take the infimum on all such pairs $(A, B)$, and invoke (2.6) to obtain for all $t > 0$, and all $u \in \dot{H}^{s_0, p}(\mathbb{R}^n) + \dot{H}^{s_1, p}(\mathbb{R}^n)$,

$$ K(t, (\dot{\Delta}_j u)_{j \in \mathbb{Z}}, L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z}))) \sim_{p, s_0, s_1, n} K(t, u, \dot{H}^{s_0, p}(\mathbb{R}^n), \dot{H}^{s_1, p}(\mathbb{R}^n)). \quad (2.11) $$

We recall that [BL76, Theorems 5.6.1 & 3.5.3] and [Tri78, Theorem, Section 1.18.4] give, all together, the well known real interpolation identity

$$ (L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z})))_{\theta, q} = \ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n)). \quad (2.12) $$

Thus, up to multiply the estimate (2.11) by $t^{-\theta}$ and taking its $L_*^q$-norm, it can be turned into

$$ \|u\|_{\dot{H}_{p,q}^s(\mathbb{R}^n)} = \|(\dot{\Delta}_j u)_{j \in \mathbb{Z}}\|_{\ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n))} \sim_{p, s_0, s_1, \theta, n} \|(\dot{\Delta}_j u)_{j \in \mathbb{Z}}\|_{(L^p(\mathbb{R}^n, \ell_{s_0}^2(\mathbb{Z})), L^p(\mathbb{R}^n, \ell_{s_1}^2(\mathbb{Z})))_{\theta, q}} \\ \sim_{p, s_0, s_1, \theta, n} \|u\|_{(\dot{H}^{s_0, p}(\mathbb{R}^n), \dot{H}^{s_1, p}(\mathbb{R}^n))_{\theta, q}}. $$

Therefore (2.3) is proved.

**Step 2:** For $p \in (1, +\infty)$, $q \in [1, +\infty]$ and $s \in \mathbb{R}$ such that $(C_{s, p, q})$ is satisfied, for $\tilde{\Sigma}$ introduced in (2.8), we want to show the boundedness of

$$ \tilde{\Sigma} : \ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n)) \to \dot{H}_{p,q}^s(\mathbb{R}^n). \quad (2.13) $$

The idea is to consider instead the operator $(\dot{\Delta}_j \tilde{\Sigma})_{j \in \mathbb{Z}}$ and to show that it is a bounded operator, seen as

$$ (\dot{\Delta}_j \tilde{\Sigma})_{j \in \mathbb{Z}} : \ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n)) \to \ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n)), \quad p \in (1, +\infty), \quad q \in [1, +\infty], \quad s \in \mathbb{R}. \quad (2.14) $$

In fact, as for (2.9) and (2.10), by [Gra14a, Proposition 6.1.4], for given $p \in (1, +\infty)$ and
• thanks to its definition, for all $s \in \mathbb{R}$, $p \in (1, +\infty)$, $q \in [1, +\infty]$, when $(C_{s,p,q})$ is satisfied, $\dot{B}_{p,q}^s(\mathbb{R}^n)$ is a retraction of $\ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n))$ on $S'_h(\mathbb{R}^n)$ through the homogeneous Littlewood-Paley decomposition $(\dot{\Delta}_j)_{j \in \mathbb{Z}}$, and projection map $\bar{\Sigma}$;

• similarly, due to point (v) in Proposition 2.2, for all $s \in \mathbb{R}$, $p \in (1, +\infty)$, when $(C_{s,p})$ is satisfied $\dot{H}^{s,p}(\mathbb{R}^n)$ is a retraction of $L^p(\mathbb{R}^n, \ell_s^2(\mathbb{Z}))$ on $S'_h(\mathbb{R}^n)$ through the homogeneous Littlewood-Paley decomposition $(\dot{\Delta}_j)_{j \in \mathbb{Z}}$, and projection map $\bar{\Sigma}$.

Thus, one may apply [BL76, Theorem 6.4.2], with [BL76, Theorem 5.6.3] for complex interpolation of Besov spaces and [Tri78, Theorem, Section 1.18.4] for complex interpolation of Sobolev spaces, to obtain respectively (2.5) and (2.4).

The completeness assumption is necessary in the case of complex interpolation, since one can not provide in general an appropriate sense of holomorphic functions (then of the definition of complex interpolation spaces) in non-complete normed vector spaces. ■

**Proposition 2.7** For any $s \in \mathbb{R}$, $p \in (1, +\infty)$,

$$ \left\{ \begin{array}{l} \dot{H}^{s,p} \times \dot{H}^{-s,p'} \longrightarrow \mathbb{C} \\ (u, v) \quad \longmapsto \quad \sum_{|j-j'| \le 1} \langle \dot{\Delta}_j u, \dot{\Delta}_{j'} v \rangle_{\mathbb{R}^n} \end{array} \right. $$

defines a continuous bilinear functional on $\dot{H}^{s,p}(\mathbb{R}^n) \times \dot{H}^{-s,p'}(\mathbb{R}^n)$. Denote by $\mathcal{V}^{-s,p'}$ the set of functions $v \in S(\mathbb{R}^n) \cap \dot{H}^{-s,p'}(\mathbb{R}^n)$ such that $\|v\|_{\dot{H}^{-s,p'}(\mathbb{R}^n)} \le 1$. If $u \in S'_h(\mathbb{R}^n)$, then we have

$$ \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} = \sup_{v \in \mathcal{V}^{-s,p'}} |\langle u, v \rangle_{\mathbb{R}^n}|. $$

Moreover, if $(C_{s,p})$ is satisfied, $\dot{H}^{s,p}(\mathbb{R}^n)$ is reflexive and we have

$$ (\dot{H}^{-s,p'}(\mathbb{R}^n))' = \dot{H}^{s,p}(\mathbb{R}^n). \qquad (2.15) $$

*Proof.* — For simplicity, we will first work with the norm provided point (v) in Proposition 2.2, by equivalence of norms, the result will remain true. Let $(u, v) \in \dot{H}^{s,p}(\mathbb{R}^n) \times \dot{H}^{-s,p'}(\mathbb{R}^n)$, the $L^p(\ell^2) - L^{p'}(\ell^2)$ Hölder's inequality gives,

$$ \begin{aligned} |\langle u, v \rangle_{\mathbb{R}^n}| &\le \|u\|_{\dot{H}_{p,2}^s(\mathbb{R}^n)} \|\|(2^{-js}[\dot{\Delta}_{j-1} + \dot{\Delta}_j + \dot{\Delta}_{j+1}]v)_{j \in \mathbb{Z}}\|_{\ell^2(\mathbb{Z})} \|_{L^{p'}(\mathbb{R}^n)} \\ &\le (2^{|s|+1} + 1) \|u\|_{\dot{H}_{p,2}^s(\mathbb{R}^n)} \|v\|_{\dot{H}_{p',2}^{-s}(\mathbb{R}^n)}. \end{aligned} $$

Now, we know that it is a well defined quantity, we can compute

$$ \begin{aligned} \langle u, v \rangle_{\mathbb{R}^n} &= \sum_{|j-j'| \le 1} \langle \dot{\Delta}_j u, \dot{\Delta}_{j'} v \rangle_{\mathbb{R}^n} \\ &= \sum_{|j-j'| \le 1} \langle (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u, (-\Delta)^{-\frac{s}{2}} \dot{\Delta}_{j'} v \rangle_{\mathbb{R}^n} \\ &= \langle (-\Delta)^{\frac{s}{2}} u, (-\Delta)^{-\frac{s}{2}} v \rangle_{\mathbb{R}^n}. \end{aligned} $$

Hence, Hölder's inequality gives

$$ |\langle u, v \rangle_{\mathbb{R}^n}| \le \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \|v\|_{\dot{H}^{-s,p'}(\mathbb{R}^n)}, $$

which can be turned effortless into

$$ \sup_{v \in \mathcal{V}^{-s,p'}} |\langle u, v \rangle_{\mathbb{R}^n}| \le \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

This also proves the continuous embedding $\dot{H}^{s,p}(\mathbb{R}^n) \hookrightarrow (\dot{H}^{-s,p'}(\mathbb{R}^n))'$. For the reverse inequality, but not the reverse embedding, from $L^p - L^{p'}$ duality, by density of $S_0(\mathbb{R}^n)$, we have

$$ \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} = \sup_{\substack{v \in S_0(\mathbb{R}^n), \\ \|v\|_{L^{p'}} \le 1}} |\langle (-\Delta)^{\frac{s}{2}} u, v \rangle_{\mathbb{R}^n}| = \sup_{\substack{w \in S_0(\mathbb{R}^n), \\ \|w\|_{\dot{H}^{-s,p'}} \le 1}} |\langle u, w \rangle_{\mathbb{R}^n}| \le \sup_{v \in \mathcal{V}^{-s,p'}} |\langle u, v \rangle_{\mathbb{R}^n}|. $$
$s, s_0, s_1 \in \mathbb{R}$ with $s_0 < s < s_1$, one has the following boundedness property:

$$ (\dot{\Delta}_j \tilde{\Sigma})_{j \in \mathbb{Z}} : L^p(\mathbb{R}^n, \ell_{s_k}^2(\mathbb{Z})) \to L^p(\mathbb{R}^n, \ell_{s_k}^2(\mathbb{Z})), \quad k \in \{0, 1\}. $$

Therefore, (2.14) follows from real interpolation, thanks to the previous identity (2.12).

Now, in order to obtain (2.13) when $(C_{s,p,q})$ is satisfied, it suffices to consider an element $(u_j)_{j \in \mathbb{Z}}$ in the space $\ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n))$, then to approximate it by truncation with respect to the discrete variable, so that

$$ (\tilde{\Sigma}(u_j)_{j \in [-N, N]})_{N \in \mathbb{N}} \subset L^p(\mathbb{R}^n) \cap \dot{B}_{p,q}^s(\mathbb{R}^n). $$

Then, the map $\tilde{\Sigma}$ extends uniquely to a bounded map with values in $\dot{B}_{p,q}^s(\mathbb{R}^n)$ whenever $(C_{s,p,q})$ is satisfied and $q < +\infty$.

For the case $q = +\infty$ when $(C_{s,p,q})$ is satisfied, i.e. when $s < n/p$ is satisfied, the result follows in fact directly from **Step 1**.

In fact, the above manual real interpolation procedure was mainly performed to reach the endpoint couple $(\dot{B}_{p,1}^{n/p}(\mathbb{R}^n), \ell_{n/p}^1(\mathbb{Z}, L^p(\mathbb{R}^n)))$.

**Step 3:** For the real interpolation identity (2.3) in the case of Besov spaces, by the previous **Step 2**, the proof presented in **Step 1** is still valid if we replace $(\dot{H}^{s_0,p}, \dot{H}^{s_1,p})$ and the condition $(C_{s_0,p})$ by $(\dot{B}_{p,q_0}^{s_0}, \dot{B}_{p,q_1}^{s_1})$ with the condition $(C_{s_0,p,q_0})$.

**Step 4:** As in the proof of [BL76, Theorem 6.4.5], being aware of [BL76, Definition 6.4.1], we can claim, thanks to previous steps, that

* thanks to its definition, for all $s \in \mathbb{R}$, $p \in (1, +\infty)$, $q \in [1, +\infty]$, when $(C_{s,p,q})$ is satisfied, $\dot{B}_{p,q}^s(\mathbb{R}^n)$ is a retraction of $\ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n))$ on $S_h'(\mathbb{R}^n)$ through the homogeneous Littlewood-Paley decomposition $(\dot{\Delta}_j)_{j \in \mathbb{Z}}$, and projection map $\tilde{\Sigma}$;

* similarly, due to point (v) in Proposition 2.2, for all $s \in \mathbb{R}$, $p \in (1, +\infty)$, when $(C_{s,p})$ is satisfied $\dot{H}^{s,p}(\mathbb{R}^n)$ is a retraction of $L^p(\mathbb{R}^n, \ell_s^2(\mathbb{Z}))$ on $S_h'(\mathbb{R}^n)$ through the homogeneous Littlewood-Paley decomposition $(\dot{\Delta}_j)_{j \in \mathbb{Z}}$, and projection map $\tilde{\Sigma}$.

Thus, one may apply [BL76, Theorem 6.4.2], with [BL76, Theorem 5.6.3] for complex interpolation of Besov spaces and [Tri78, Theorem, Section 1.18.4] for complex interpolation of Sobolev spaces, to obtain respectively (2.5) and (2.4).

The completeness assumption is necessary in the case of complex interpolation, since one can not provide in general an appropriate sense of holomorphic functions (then of the definition of complex interpolation spaces) in non-complete normed vector spaces. ■

**Proposition 2.7** For any $s \in \mathbb{R}$, $p \in (1, +\infty)$,

$$ \left\{ \begin{array}{ll} \dot{H}^{s,p} \times \dot{H}^{-s,p'} & \to \mathbb{C} \\ (u,v) & \mapsto \sum_{|j-j'| \le 1} \langle \dot{\Delta}_j u, \dot{\Delta}_{j'} v \rangle_{\mathbb{R}^n} \end{array} \right. $$

defines a continuous bilinear functional on $\dot{H}^{s,p}(\mathbb{R}^n) \times \dot{H}^{-s,p'}(\mathbb{R}^n)$. Denote by $\mathcal{V}^{-s,p'}$ the set of functions $v \in S(\mathbb{R}^n) \cap \dot{H}^{-s,p'}(\mathbb{R}^n)$ such that $\|v\|_{\dot{H}^{-s,p'}(\mathbb{R}^n)} \le 1$. If $u \in S_h'(\mathbb{R}^n)$, then we have

$$ \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} = \sup_{v \in \mathcal{V}^{-s,p'}} |\langle u, v \rangle_{\mathbb{R}^n}|. $$

Moreover, if $(C_{s,p})$ is satisfied, $\dot{H}^{s,p}(\mathbb{R}^n)$ is reflexive and we have

$$ (\dot{H}^{-s,p'}(\mathbb{R}^n))' = \dot{H}^{s,p}(\mathbb{R}^n). \qquad (2.15) $$

*Proof.* — For simplicity, we will first work with the norm provided point (v) in Proposition 2.2, by equivalence of norms, the result will remain true. Let $(u, v) \in \dot{H}^{s,p}(\mathbb{R}^n) \times \dot{H}^{-s,p'}(\mathbb{R}^n)$, the $L^p(\ell^2)$-
In particular, the embedding $\dot{H}^{s,p}(\mathbb{R}^n) \hookrightarrow (\dot{H}^{-s,p'}(\mathbb{R}^n))'$ always holds and is isometric.

Now, assume that $(C_{s,p})$ holds. We recall that point (ii) of Proposition 2.2 yields the reflexivity of $\dot{H}^{s,p}(\mathbb{R}^n)$. Let $\tilde{U} \in (\dot{H}^{-s,p'}(\mathbb{R}^n))'$, we have

$$|\langle \tilde{U}, (-\Delta)^{\frac{s}{2}} v \rangle| \le \|\tilde{U}\|_{(\dot{H}^{-s,p'}(\mathbb{R}^n))'} \|v\|_{L^{p'}(\mathbb{R}^n)}, \quad v \in S_0(\mathbb{R}^n).$$

Since the space $S_0(\mathbb{R}^n)$ is dense in $L^{p'}(\mathbb{R}^n)$, we deduce there exists a unique function $w \in L^p(\mathbb{R}^n)$ such that,

$$\langle \tilde{U}, v \rangle = \langle w, (-\Delta)^{-\frac{s}{2}} v \rangle_{\mathbb{R}^n}, \quad v \in S(\mathbb{R}^n).$$

Thus $u := (-\Delta)^{-\frac{s}{2}} w \in \dot{H}^{s,p}(\mathbb{R}^n)$ by point (ii) in Proposition 2.2, and yields that the canonical embedding $\dot{H}^{s,p}(\mathbb{R}^n) \hookrightarrow (\dot{H}^{-s,p'}(\mathbb{R}^n))'$ is surjective. ■

**Proposition 2.8** For any $s \in \mathbb{R}$, $p \in (1, +\infty)$, $q \in [1, +\infty]$,

$$\left\{ \begin{array}{l} \dot{B}_{p,q}^s \times \dot{B}_{p',q'}^{-s} \quad \longrightarrow \mathbb{C} \\ (u,v) \quad \longmapsto \quad \sum_{|j-j'| \le 1} \langle \dot{\Delta}_j u, \dot{\Delta}_{j'} v \rangle_{\mathbb{R}^n} \end{array} \right. $$

defines a continuous bilinear functional on $\dot{B}_{p,q}^s(\mathbb{R}^n) \times \dot{B}_{p',q'}^{-s}(\mathbb{R}^n)$. Denote by $Q_{p',q'}^{-s}$ the set of functions $v \in S(\mathbb{R}^n) \cap \dot{B}_{p',q'}^{-s}(\mathbb{R}^n)$ such that $\|v\|_{\dot{B}_{p',q'}^{-s}(\mathbb{R}^n)} \le 1$. If $u \in S_h'(\mathbb{R}^n)$, then we have

$$\|u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{p,s,n} \sup_{v \in Q_{p',q'}^{-s}} |\langle u, v \rangle_{\mathbb{R}^n}|.$$

Moreover, if $-n/p' < s < n/p$ is satisfied and $q \in (1, +\infty]$ then

$$(\dot{B}_{p',q'}^{-s}(\mathbb{R}^n))' = \dot{B}_{p,q}^s(\mathbb{R}^n) \quad \text{and} \quad (\dot{B}_{p',\infty}^{-s}(\mathbb{R}^n))' = \dot{B}_{p,1}^s(\mathbb{R}^n). \qquad (2.16)$$

The space $\dot{B}_{p,q}^s(\mathbb{R}^n)$ is reflexive whenever both $(C_{s,p,q})$ and $q \ne 1, +\infty$ are satisfied.

**Proof.** — The first part of the claim is just [BCD11, Proposition 2.29]. The claimed part about reflexivity and duality follows directly from the application of [BL76, Theorem 3.7.1] and of Theorem 2.6 and Proposition 2.7. ■

We also have a Sobolev-Besov multiplier result, which is useful for the construction of homogeneous Sobolev and Besov space on domains. The first presentation of the next result in the setting of inhomogeneous function spaces is due to Strichartz [Str67, Chapter II, Corollary 3.7], one may all so check [JK95, Proposition 3.5]. We are going to use it to prove a straightforward generalization. The next result was also known but only stated for homogeneous Besov spaces up to now, see e.g. [DM09, Appendix].

**Proposition 2.9** For all $p \in (1, +\infty)$, $q \in [1, +\infty]$, for all $s \in (-1 + \frac{1}{p}, \frac{1}{p})$, for all $u \in \dot{H}^{s,p}(\mathbb{R}^n)$ (resp. $\dot{B}_{p,q}^s(\mathbb{R}^n)$),

$$\|1_{\mathbb{R}^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \quad (\text{resp. } \|1_{\mathbb{R}^n} u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)}).$$

The same results still holds with $(H, B)$ instead of $(\dot{H}, \dot{B})$.

**Proof.** — We start from the result stated in the inhomogeneous case [Str67, Chapter II, Corollary 3.7], which states the following: for all $p \in (1, +\infty)$, for all $s \in [0, \frac{1}{p})$, for all $u \in \dot{H}^{s,p}(\mathbb{R}^n)$

$$\|1_{\mathbb{R}^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}.$$

If $s = 0$, there is nothing to achieve since $\dot{H}^{0,p}(\mathbb{R}^n) = \dot{H}^{0,p'}(\mathbb{R}^n) = L^p(\mathbb{R}^n)$ with equality of norms. Now for $s > 0$, by the equivalence of norms, we obtain

$$\|1_{\mathbb{R}^n} u\|_{L^p(\mathbb{R}^n)} + \|1_{\mathbb{R}^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{L^p(\mathbb{R}^n)} + \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}.$$
L$^{p'}$($\ell^2$) Hölder's inequality gives,

$$
\begin{aligned}
& \left|\langle u, v \rangle_{\mathbb{R}^n}\right| \le \|u\|_{\dot{\mathbb{F}}_{p,2}^s(\mathbb{R}^n)} \left\|\|(2^{-js}[\dot{\Delta}_{j-1} + \dot{\Delta}_j + \dot{\Delta}_{j+1}]v)_{j \in \mathbb{Z}}\|_{\ell^2(\mathbb{Z})}\right\|_{L^{p'}(\mathbb{R}^n)} \\
& \le (2^{|s|+1} + 1) \|u\|_{\dot{\mathbb{F}}_{p,2}^s(\mathbb{R}^n)} \|v\|_{\dot{\mathbb{F}}_{p',2}^{-s}(\mathbb{R}^n)}.
\end{aligned}
$$

Now, we know that it is a well-defined quantity, we can compute

$$
\begin{aligned}
\langle u, v \rangle_{\mathbb{R}^n} &= \sum_{|j-j'| \le 1} \langle \dot{\Delta}_j u, \dot{\Delta}_{j'} v \rangle_{\mathbb{R}^n} \\
&= \sum_{|j-j'| \le 1} \langle (-\Delta)^{\frac{s}{2}} \dot{\Delta}_j u, (-\Delta)^{-\frac{s}{2}} \dot{\Delta}_{j'} v \rangle_{\mathbb{R}^n} \\
&= \langle (-\Delta)^{\frac{s}{2}} u, (-\Delta)^{-\frac{s}{2}} v \rangle_{\mathbb{R}^n}.
\end{aligned}
$$

Hence, Hölder's inequality gives

$$
\left|\langle u, v \rangle_{\mathbb{R}^n}\right| \le \|u\|_{\dot{\mathbb{H}}^{s,p}(\mathbb{R}^n)} \|v\|_{\dot{\mathbb{H}}^{-s,p'}(\mathbb{R}^n)},
$$
which can be turned effortless into
$$
\sup_{v \in \mathcal{V}^{-s,p'}} \left|\langle u, v \rangle_{\mathbb{R}^n}\right| \le \|u\|_{\dot{\mathbb{H}}^{s,p}(\mathbb{R}^n)}.
$$

This also proves the continuous embedding $\dot{\mathbb{H}}^{s,p}(\mathbb{R}^n) \hookrightarrow (\dot{\mathbb{H}}^{-s,p'}(\mathbb{R}^n))'$. For the reverse inequality, but not the reverse embedding, from $L^p - L^{p'}$ duality, by density of $S_0(\mathbb{R}^n)$, we have

$$
\|u\|_{\dot{\mathbb{H}}^{s,p}(\mathbb{R}^n)} = \sup_{\substack{v \in S_0(\mathbb{R}^n), \\ \|v\|_{L^{p'}} \le 1}} \left| \langle (-\Delta)^{\frac{s}{2}} u, v \rangle_{\mathbb{R}^n} \right| = \sup_{\substack{w \in S_0(\mathbb{R}^n), \\ \|w\|_{\dot{\mathbb{H}}^{-s,p'}} \le 1}} \left| \langle u, w \rangle_{\mathbb{R}^n} \right| \le \sup_{v \in \mathcal{V}^{-s,p'}} \left| \langle u, v \rangle_{\mathbb{R}^n} \right|.
$$

In particular, the embedding $\dot{\mathbb{H}}^{s,p}(\mathbb{R}^n) \hookrightarrow (\dot{\mathbb{H}}^{-s,p'}(\mathbb{R}^n))'$ always holds and is isometric.

Now, assume that ($C_{s,p}$) holds. We recall that point (ii) of Proposition 2.2 yields the reflexivity of $\dot{\mathbb{H}}^{s,p}(\mathbb{R}^n)$. Let $\tilde{U} \in (\dot{\mathbb{H}}^{-s,p'}(\mathbb{R}^n))'$, we have

$$
|\langle \tilde{U}, (-\Delta)^{\frac{s}{2}} v \rangle| \le \|\tilde{U}\|_{(\dot{\mathbb{H}}^{-s,p'}(\mathbb{R}^n))'} \|v\|_{L^{p'}(\mathbb{R}^n)}, \quad v \in S_0(\mathbb{R}^n).
$$

Since the space $S_0(\mathbb{R}^n)$ is dense in $L^{p'}(\mathbb{R}^n)$, we deduce there exists a unique function $w \in L^p(\mathbb{R}^n)$ such that,

$$
\langle \tilde{U}, v \rangle = \langle w, (-\Delta)^{-\frac{s}{2}} v \rangle_{\mathbb{R}^n}, \quad v \in S(\mathbb{R}^n).
$$

Thus $u := (-\Delta)^{-\frac{s}{2}} w \in \dot{\mathbb{H}}^{s,p}(\mathbb{R}^n)$ by point (ii) in Proposition 2.2, and yields that the canonical embedding $\dot{\mathbb{H}}^{s,p}(\mathbb{R}^n) \hookrightarrow (\dot{\mathbb{H}}^{-s,p'}(\mathbb{R}^n))'$ is surjective. $\blacksquare$

**Proposition 2.8** For any $s \in \mathbb{R}$, $p \in (1, +\infty)$, $q \in [1, +\infty]$,

$$
\left\{
\begin{array}{l}
\dot{\mathbb{B}}_{p,q}^{s} \times \dot{\mathbb{B}}_{p',q'}^{-s} \quad \longrightarrow \mathbb{C} \\
\\
(u,v) \quad \longmapsto \quad \sum_{|j-j'| \le 1} \langle \dot{\Delta}_j u, \dot{\Delta}_{j'} v \rangle_{\mathbb{R}^n}
\end{array}
\right.
$$

defines a continuous bilinear functional on $\dot{\mathbb{B}}_{p,q}^s(\mathbb{R}^n) \times \dot{\mathbb{B}}_{p',q'}^{-s}(\mathbb{R}^n)$. Denote by $\mathcal{Q}_{p',q'}^{-s}$ the set of func-
tions $v \in S(\mathbb{R}^n) \cap \dot{\mathbb{B}}_{p',q'}^{-s}(\mathbb{R}^n)$ such that $\|v\|_{\dot{\mathbb{B}}_{p',q'}^{-s}(\mathbb{R}^n)} \le 1$. If $u \in S_h'(\mathbb{R}^n)$, then we have

$$
\|u\|_{\dot{\mathbb{B}}_{p,q}^s(\mathbb{R}^n)} \lesssim_{p,s,n} \sup_{v \in \mathcal{Q}_{p',q'}^{-s}} |\langle u, v \rangle_{\mathbb{R}^n}|.
$$

Moreover, if $-n/p' < s < n/p$ is satisfied and $q \in (1, +\infty]$ then

$$
(\dot{\mathbb{B}}_{p',q'}^{-s}(\mathbb{R}^n))' = \dot{\mathbb{B}}_{p,q}^s(\mathbb{R}^n) \text{ and } (\dot{\mathbb{B}}_{p',\infty}^{-s}(\mathbb{R}^n))' = \dot{\mathbb{B}}_{p,1}^s(\mathbb{R}^n). \quad (2.16)
$$

The space $\dot{\mathbb{B}}_{p,q}^s(\mathbb{R}^n)$ is reflexive whenever both $(C_{s,p,q})$ and $q \neq 1, +\infty$ are satisfied.

*Proof*. — The first part of the claim is just [BCD11, Proposition 2.29]. The claimed part about
Plugging $u_\lambda := u(\lambda\cdot)$ in above inequality, provided $\lambda$ is a positive real number, since one has $1_{\mathbb{R}_+^n}(\lambda\cdot)u_\lambda = 1_{\mathbb{R}_+^n}u_\lambda$, we obtain that

$$ \lambda^{-\frac{n}{p}} \|1_{\mathbb{R}_+^n} u\|_{L^p(\mathbb{R}^n)} + \lambda^{s-\frac{n}{p}} \|1_{\mathbb{R}_+^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \lambda^{-\frac{n}{p}} \|u\|_{L^p(\mathbb{R}^n)} + \lambda^{s-\frac{n}{p}} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

Thus one may divide by $\lambda^{s-\frac{n}{p}}$, and then as $\lambda$ tends to infinity, we deduce

$$ \|1_{\mathbb{R}_+^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

Therefore, the result follows by density argument.

The result for $s \in (-1 + \frac{1}{p}, 0)$ is a consequence of duality and density using the duality bracket defined on $S_0(\mathbb{R}^n) \times S_0(\mathbb{R}^n)$.

The Besov space case follows by real interpolation.

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

### 3 Function spaces on the upper half-space

Let $s \in \mathbb{R}$, $p \in (1, +\infty)$, $q \in [1, +\infty]$. Then for any $X \in \{\mathbb{B}_{p,q}^s, \dot{\mathbb{B}}_{p,q}^s, \mathbb{H}^{s,p}, \dot{\mathbb{H}}^{s,p}\}$, and we define

$$ X(\mathbb{R}_+^n) := X(\mathbb{R}^n)|_{\mathbb{R}_+^n}, $$

with the quotient norm $\|u\|_{X(\mathbb{R}_+^n)} := \inf_{\substack{\tilde{u} \in X(\mathbb{R}^n), \\ \tilde{u}|_{\mathbb{R}_+^n} = u.}} \|\tilde{u}\|_{X(\mathbb{R}^n)}$. A direct consequence from the definition of those spaces is the density of $S_0(\mathbb{R}_+^n) \subset S(\mathbb{R}_+^n)$ in each of them. The completeness and convexity is also carried over when their counterpart on $\mathbb{R}^n$ are respectively complete and reflexive. We also define

$$ X_0(\mathbb{R}_+^n) := \{ u \in X(\mathbb{R}^n) \mid \text{supp } u \subset \overline{\mathbb{R}_+^n} \}, $$

with natural induced norm $\|u\|_{X_0(\mathbb{R}_+^n)} := \|u\|_{X(\mathbb{R}^n)}$. We always have the canonical continuous injection,

$$ X_0(\mathbb{R}_+^n) \hookrightarrow X(\mathbb{R}_+^n). $$

Since there is a natural embedding $S'(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}_+^n)$, we also have the inclusion

$$ X(\mathbb{R}_+^n) \subset \mathcal{D}'(\mathbb{R}_+^n). $$

If X and Y are different function spaces

* if one has continuous embedding

$$ Y(\mathbb{R}^n) \hookrightarrow X(\mathbb{R}^n). $$

a direct consequence from the definition is

$$ Y(\mathbb{R}_+^n) \hookrightarrow X(\mathbb{R}_+^n), $$

and similarly with $X_0$ and $Y_0$.

* We write $[X \cap Y](\mathbb{R}_+^n)$ the restriction of $X(\mathbb{R}^n) \cap Y(\mathbb{R}^n)$ to $\mathbb{R}_+^n$, in general there is nothing to ensure more than

$$ [X \cap Y](\mathbb{R}_+^n) \hookrightarrow X(\mathbb{R}_+^n) \cap Y(\mathbb{R}_+^n). $$

Results corresponding to those obtained for the whole space $\mathbb{R}^n$ in previous section are usually carried over by the existence of an appropriate extension operator

$$ \mathcal{E} : S'(\mathbb{R}_+^n) \to S'(\mathbb{R}^n), $$

bounded from $X(\mathbb{R}_+^n)$ to $X(\mathbb{R}^n)$.
reflexivity and duality follows directly from the application of [BL76, Theorem 3.7.1] and of Theorem 2.6 and Proposition 2.7.

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

We also have a Sobolev-Besov multiplier result, which is useful for the construction of homogeneous Sobolev and Besov space on domains. The first presentation of the next result in the setting of inhomogeneous function spaces is due to Strichartz [Str67, Chapter II, Corollary 3.7], one may also check [JK95, Proposition 3.5]. We are going to use it to prove a straightforward generalization. The next result was also known but only stated for homogeneous Besov spaces up to now, see e.g. [DM09, Appendix].

**Proposition 2.9** For all $p \in (1, +\infty)$, $q \in [1, +\infty]$, for all $s \in (-1 + \frac{1}{p}, \frac{1}{p})$, for all $u \in \dot{H}^{s,p}(\mathbb{R}^n)$ (resp. $\dot{B}_{p,q}^s(\mathbb{R}^n)$),

$$ \|1_{\mathbb{R}_+^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \quad (\text{resp. } \|1_{\mathbb{R}_+^n} u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)}). $$

The same results still hold with $(H, B)$ instead of $(\dot{H}, \dot{B})$.

*Proof.* — We start from the result stated in the inhomogeneous case [Str67, Chapter II, Corollary 3.7], which states the following: for all $p \in (1, +\infty)$, for all $s \in [0, \frac{1}{p})$, for all $u \in H^{s,p}(\mathbb{R}^n)$

$$ \|1_{\mathbb{R}_+^n} u\|_{H^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{H^{s,p}(\mathbb{R}^n)}. $$

If $s = 0$, there is nothing to achieve since $H^{0,p}(\mathbb{R}^n) = \dot{H}^{0,p}(\mathbb{R}^n) = L^p(\mathbb{R}^n)$ with equality of norms. Now for $s > 0$, by the equivalence of norms, we obtain

$$ \|1_{\mathbb{R}_+^n} u\|_{L^p(\mathbb{R}^n)} + \|1_{\mathbb{R}_+^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{L^p(\mathbb{R}^n)} + \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

Plugging $u_\lambda := u(\lambda \cdot)$ in the above inequality, provided $\lambda$ is a positive real number, since one has $1_{\mathbb{R}_+^n}(\lambda \cdot)u_\lambda = 1_{\mathbb{R}_+^n}u_\lambda$, we obtain that

$$ \lambda^{-\frac{n}{p}} \|1_{\mathbb{R}_+^n} u\|_{L^p(\mathbb{R}^n)} + \lambda^{s-\frac{n}{p}} \|1_{\mathbb{R}_+^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \lambda^{-\frac{n}{p}} \|u\|_{L^p(\mathbb{R}^n)} + \lambda^{s-\frac{n}{p}} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

Thus one may divide by $\lambda^{s-\frac{n}{p}}$, and then as $\lambda$ tends to infinity, we deduce

$$ \|1_{\mathbb{R}_+^n} u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

Therefore, the result follows by density argument.

The result for $s \in (-1 + \frac{1}{p}, 0)$ is a consequence of duality and density using the duality bracket defined on $S_0(\mathbb{R}^n) \times S_0(\mathbb{R}^n)$.

The Besov space case follows by real interpolation.

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

# 3 Function spaces on the upper half-space

From now on, until the end of the paper, we assume that the dimension $n$ satisfies $n \ge 2$.

Let $s \in \mathbb{R}$, $p \in (1, +\infty)$, $q \in [1, +\infty]$. Then for any $X \in \{B_{p,q}^s, \dot{B}_{p,q}^s, H^{s,p}, \dot{H}^{s,p}\}$, we define

$$ X(\mathbb{R}_+^n) := X(\mathbb{R}^n)|_{\mathbb{R}_+^n}, $$

with the quotient norm $\|u\|_{X(\mathbb{R}_+^n)} := \inf_{\substack{\tilde{u} \in X(\mathbb{R}^n), \\ \tilde{u}|_{\mathbb{R}_+^n} = u.}} \|\tilde{u}\|_{X(\mathbb{R}^n)}$. A direct consequence of the definition of

those spaces is the density of $S_0(\overline{\mathbb{R}_+^n}) \subset S(\overline{\mathbb{R}_+^n})$ in each of them. The completeness and convexity is also carried over when their counterpart on $\mathbb{R}^n$ are respectively complete and reflexive. We also define

$$ X_0(\mathbb{R}_+^n) := \{ u \in X(\mathbb{R}^n) \mid \text{supp } u \subset \overline{\mathbb{R}_+^n} \}, $$

with natural induced norm $\|u\|_{X_0(\mathbb{R}_+^n)} := \|u\|_{X(\mathbb{R}^n)}$. We always have the canonical continuous injection,

$$ X_0(\mathbb{R}_+^n) \hookrightarrow X(\mathbb{R}_+^n). $$
## 3.1 Quick overview of inhomogeneous function spaces on $\mathbb{R}_+^n$

For inhomogeneous spaces on special Lipschitz domains (in particular on $\mathbb{R}_+^n$), an approach was done by Stein in [Ste70, Chapter VI], for Sobolev spaces with non-negative index, and Besov spaces of positive index of regularity (this follows by real interpolation). A full and definitive result for the inhomogeneous case on Lipschitz domains, and even in a more general case (allowing $p, q$ to be less than 1 considering the whole Besov and Triebel-Lizorkin scales), was given by Rychkov in [Ryc99] where the extension operator is known to be universal and to cover even negative regularity index.

The extension operator provided by Rychkov can be used to prove, thanks to [BL76, Theorem 6.4.2], if $(\mathfrak{h}, \mathfrak{b}) \in \{(H, B), (H_0, B_{.,.}, 0)\}$,

$$ [\mathfrak{h}^{s_0, p_0}(\mathbb{R}_+^n), \mathfrak{h}^{s_1, p_1}(\mathbb{R}_+^n)]_\theta = \mathfrak{h}^{s, p_\theta}(\mathbb{R}_+^n), \quad (\mathfrak{b}_{p, q_0}^{s_0}(\mathbb{R}_+^n), \mathfrak{b}_{p, q_1}^{s_1}(\mathbb{R}_+^n))_{\theta, q} = \mathfrak{b}_{p, q}^{s}(\mathbb{R}_+^n), \quad (3.1) $$

$$ (\mathfrak{h}^{s_0, p}(\mathbb{R}_+^n), \mathfrak{h}^{s_1, p}(\mathbb{R}_+^n))_{\theta, q} = \mathfrak{b}_{p, q}^s(\mathbb{R}_+^n), \quad [\mathfrak{b}_{p_0, q_0}^{s_0}(\mathbb{R}_+^n), \mathfrak{b}_{p_1, q_1}^{s_1}(\mathbb{R}_+^n)]_{\theta} = \mathfrak{b}_{p_\theta, q_\theta}^s(\mathbb{R}_+^n), \quad (3.2) $$

whenever $(p_0, q_0), (p_1, q_1), (p, q) \in [1, +\infty]^2$ ($p \neq 1, +\infty$), when dealing with Sobolev (Bessel potential) spaces), $\theta \in (0, 1)$, $s_0 \neq s_1$ two real numbers, such that

$$ \left(s, \frac{1}{p_\theta}, \frac{1}{q_\theta}\right) := \left(1 - \theta\right) \left(s_0, \frac{1}{p_0}, \frac{1}{q_0}\right) + \theta \left(s_1, \frac{1}{p_1}, \frac{1}{q_1}\right). $$

A nice property is that the description of the boundary yields the following density results, for all $p \in (1, +\infty)$, $q \in [1, +\infty)$, $s \in \mathbb{R}$,

$$ H_0^{s,p}(\mathbb{R}_+^n) = \overline{C_c^\infty(\mathbb{R}_+^n)}^{\|\cdot\|_{H^{s,p}(\mathbb{R}^n)}}, \quad \text{and} \quad B_{p,q,0}^s(\mathbb{R}_+^n) = \overline{C_c^\infty(\mathbb{R}_+^n)}^{\|\cdot\|_{B_{p,q}^s(\mathbb{R}^n)}}. \quad (3.3) $$

One may check [JK95, Section 2] for the treatment of Sobolev spaces case, the Besov spaces case follows by interpolation argument, see [BL76, Theorem 3.4.2]. As a direct consequence, one has from [JK95, Proposition 2.9] and [BL76, Theorem 3.7.1], that for all $s \in \mathbb{R}$, $p \in (1, +\infty)$, $q \in [1, +\infty)$,

$$ (H^{s,p}(\mathbb{R}_+^n))' = H_0^{-s, p'}(\mathbb{R}_+^n), \quad (B_{p,q}^s(\mathbb{R}_+^n))' = B_{p', q', 0}^{-s}(\mathbb{R}_+^n), \quad (3.4) $$

$$ (B_{p,q,0}^s(\mathbb{R}_+^n))' = B_{p',q'}^{-s}(\mathbb{R}_+^n). \quad (3.5) $$

And finally, thanks to the inhomogeneous version of Proposition 2.9, we also have a particular case of equality of Sobolev spaces, with equivalent norms, for all $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$,

$$ H^{s,p}(\mathbb{R}_+^n) = H_0^{s,p}(\mathbb{R}_+^n), \quad B_{p,q}^s(\mathbb{R}_+^n) = B_{p,q,0}^s(\mathbb{R}_+^n). \quad (3.6) $$

The interested reader may also found an explicit and way more general (and still valid, for the most part of it, in the case of the half-space) treatment for bounded Lipschitz domains in [KMM07], where the Triebel-Lizorkin scale, including Hardy spaces, and other endpoint function spaces are also treated.

All the results presented above will be used without being mentioned and are assumed to be well known to the reader.

## 3.2 Homogeneous function spaces on $\mathbb{R}_+^n$

One may expect to recover similar results for the scale of homogeneous Sobolev and Besov as the one mentioned in the subsection 3.1. However, due to the setting involving the use of $S'_h(\mathbb{R}^n)$, we have a lack of completeness so that one can no longer use complex interpolation theory and density argument on the whole scale to provide boundedness of linear operators. A first approach we could have in mind is that one would expect Rychkov's extension operator to preserve $S'_h$, say $\mathcal{E}(S'_h(\mathbb{R}_+^n)) \subset S'_h(\mathbb{R}^n)$ with homogeneous estimates, which is not known yet.

However, if we consider a more naive extension operator like by reflection around the boundary, as in [DHMT21, Chapter 3], a certain amount of results remains true, up to consider index $s > -1 + \frac{1}{p}$, provided $p \in (1, +\infty)$. This is what we are going to achieve here: this subsection is devoted to proofs of usual results on homogeneous Sobolev and Besov spaces on $\mathbb{R}_+^n$.

This subsection contains 3 subparts: the first one is about extension-restriction and density results for our homogeneous Sobolev spaces, from which for the second, we are going to build
Since there is a natural embedding $S'(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}_+^n)$, we also have the inclusion

$$X(\mathbb{R}_+^n) \subset \mathcal{D}'(\mathbb{R}_+^n).$$

If X and Y are different function spaces

* if one has continuous embedding

$$Y(\mathbb{R}^n) \hookrightarrow X(\mathbb{R}^n).$$

a direct consequence of the definition is

$$Y(\mathbb{R}_+^n) \hookrightarrow X(\mathbb{R}_+^n),$$

and similarly with $X_0$ and $Y_0$.

* We write $[X \cap Y](\mathbb{R}_+^n)$ the restriction of $X(\mathbb{R}^n) \cap Y(\mathbb{R}^n)$ to $\mathbb{R}_+^n$, in general there is nothing to ensure more than

$$[X \cap Y](\mathbb{R}_+^n) \hookrightarrow X(\mathbb{R}_+^n) \cap Y(\mathbb{R}_+^n).$$

Results corresponding to those obtained for the whole space $\mathbb{R}^n$ in the previous section are usually carried over by the existence of an appropriate extension operator

$$\mathcal{E}: S'(\mathbb{R}_+^n) \to S'(\mathbb{R}^n),$$

bounded from $X(\mathbb{R}_+^n)$ to $X(\mathbb{R}^n)$.

## 3.1 Quick overview of inhomogeneous function spaces on $\mathbb{R}_+^n$

For inhomogeneous spaces on special Lipschitz domains (in particular on $\mathbb{R}_+^n$), an approach was done by Stein in [Ste70, Chapter VI], for Sobolev spaces with non-negative index, and Besov spaces of positive index of regularity (this follows by real interpolation). A full and definitive result for the inhomogeneous case on Lipschitz domains, and even in a more general case (allowing $p, q$ to be less than 1 considering the whole Besov and Triebel-Lizorkin scales), was given by Rychkov in [Ryc99] where the extension operator is known to be universal and to cover even negative regularity index.

The extension operator provided by Rychkov can be used to prove, thanks to [BL76, Theorem 6.4.2], if $(h, b) \in \{(H, B), (H_0, B_{.,.}, 0)\}$,

$$[h^{s_0, p_0}(\mathbb{R}_+^n), h^{s_1, p_1}(\mathbb{R}_+^n)]_{\theta} = h^{s, p_{\theta}}(\mathbb{R}_+^n), \quad (h_{p_0, q_0}^{s_0}(\mathbb{R}_+^n), h_{p, q_1}^{s_1}(\mathbb{R}_+^n))_{\theta, q} = h_{p, q}^{s}(\mathbb{R}_+^n), \quad (3.1)$$

$$(h^{s_0, p}(\mathbb{R}_+^n), h^{s_1, p}(\mathbb{R}_+^n))_{\theta, q} = h_{p, q}^s(\mathbb{R}_+^n), \quad [h_{p_0, q_0}^{s_0}(\mathbb{R}_+^n), h_{p_1, q_1}^{s_1}(\mathbb{R}_+^n)]_{\theta} = h_{p_{\theta}, q_{\theta}}^s(\mathbb{R}_+^n), \quad (3.2)$$

whenever $(p_0, q_0), (p_1, q_1), (p, q) \in [1, +\infty]^2$ ($p \neq 1, +\infty$, when dealing with Sobolev (Bessel potential) spaces), $\theta \in (0, 1)$, $s_0 \neq s_1$ two real numbers, such that

$$\left(s, \frac{1}{p_{\theta}}, \frac{1}{q_{\theta}}\right) := (1 - \theta) \left(s_0, \frac{1}{p_0}, \frac{1}{q_0}\right) + \theta \left(s_1, \frac{1}{p_1}, \frac{1}{q_1}\right),$$

with $q_{\theta} < +\infty$.

A nice property is that the description of the boundary yields the following density results, for all $p \in (1, +\infty)$, $q \in [1, +\infty)$, $s \in \mathbb{R}$,

$$H_0^{s, p}(\mathbb{R}_+^n) = \overline{C_c^{\infty}(\mathbb{R}_+^n)}^{\|\cdot\|_{H^{s, p}(\mathbb{R}^n)}}, \quad \text{and} \quad B_{p, q, 0}^s(\mathbb{R}_+^n) = \overline{C_c^{\infty}(\mathbb{R}_+^n)}^{\|\cdot\|_{B_{p, q}(\mathbb{R}^n)}}, \quad (3.3)$$

One may check [JK95, Section 2] for the treatment of Sobolev spaces case, the Besov spaces case follows by interpolation argument, see [BL76, Theorem 3.4.2]. As a direct consequence, one has from [JK95, Proposition 2.9] and [BL76, Theorem 3.7.1], that for all $s \in \mathbb{R}$, $p \in (1, +\infty)$, $q \in [1, +\infty)$,

$$(H^{s, p}(\mathbb{R}_+^n))' = H_0^{-s, p'}(\mathbb{R}_+^n), \quad (B_{p, q}^s(\mathbb{R}_+^n))' = B_{p', q', 0}^{-s}(\mathbb{R}_+^n), \quad (3.4)$$

$$(B_{p, q, 0}^s(\mathbb{R}_+^n))' = B_{p', q', 0}^{-s}(\mathbb{R}_+^n). \quad (3.5)$$

And finally, thanks to the inhomogeneous version of Proposition 2.9, we also have a particular case
corresponding ones for Besov spaces, via some ersatz of real interpolation procedure. Both will be used to build the third subpart which concerns effective interpolation results for our homogeneous Sobolev and Besov spaces.

### 3.2.1 Homogeneous Sobolev spaces

We start proving the boundedness of extension operators defined by higher order reflection principle but for homogeneous Sobolev spaces with fractional index of regularity. This is done as in [DHMT21, Lemma 3.15, Proposition 3.19], where it was achieved only for homogeneous Besov spaces.

**Proposition 3.1** For $m \in \mathbb{N}$, there exists a linear extension operator E, depending on $m$, such that for all $p \in (1, +\infty)$, $-1 + \frac{1}{p} < s < m + 1 + \frac{1}{p}$, so that if either,

* $s \ge 0$ and $u \in H^{s,p}(\mathbb{R}_+^n)$;

* $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ and $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$;

we have

$$Eu_{|\mathbb{R}_+^n} = u,$$

with the estimate

$$\|Eu\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{p,s,n,m} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}.$$

In particular, E: $\dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$ extends uniquely to a bounded operator whenever $(C_{s,p})$ is satisfied.

*Proof.* — As in [DHMT21, Lemma 3.15], let us introduce the higher order reflexion operator E, defined for all measurable function $u: \mathbb{R}_+^n \to \mathbb{C}$ by

$$Eu(x) := \begin{cases} u(x) & , \text{ if } x \in \mathbb{R}_+^n, \\ \sum_{j=0}^{m} \alpha_j u(x', -\frac{x_n}{j+1}) & , \text{ if } x \in \mathbb{R}^n \setminus \mathbb{R}_+^n. \end{cases}$$

where, as in [DHMT21, Lemma 3.15], $x = (x_1, \dots, x_{n-1}, x_n) = (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, and $(\alpha_j)_{j \in [0, m]}$ is such that E maps $C^m$-functions on $\mathbb{R}_+^n$ to $C^m$-functions on $\mathbb{R}^n$. This is indeed true since $\alpha_j$, $j \in [0, m]$, is chosen so that it satisfies for all $\kappa \in [0, m]$,

$$\sum_{j=0}^{m} \left( \frac{-1}{j+1} \right)^{\kappa} \alpha_j = 1.$$

By construction, the operator E also maps boundedly $H^{k,p}(\mathbb{R}_+^n)$ to $H^{k,p}(\mathbb{R}^n)$ for all $k \in [0, m+1]$. The boundedness of the operator E from $H^{s,p}(\mathbb{R}_+^n)$ to $H^{s,p}(\mathbb{R}^n)$ for all $s \in [0, m+1]$ follows from complex interpolation.

Notice also that Proposition 2.9 and the formulation, given for $x \in \mathbb{R}^n$,

$$Eu(x) = [1_{\mathbb{R}_+^n} u](x) + \sum_{j=0}^{m} \alpha_j [1_{\mathbb{R}_+^n} u](x', -\frac{x_n}{j+1})$$

implies that E: $\dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$ is bounded for all $s \in (-1 + \frac{1}{p}, \frac{1}{p})$.

Now for $p \in (1, +\infty)$, $s \in [0, m + 1 + \frac{1}{p})$, $s - \frac{1}{p} \notin \mathbb{N}$, $u \in H^{s,p}(\mathbb{R}_+^n)$, E: $H^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$, we can whose $\ell \in \mathbb{N}$ such that $s - \ell \in (-1 + \frac{1}{p}, \frac{1}{p})$ so that

$$\partial_{x_k}^\ell Eu = E[\partial_{x_\ell}^\ell u], \text{ provided } k \in [1, n-1],$$

$$\partial_{x_n}^\ell Eu = E^{(\ell)} \partial_{x_n}^\ell u = \sum_{j=0}^{m} \alpha_j \left(\frac{-1}{j+1}\right)^\ell \partial_{x_n}^\ell u(x', -\frac{x_n}{j+1}).$$

For the same reasons as in the beginning of the present proof, $E^{(\ell)}$ maps $H^{s,p}(\mathbb{R}_+^n)$ to $H^{s,p}(\mathbb{R}^n)$ for all $s \in [0, m - \ell + 1]$, and $\dot{H}^{s,p}(\mathbb{R}_+^n)$ to $\dot{H}^{s,p}(\mathbb{R}^n)$ for $s \in (-1 + 1/p, 1/p)$, thanks to Proposition 2.9.
of equality of Sobolev spaces, with equivalent norms, for all $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (-1+\frac{1}{p}, \frac{1}{p})$,

$$H^{s,p}(\mathbb{R}_+^n) = H_0^{s,p}(\mathbb{R}_+^n), \quad B_{p,q}^s(\mathbb{R}_+^n) = B_{p,q,0}^s(\mathbb{R}_+^n). \qquad (3.6)$$

The interested reader may also find an explicit and way more general (and still valid, for the most part of it, in the case of the half-space) treatment for bounded Lipschitz domains in [KMM07], where the Triebel-Lizorkin scale, including Hardy spaces, and other endpoint function spaces are also treated.

All the results presented above will be used without being mentioned and are assumed to be well known to the reader.

## 3.2 Homogeneous function spaces on $\mathbb{R}_+^n$

One may expect to recover similar results for the scale of homogeneous Sobolev and Besov as the one mentioned in the subsection 3.1. However, due to the setting involving the use of $S'_h(\mathbb{R}^n)$, we have a lack of completeness so that one can no longer use complex interpolation theory and density argument on the whole scale to provide boundedness of linear operators. A first approach we could have in mind is that one would expect Rychkov's extension operator to preserve $S'_h$, say $\mathcal{E}(S'_h(\mathbb{R}_+^n)) \subset S'_h(\mathbb{R}^n)$ with **homogeneous** estimates, which is not known yet.

However, if we consider a more naive extension operator like by reflection around the boundary, as in [DHMT21, Chapter 3], a certain amount of results remains true, up to consider index $s > -1+\frac{1}{p}$, provided $p \in (1, +\infty)$. This is what we are going to achieve here: this subsection is devoted to proofs of usual results on homogeneous Sobolev and Besov spaces on $\mathbb{R}_+^n$.

This subsection contains 3 subparts: the first one is about extension-restriction and density results for our homogeneous Sobolev spaces, from which for the second, we are going to build corresponding ones for Besov spaces, via some ersatz of real interpolation procedure. Both will be used to build the third subpart, which concerns effective interpolation results for our homogeneous Sobolev and Besov spaces.

### 3.2.1 Homogeneous Sobolev spaces

We start proving the boundedness of extension operators defined by higher order reflection principle but for homogeneous Sobolev spaces with fractional index of regularity. This is done as in [DHMT21, Lemma 3.15, Proposition 3.19], where it was achieved only for homogeneous Besov spaces.

**Proposition 3.1** For $m \in \mathbb{N}$, there exists a linear extension operator E, depending on m, such that for all $p \in (1, +\infty)$, $-1+\frac{1}{p} < s < m+1+\frac{1}{p}$, so that if either,

* $s \ge 0$ and $u \in H^{s,p}(\mathbb{R}_+^n)$;

* $s \in (-1+\frac{1}{p}, \frac{1}{p})$ and $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$;

we have

$$Eu|_{\mathbb{R}_+^n} = u,$$

with the estimate

$$\|Eu\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{p,s,n,m} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}.$$

In particular, $E: \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$ extends uniquely to a bounded operator whenever $(C_{s,p})$ is satisfied.

*Proof.* — As in [DHMT21, Lemma 3.15], let us introduce the higher order reflection operator E, defined for all measurable function $u: \mathbb{R}_+^n \to \mathbb{C}$ by

$$Eu(x) := \begin{cases} u(x) & , \text{ if } x \in \mathbb{R}_+^n, \\ \sum_{j=0}^{m} \alpha_j u(x', -\frac{x_j}{j+1}) & , \text{ if } x \in \mathbb{R}^n \setminus \mathbb{R}_+^n. \end{cases}$$
From the fact that $\partial_{x_j}^\ell u \in \dot{H}^{s-\ell,p}(\mathbb{R}_+^n)$, we deduce

$$ \|Eu\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \sim_{\ell,p,n} \sum_{j=1}^{n-1} \|\partial_{x_j}^\ell Eu\|_{\dot{H}^{s-\ell,p}(\mathbb{R}^n)} + \|E^{(\ell)}\partial_{x_n}^\ell u\|_{\dot{H}^{s-\ell,p}(\mathbb{R}^n)} \lesssim_{s,\ell,p,n,m} \sum_{j=1}^n \|\partial_{x_j}^\ell u\|_{\dot{H}^{s-\ell,p}(\mathbb{R}^n)}. \quad (3.7) $$

To be more synthetic, we have obtained

$$ \|Eu\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{p,k,n,m} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}, $$

so that $E: \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$ is bounded on the subspace $H^{s,p}(\mathbb{R}_+^n)$, in particular it extends uniquely to a bounded linear operator on whole $\dot{H}^{s,p}(\mathbb{R}_+^n)$ when it is complete, i.e. $s < \frac{n}{p}$, this follows from the fact that $S(\mathbb{R}_+^n) \subset H^{s,p}(\mathbb{R}_+^n)$ is dense in $\dot{H}^{s,p}(\mathbb{R}_+^n)$.

It remains to cover cases when $s - \frac{1}{p} \in [0, m]$. To do so, we want to reproduce the above procedure, proving first that $E(\text{resp. } E^{(\ell)}, \ell \in [1, m])$ is bounded from $\dot{H}^{\frac{1}{p},p}(\mathbb{R}_+^n)$ to $\dot{H}^{\frac{1}{p},p}(\mathbb{R}^n)$, via some complex interpolation scheme.

Now let $p_0, p_1 \in (1, +\infty)$, $p_1 < n$, $\theta \in (0, 1)$. Consider $u \in [L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1, p_1}(\mathbb{R}_+^n)]_\theta$. Let $f \in F(L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1, p_1}(\mathbb{R}_+^n))$, such that $f(\theta) = u$, it follows from the previous considerations that $Ef \in F(L^{p_0}(\mathbb{R}^n), \dot{H}^{1, p_1}(\mathbb{R}^n))$. Thus, from Theorem 2.6, one has

$$ \mathrm{E}f(\theta) \in \dot{H}^{\theta,p}(\mathbb{R}^n), \text{ where } \left(\theta, \frac{1}{p}\right) := (1-\theta)\left(0, \frac{1}{p_0}\right) + \theta\left(1, \frac{1}{p_1}\right). $$

So $u = \mathrm{E}f(\theta)|_{\mathbb{R}_+^n} \in \dot{H}^{\theta,p}(\mathbb{R}_+^n)$ with the norm estimate

$$ \|u\|_{\dot{H}^{\theta,p}(\mathbb{R}_+^n)} \lesssim_{m_1,p,n} \|u\|_{[L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1,p_1}(\mathbb{R}_+^n)]_\theta} $$

which is a direct consequence of the definition of restriction space, the equivalence of the complex interpolation norm (2.4) from Theorem 2.6, the definition of the complex interpolation norm, and then of the boundedness of E from $L^{p_0}(\mathbb{R}^n)$ to $L^{p_0}(\mathbb{R}_+^n)$ and from $\dot{H}^{1, p_1}(\mathbb{R}^n)$ to $\dot{H}^{1, p_1}(\mathbb{R}_+^n)$. Now, if $u \in \dot{H}^{\theta,p}(\mathbb{R}_+^n)$, by definition of restriction spaces there exists $U \in \dot{H}^{\theta,p}(\mathbb{R}^n)$, such that

$$ U|_{\mathbb{R}_+^n} = u, \quad \text{and} \quad \frac{1}{2} \|U\|_{\dot{H}^{\theta,p}(\mathbb{R}^n)} \le \|u\|_{\dot{H}^{\theta,p}(\mathbb{R}_+^n)} \le \|U\|_{\dot{H}^{\theta,p}(\mathbb{R}^n)}. $$

By Theorem 2.6, there exists $f \in F(L^{p_0}(\mathbb{R}^n), \dot{H}^{1, p_1}(\mathbb{R}^n))$ such that $f(\theta) = U$, we deduce $f(\cdot)|_{\mathbb{R}_+^n} \in F(L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1, p_1}(\mathbb{R}_+^n))$, so $u = f(\theta)|_{\mathbb{R}_+^n} \in [L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1, p_1}(\mathbb{R}_+^n)]_\theta$ with the following estimate which is a direct consequence from the definition of function spaces by restriction, and complex interpolation spaces,

$$ \|u\|_{[L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1, p_1}(\mathbb{R}_+^n)]_\theta} \lesssim \|u\|_{\dot{H}^{\theta,p}(\mathbb{R}_+^n)}. $$

Hence, homogeneous (Riesz potential) Sobolev spaces on the half-space are still a complex interpolation scale provided that $p \in (1, +\infty)$, $s \in [0, 1]$, ($C_{s,p}$) being satisfied, so the boundedness of $E: \dot{H}^{\theta,p}(\mathbb{R}_+^n) \to \dot{H}^{\theta,p}(\mathbb{R}^n)$ follows by interpolation.

In particular, $E: \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$ is bounded for all $s \in (-1 + \frac{1}{p}, \frac{1}{p}]$. Hence the result has been proved for $s - \frac{1}{p} = 0$. The same result is obtained for $E^{(\ell)}$, provided $\ell \in [1, m]$.

Now, let $p \in (1, +\infty)$, $s - \frac{1}{p} \in [1, m]$, for $u \in H^{s,p}(\mathbb{R}_+^n)$, we have $Eu \in H^{s,p}(\mathbb{R}^n)$, $\nabla^\ell Eu \in \dot{H}^{s-\ell,p}(\mathbb{R}^n)$, $s - \ell = \frac{1}{p}$, so that, similarly as in (3.7),

$$ \|Eu\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n,\ell} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

Therefore, we have obtained the desired estimate and can conclude about the boundedness of E via density argument whenever ($C_{s,p}$) is satisfied. ■

In the proof of Proposition 3.1, we used boundedness of derivatives, i.e. for all $p \in (1, +\infty)$,
where, as in [DHMT21, Lemma 3.15], $x = (x_1, \dots, x_{n-1}, x_n) = (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, and $(\alpha_j)_{j \in [0, m]}$ is such that E maps $\mathbb{C}^m$-functions on $\mathbb{R}_+^n$ to $\mathbb{C}^m$-functions on $\mathbb{R}^n$. This is indeed true since $\alpha_j$, $j \in [0, m]$, is chosen so that it satisfies for all $\kappa \in [0, m]$,

$$ \sum_{j=0}^{m} \left( \frac{-1}{j+1} \right)^{\kappa} \alpha_j = 1. $$

By construction, the operator E also maps boundedly $H^{k,p}(\mathbb{R}_+^n)$ to $H^{k,p}(\mathbb{R}^n)$ for all $k \in [0, m+1]$. The boundedness of the operator E from $H^{s,p}(\mathbb{R}_+^n)$ to $H^{s,p}(\mathbb{R}^n)$ for all $s \in [0, m+1]$ follows from complex interpolation.

Notice also that Proposition 2.9 and the formulation, given for $x \in \mathbb{R}^n$,

$$ \mathrm{E}u(x) = [1_{\mathbb{R}_+^n} u](x) + \sum_{j=0}^{m} \alpha_j [1_{\mathbb{R}_+^n} u](x', -\frac{x_n}{j+1}). $$

implies that $E : \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$ is bounded for all $s \in (-1 + \frac{1}{p}, \frac{1}{p})$.

Now, for $p \in (1, +\infty)$, $s \in [0, m+1+\frac{1}{p})$, $s-\frac{1}{p} \notin \mathbb{N}$, $u \in H^{s,p}(\mathbb{R}_+^n)$, and $E : H^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$, we can choose $\ell \in \mathbb{N}$ such that $s - \ell \in (-1 + \frac{1}{p}, \frac{1}{p})$, so that

$$ \partial_{x_k}^\ell \mathrm{E}u = \mathrm{E}[\partial_{x_\ell}^\ell u], \text{ provided } k \in [1, n-1], $$

$$ \partial_{x_n}^\ell \mathrm{E}u = \mathrm{E}^{(\ell)} \partial_{x_n}^\ell u = \sum_{j=0}^{m} \alpha_j \left( \frac{-1}{j+1} \right)^\ell \partial_{x_n}^\ell u(x', -\frac{x_n}{j+1}). $$

For the same reasons as in the beginning of the present proof, $E^{(\ell)}$ maps $H^{s,p}(\mathbb{R}_+^n)$ to $H^{s,p}(\mathbb{R}^n)$ for all $s \in [0, m - \ell + 1]$, and $\dot{H}^{s,p}(\mathbb{R}_+^n)$ to $\dot{H}^{s,p}(\mathbb{R}^n)$ for $s \in (-1 + 1/p, 1/p)$, thanks to Proposition 2.9.

From the fact that $\partial_{x_j}^\ell u \in \dot{H}^{s-\ell,p}(\mathbb{R}_+^n)$, we deduce

$$ \|\mathrm{E}u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \sim_{\ell,p,n} \sum_{j=1}^{n-1} \|\partial_{x_j}^\ell \mathrm{E}u\|_{\dot{H}^{s-\ell,p}(\mathbb{R}^n)} + \|\mathrm{E}^{(\ell)}\partial_{x_n}^\ell u\|_{\dot{H}^{s-\ell,p}(\mathbb{R}^n)} \lesssim_{s,\ell,p,n,m} \sum_{j=1}^{n} \|\partial_{x_j}^\ell u\|_{\dot{H}^{s-\ell,p}(\mathbb{R}^n)}. \quad (3.7) $$

To be more synthetic, we have obtained

$$ \|\mathrm{E}u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{p,k,n,m} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}, $$

so that $E : \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$ is bounded on the subspace $H^{s,p}(\mathbb{R}_+^n)$, in particular it extends uniquely to a bounded linear operator on the whole $\dot{H}^{s,p}(\mathbb{R}_+^n)$ when it is complete, i.e. $s < \frac{n}{p}$, this follows from the fact that $S(\mathbb{R}_+^n) \subset H^{s,p}(\mathbb{R}_+^n)$ is dense in $\dot{H}^{s,p}(\mathbb{R}_+^n)$.

It remains to cover cases when $s - \frac{1}{p} \in [0, m]$. To do so, we want to reproduce the above procedure, proving first that $E(\mathrm{resp. } E^{(\ell)}, \ell \in [1, m])$ is bounded from $\dot{H}^{\frac{1}{p},p}(\mathbb{R}_+^n)$ to $\dot{H}^{\frac{1}{p},p}(\mathbb{R}^n)$, via some complex interpolation scheme.

Now let $p_0, p_1 \in (1, +\infty)$, $p_1 < n$, $\theta \in (0, 1)$. Notice that, here, $n \ge 2$ is a really important assumption, otherwise there wouldn't any such $p_1$. Consider $u \in [L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1,p_1}(\mathbb{R}_+^n)]_\theta$. Let $f \in F(L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1,p_1}(\mathbb{R}_+^n))$, such that $f(\theta) = u$, it follows from the previous considerations that $Ef \in F(L^{p_0}(\mathbb{R}^n), \dot{H}^{1,p_1}(\mathbb{R}^n))$. Thus, from Theorem 2.6, one has

$$ \mathrm{E}f(\theta) \in \dot{H}^{\theta,p}(\mathbb{R}^n), \text{ where } \left(\theta, \frac{1}{p}\right) := (1-\theta) \left(0, \frac{1}{p_0}\right) + \theta \left(1, \frac{1}{p_1}\right). $$

So $u = \mathrm{E}f(\theta)|_{\mathbb{R}_+^n} \in \dot{H}^{\theta,p}(\mathbb{R}_+^n)$ with the norm estimate

$$ \|u\|_{\dot{H}^{\theta,p}(\mathbb{R}_+^n)} \lesssim_{m_1,p,n} \|u\|_{[L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1,p_1}(\mathbb{R}_+^n)]_\theta} $$

which is a direct consequence of the definition of restriction space, the equivalence of the complex interpolation norm (2.4) from Theorem 2.6, the definition of the complex interpolation norm, and then of the boundedness of E from $L^{p_0}(\mathbb{R}^n)$ to $L^{p_0}(\mathbb{R}_+^n)$ and from $\dot{H}^{1,p_1}(\mathbb{R}^n)$ to $\dot{H}^{1,p_1}(\mathbb{R}_+^n)$. Now, if
$s \in \mathbb{R}$, $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$, $m \in \mathbb{N}$,

$$ \|\nabla^m u\|_{\dot{H}^{s-m,p}(\mathbb{R}_+^n)} \lesssim_{p,s,n,m} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. \quad (3.8) $$

The estimate above is a direct consequence of definition of function spaces by restriction and can be turned into an equivalence under some additional assumptions.

**Proposition 3.2** Let $p \in (1, +\infty)$, $k \in \mathbb{N}$, $k > 1$, $s > k - 1 + \frac{1}{p}$, for all $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$,

$$ \sum_{j=1}^{n} \| \partial_{x_j}^k u \|_{\dot{H}^{s-k,p}(\mathbb{R}_+^n)} \sim_{s,k,p,n} \| \nabla^k u \|_{\dot{H}^{s-k,p}(\mathbb{R}_+^n)} \sim_{s,k,p,n} \| u \|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

In particular, $\|\nabla^k \cdot \dot{H}^{s-k,p}(\mathbb{R}_+^n) \|$ and $\sum_{j=1}^n \|\partial_{x_j}^k \cdot \dot{H}^{s-k,p}(\mathbb{R}_+^n) \|$ provide equivalent norms on $\dot{H}^{s,p}(\mathbb{R}_+^n)$, whenever $(C_{s-k,p})$ is satisfied.

*Proof*. — Let us prove it for $k = 1$, the higher order case can be achieved in a similar manner. Consider $p \in (1, +\infty)$, $s > \frac{1}{p}$, for $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$, we have $Eu \in \dot{H}^{s,p}(\mathbb{R}^n)$, where E is an extension operator provided by Proposition 3.1 (for some big enough $m \ge 1$), $\nabla Eu \in \dot{H}^{s-1,p}(\mathbb{R}^n)$, with $s-1 > -1 + \frac{1}{p}$. We can write on $\mathbb{R}_+^n$

$$ \partial_{x_\ell} Eu = E[\partial_{x_\ell} u], \text{ provided } \ell \in [1, n-1], \text{ and } \partial_{x_n} Eu = \sum_{j=0}^{m} \alpha_j \left(\frac{-1}{j+1}\right) \partial_{x_n} u(x', -\frac{x_n}{j+1}). $$

Hence, we can use definition of restriction space, apply point (iii) in Proposition 2.2, and the boundedness of E, since m is large enough, to obtain,

$$ \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim \|Eu\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|\nabla Eu\|_{\dot{H}^{s-1,p}(\mathbb{R}^n)} \lesssim_{s,p,n,m} \|\nabla u\|_{\dot{H}^{s-1,p}(\mathbb{R}_+^n)}. $$

Therefore by (3.8), the equivalence of norms on $\dot{H}^{s,p}(\mathbb{R}_+^n)$ holds by density when $(C_{s-k,p})$ is true. ■

The next proposition is about identifying intersection of homogeneous Sobolev spaces on $\mathbb{R}_+^n$, and give a dense subspace. As we can see later this will help for real interpolation.

**Proposition 3.3** Let $p_j \in (1, +\infty)$, $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, if $(C_{s_0, p_0})$ is satisfied then the following equality of vector spaces holds with equivalence of norms

$$ \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n) = [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n). $$

In particular, $\dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n)$ is a Banach space which admits $S_0(\mathbb{R}_+^n)$ as a dense subspace.

*Proof*. — Let $p \in (1, +\infty)$, $s_0, s_1 \in \mathbb{R}$, such that $(C_{s_0, p_0})$. By definition of restriction spaces and Lemma 2.5, $[\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n)$ is complete and admits $S_0(\mathbb{R}_+^n)$ as a dense subspace. The following continuous embedding also holds by definition,

$$ [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n) \hookrightarrow \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n). $$

Hence, it suffices to prove the reverse one. To do so, let us choose $\ell \in \mathbb{N}$ such that $(C_{s_1-\ell, p_1})$ is satisfied, and $s_1 - \ell > -1 + \frac{1}{p_1}$, then choosing E from Proposition 3.1 with $m + 1 + \frac{1}{p_j} > s_j$, $j \in \{0, 1\}$ ($m$ big enough), for all $j \in [1, n]$, and all $u \in \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n)$, Eu makes sense in $\dot{H}^{s_0, p_0}(\mathbb{R}^n)$ then in $S'_h(\mathbb{R}^n)$ and one may use an estimate similar to (3.7), to deduce

$$ \sum_{k=1}^{n} \| \partial_{x_k}^\ell Eu \|_{\dot{H}^{s_1-\ell, p_1}(\mathbb{R}^n)} = \sum_{k=1}^{n-1} \| E \partial_{x_k}^\ell u \|_{\dot{H}^{s_1-\ell, p_1}(\mathbb{R}^n)} + \| E^{(\ell)} \partial_{x_n}^\ell u \|_{\dot{H}^{s_1-\ell, p_1}(\mathbb{R}^n)} \lesssim_{s_1, m, \ell} \| u \|_{\dot{H}^{s_1, p_1}(\mathbb{R}_+^n)}. $$

The above operator $E^{(\ell)}$ is given via the identity $\partial_{x_n}^\ell E = E^{(\ell)} \partial_{x_n}^\ell$. Hence, it follows that for all $u \in \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n)$,

$$ \|Eu\|_{\dot{H}^{s_0, p_0}(\mathbb{R}^n)} + \sum_{k=1}^{n} \|\partial_{x_k}^\ell Eu\|_{\dot{H}^{s_1-\ell, p_1}(\mathbb{R}^n)} \lesssim_{s_0, s_1, m, \ell} \|u\|_{\dot{H}^{s_0, p_0}(\mathbb{R}_+^n)} + \|u\|_{\dot{H}^{s_1, p_1}(\mathbb{R}_+^n)}. $$
$u \in \dot{H}^{\theta,p}(\mathbb{R}_+^n)$, by definition of restriction spaces there exists $U \in \dot{H}^{\theta,p}(\mathbb{R}^n)$, such that

$$U_{|\mathbb{R}_+^n} = u, \quad \text{and} \quad \frac{1}{2} \|U\|_{\dot{H}^{\theta,p}(\mathbb{R}^n)} \le \|u\|_{\dot{H}^{\theta,p}(\mathbb{R}_+^n)} \le \|U\|_{\dot{H}^{\theta,p}(\mathbb{R}^n)}.$$

By Theorem 2.6, there exists $f \in F(L^{p_0}(\mathbb{R}^n), \dot{H}^{1,p_1}(\mathbb{R}^n))$ such that $f(\theta) = U$, we deduce $f(\cdot)|_{\mathbb{R}_+^n} \in F(L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1,p_1}(\mathbb{R}_+^n))$, so $u = f(\theta)|_{\mathbb{R}_+^n} \in [L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1,p_1}(\mathbb{R}_+^n)]_\theta$ with the following estimate which is a direct consequence of the definition of function spaces by restriction, and complex interpolation spaces,

$$\|u\|_{[L^{p_0}(\mathbb{R}_+^n), \dot{H}^{1,p_1}(\mathbb{R}_+^n)]_\theta} \lesssim \|u\|_{\dot{H}^{\theta,p}(\mathbb{R}_+^n)}.$$

Hence, homogeneous (Riesz potential) Sobolev spaces on the half-space are still a complex interpolation scale provided that $p \in (1, +\infty)$, $s \in [0, 1]$, ($C_{s,p}$) being satisfied, so the boundedness of $E : \dot{H}^{\theta,p}(\mathbb{R}_+^n) \to \dot{H}^{\theta,p}(\mathbb{R}^n)$ follows by interpolation.

In particular, $E : \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n)$ is bounded for all $s \in (-1 + \frac{1}{p}, \frac{1}{p}]$. Hence, the result has been proved for $s - \frac{1}{p} = 0$. The same result is obtained for $E^{(\ell)}$, provided $\ell \in [1, m]$.

Now, let $p \in (1, +\infty)$, $s - \frac{1}{p} \in [1, m]$, for $u \in H^{s,p}(\mathbb{R}_+^n)$, we have $Eu \in H^{s,p}(\mathbb{R}^n)$, $\nabla^\ell Eu \in \dot{H}^{s-\ell,p}(\mathbb{R}^n)$, $s - \ell = \frac{1}{p}$, so that, similarly as in (3.7),

$$\|Eu\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n,\ell} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}.$$

Therefore, we have obtained the desired estimate and can conclude about the boundedness of $E$ via density argument whenever ($C_{s,p}$) is satisfied. ■

In the proof of Proposition 3.1, we used boundedness of derivatives, i.e., for all $p \in (1, +\infty)$, $s \in \mathbb{R}$, $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$, $m \in \mathbb{N}$,

$$\|\nabla^m u\|_{\dot{H}^{s-m,p}(\mathbb{R}_+^n)} \lesssim_{p,s,n,m} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. \qquad (3.8)$$

The estimate above is a direct consequence of definition of function spaces by restriction and can be turned into an equivalence under some additional assumptions.

**Proposition 3.2** Let $p \in (1, +\infty)$, $k \in [1, +\infty[$, $s > k - 1 + \frac{1}{p}$, for all $u \in H^{s,p}(\mathbb{R}_+^n)$,

$$\sum_{j=1}^{n} \| \partial_{x_j}^k u \|_{\dot{H}^{s-k,p}(\mathbb{R}_+^n)} \sim_{s,k,p,n} \| \nabla^k u \|_{\dot{H}^{s-k,p}(\mathbb{R}_+^n)} \sim_{s,k,p,n} \| u \|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}.$$

In particular, $\|\nabla^k\|_{\dot{H}^{s-k,p}(\mathbb{R}_+^n)}$ and $\sum_{j=1}^n \|\partial_{x_j}^k\|_{\dot{H}^{s-k,p}(\mathbb{R}_+^n)}$ provide equivalent norms on $\dot{H}^{s,p}(\mathbb{R}_+^n)$, whenever ($C_{s-k,p}$) is satisfied.

*Proof.* — Let us prove it for $k = 1$, the higher order case can be achieved similarly. Consider $p \in (1, +\infty)$, $s > \frac{1}{p}$, for $u \in H^{s,p}(\mathbb{R}_+^n)$, we have $Eu \in \dot{H}^{s,p}(\mathbb{R}^n)$, where $E$ is an extension operator provided by Proposition 3.1 (for some big enough $m \ge 1$), $\nabla Eu \in \dot{H}^{s-1,p}(\mathbb{R}^n)$, with $s - 1 > -1 + \frac{1}{p}$. We can write on $\mathbb{R}_+^n$

$$\partial_{x_\ell} Eu = E[\partial_{x_\ell} u], \text{ provided } \ell \in [1, n-1], \text{ and } \partial_{x_n} Eu = \sum_{j=0}^{m} \alpha_j \left(\frac{-1}{j+1}\right) \partial_{x_n} u(x', -\frac{x_n}{j+1}).$$

Hence, we can use definition of restriction space, apply point (iii) in Proposition 2.2, and the boundedness of $E$, since $m$ is large enough, to obtain,

$$\|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \le \|Eu\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|\nabla Eu\|_{\dot{H}^{s-1,p}(\mathbb{R}^n)} \lesssim_{s,p,n,m} \|\nabla u\|_{\dot{H}^{s-1,p}(\mathbb{R}_+^n)}.$$

Therefore by (3.8), the equivalence of norms on $\dot{H}^{s,p}(\mathbb{R}_+^n)$ holds by density when ($C_{s-k,p}$) is true. ■

The next proposition is about identifying intersection of homogeneous Sobolev spaces on $\mathbb{R}_+$, and give a dense subspace. As we can see later, this will help for real interpolation.
In particular, since $Eu \in S'_h(\mathbb{R}^n)$, and by uniqueness of representation of $\partial_{x_j}^\ell Eu$ in $S'(\mathbb{R}^n)$, we deduce from point (iii) in Proposition 2.2 that $Eu \in \dot{H}^{s_0,p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1,p_1}(\mathbb{R}^n)$.

Thus $u \in [\dot{H}^{s_0,p_0} \cap \dot{H}^{s_1,p_1}](\mathbb{R}_+^n)$, and by definition of restriction spaces,

$$ \|u\|_{[\dot{H}^{s_0,p_0} \cap \dot{H}^{s_1,p_1}](\mathbb{R}_+^n)} \le \|Eu\|_{\dot{H}^{s_0,p_0}(\mathbb{R}^n)} + \|Eu\|_{\dot{H}^{s_1,p_1}(\mathbb{R}^n)} \lesssim_{s_0,s_1,m,\ell} \|u\|_{\dot{H}^{s_0,p_0}(\mathbb{R}_+^n)} + \|u\|_{\dot{H}^{s_1,p_1}(\mathbb{R}_+^n)}. $$

This proves the claim.

<figure><img src="image_6.png" /></figure>

So one can deduce the following corollary which allows separate homogeneous estimates for intersection of homogeneous Sobolev spaces on $\mathbb{R}_+^n$. Since the estimates below are decoupled, it provides an ersatz of extension-restriction operators for homogeneous Sobolev spaces of higher order, thanks to the taken intersection yielding a complete space. For instance, this will be of use to circumvent the lack of completeness when we will want to (real-)interpolate between a "higher" order homogeneous Sobolev space, and one that is known to be complete.

**Corollary 3.4** Let $p_j \in (1, +\infty)$, $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, such that $(C_{s_0,p_0})$ is satisfied, consider $m \in \mathbb{N}$ such that $s_j < m + 1 + \frac{1}{p_j}$, and the extension operator E given by Proposition 3.1.

Then for all $u \in \dot{H}^{s_0,p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1,p_1}(\mathbb{R}_+^n)$, we have $Eu \in \dot{H}^{s_j,p_j}(\mathbb{R}^n)$, $j \in \{0, 1\}$, with the estimate

$$ \|Eu\|_{\dot{H}^{s_j,p_j}(\mathbb{R}^n)} \lesssim_{s_j,p_j,m,n} \|u\|_{\dot{H}^{s_j,p_j}(\mathbb{R}_+^n)}. $$

Corollary 3.4 and the proof of Proposition 3.2 lead to

**Corollary 3.5** Let $p_j \in (1, +\infty)$, $m_j \in [1, +\infty[$, $s_1 > m_1 - 1 + \frac{1}{p_1}$, $j \in \{0, 1\}$, such that $(C_{s_0,p_0})$ is satisfied. Then for all $u \in \dot{H}^{s_0,p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1,p_1}(\mathbb{R}_+^n)$,

$$ \sum_{k=1}^{n} \| \partial_{x_k}^{m_1} u \|_{\dot{H}^{s_1-m_1,p_1}(\mathbb{R}_+^n)} \sim_{s_1,m_1,p_1,n} \| \nabla^{m_1} u \|_{\dot{H}^{s_1-m_1,p_1}(\mathbb{R}_+^n)} \sim_{s_1,m_1,p_1,n} \| u \|_{\dot{H}^{s_1,p_1}(\mathbb{R}_+^n)}. $$

Since one may also be interested into Sobolev spaces with 0-boundary condition, we introduce a projection operator that allows to deal with the interpolation property, and to recover, later on, some appropriate density results.

**Lemma 3.6** Let $p \in (1, +\infty)$, $s \in \mathbb{R}$, $m \in \mathbb{N}$, such that $-1 + \frac{1}{p} < s < m + 1 + \frac{1}{p}$, then there exists a bounded projection $\mathcal{P}_0$, depending on $m$, such that it maps $\dot{H}^{s,p}(\mathbb{R}^n)$ to $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$.

If either

* $s \ge 0$ and $u \in \dot{H}^{s,p}(\mathbb{R}^n)$;

* $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ and $u \in \dot{H}^{s,p}(\mathbb{R}^n)$;

we have the estimate

$$ \|\mathcal{P}_0 u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,m,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

In particular, $\mathcal{P}_0$ extends uniquely to a bounded projection from $\dot{H}^{s,p}(\mathbb{R}^n)$ to $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$ whenever $(C_{s,p})$ is satisfied.

*Proof.* — Let $p \in (1, +\infty)$, $s > -1 + \frac{1}{p}$, $m \in \mathbb{N}$, such that $s < m + 1 + \frac{1}{p}$. Then we consider the operator E given by Proposition 3.1, but we modify it into an operator $E^-$, for any measurable function $u : \mathbb{R}^n \to \mathbb{C}$, we set for almost every $x \in \mathbb{R}^n$

$$ E^{-}u(x) := \begin{cases} u(x) & , \text{ if } x \in \mathbb{R}^{n}_{-}, \\ \sum_{j=0}^{m} \alpha_{j} u(x', -\frac{x_{n}}{j+1}) & , \text{ if } x \in \mathbb{R}^{n} \setminus \mathbb{R}^{n}_{-}. \end{cases} $$

Hence for any measurable function $u : \mathbb{R}^n \to \mathbb{C}$, we set for almost every $x \in \mathbb{R}^n$,

$$ \mathcal{P}_0 u := u - E^{-}[1_{\mathbb{R}^n} u]. $$
**Proposition 3.3** Let $p_j \in (1, +\infty)$, $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, if $(C_{s_0, p_0})$ is satisfied then the following equality of vector spaces holds with equivalence of norms

$$ \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n) = [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n). $$

In particular, $\dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n)$ is a Banach space which admits $S_0(\mathbb{R}_+^n)$ as a dense subspace.

*Proof.* — Let $p \in (1, +\infty)$, $s_0, s_1 \in \mathbb{R}$, such that $(C_{s_0, p_0})$. By definition of restriction spaces and Lemma 2.5, $[\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n)$ is complete and admits $S_0(\mathbb{R}_+^n)$ as a dense subspace. The following continuous embedding also holds by definition,

$$ [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n) \hookrightarrow \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n). $$

Hence, it suffices to prove the reverse one. To do so, let us choose $\ell \in \mathbb{N}$ such that $(C_{s_1-\ell, p_1})$ is satisfied, and $s_1 - \ell > -1 + \frac{1}{p_1}$, then choosing E from Proposition 3.1 with $m + 1 + \frac{1}{p_j} > s_j$, $j \in \{0, 1\}$ ($m$ big enough), for all $j \in [1, n]$, and all $u \in \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n)$, Eu makes sense in $\dot{H}^{s_0, p_0}(\mathbb{R}^n)$, then in $S'_h(\mathbb{R}^n)$, and one may use an estimate similar to (3.7), to deduce

$$ \sum_{k=1}^{n} \| \partial_{x_k}^{\ell} \mathrm{E} u \|_{\dot{H}^{s_1-\ell, p_1}(\mathbb{R}^n)} = \sum_{k=1}^{n-1} \| \mathrm{E} \partial_{x_k}^{\ell} u \|_{\dot{H}^{s_1-\ell, p_1}(\mathbb{R}^n)} + \| \mathrm{E}^{(\ell)} \partial_{x_n}^{\ell} u \|_{\dot{H}^{s_1-\ell, p_1}(\mathbb{R}^n)} \lesssim_{s_1, m, \ell} p_1 \cdot n \| u \|_{\dot{H}^{s_1, p_1}(\mathbb{R}_+^n)}. $$

The above operator $E^{(\ell)}$ is given via the identity $\partial_{x_n}^{\ell} E = E^{(\ell)} \partial_{x_n}^{\ell}$. Hence, it follows that for all $u \in \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n)$,

$$ \|Eu\|_{\dot{H}^{s_0, p_0}(\mathbb{R}^n)} + \sum_{k=1}^{n} \|\partial_{x_k}^{\ell} Eu\|_{\dot{H}^{s_1-\ell, p_1}(\mathbb{R}^n)} \lesssim_{s_0, s_1, m, \ell} p_0 \cdot p_1 \cdot n \|u\|_{\dot{H}^{s_0, p_0}(\mathbb{R}_+^n)} + \|u\|_{\dot{H}^{s_1, p_1}(\mathbb{R}_+^n)}. $$

In particular, since $Eu \in S'_h(\mathbb{R}^n)$, and by uniqueness of representation of $\partial_{x_j}^{\ell} Eu$ in $S'(\mathbb{R}^n)$, we deduce from point (iii) in Proposition 2.2 that $Eu \in \dot{H}^{s_0, p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}^n)$.

Thus $u \in [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n)$, and by definition of restriction spaces,

$$ \|u\|_{[\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n)} \lesssim \|Eu\|_{\dot{H}^{s_0, p_0}(\mathbb{R}^n)} + \|Eu\|_{\dot{H}^{s_1, p_1}(\mathbb{R}^n)} \lesssim_{s_0, s_1, m, \ell} p_0 \cdot p_1 \cdot n \|u\|_{\dot{H}^{s_0, p_0}(\mathbb{R}_+^n)} + \|u\|_{\dot{H}^{s_1, p_1}(\mathbb{R}_+^n)}. $$

This proves the claim.

<figure><img src="image_6.png" /></figure>

So one can deduce the following corollary, which allows separate homogeneous estimates for intersection of homogeneous Sobolev spaces on $\mathbb{R}_+^n$. Since the estimates below are decoupled, it provides an ersatz of extension-restriction operators for homogeneous Sobolev spaces of higher order, thanks to the taken intersection yielding a complete space. For instance, this will be of use to circumvent the lack of completeness when we will want to (real-)interpolate between a “higher” order homogeneous Sobolev space, and one that is known to be complete.

**Corollary 3.4** Let $p_j \in (1, +\infty)$, $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0})$ is satisfied, consider $m \in \mathbb{N}$ such that $s_j < m + 1 + \frac{1}{p_j}$, and the extension operator E given by Proposition 3.1.

Then for all $u \in \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n)$, we have $Eu \in \dot{H}^{s_j, p_j}(\mathbb{R}^n)$, $j \in \{0, 1\}$, with the estimate

$$ \|Eu\|_{\dot{H}^{s_j, p_j}(\mathbb{R}^n)} \lesssim_{s_j, p_j, m, n} p_0 \cdot p_1 \cdot n \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}_+^n)}. $$

Corollary 3.4 and the proof of Proposition 3.2 lead to

**Corollary 3.5** Let $p_j \in (1, +\infty)$, $m_j \in [1, +\infty[$, $s_1 > m_1 - 1 + \frac{1}{p_1}$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0})$ is satisfied. Then for all $u \in \dot{H}^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}_+^n)$,

$$ \sum_{k=1}^{n} \| \partial_{x_k}^{m_1} u \|_{\dot{H}^{s_1-m_1, p_1}(\mathbb{R}_+^n)} \sim_{s_1, m_1, p_1, n} \| \nabla^{m_1} u \|_{\dot{H}^{s_1-m_1, p_1}(\mathbb{R}_+^n)} \sim_{s_1, m_1, p_1, n} \| u \|_{\dot{H}^{s_1, p_1}(\mathbb{R}_+^n)}. $$

Now, one may also be interested into Sobolev spaces with 0-boundary condition, we introduce a projection operator that allows to deal with the interpolation property, and to recover, later on, some appropriate density results.
The fact that $\mathcal{P}_0^2 = \mathcal{P}_0$ is clear by definition, and we have $\mathcal{P}_0 H^{s,p}(\mathbb{R}^n) \subset \mathcal{H}_0^{s,p}(\mathbb{R}_+^n)$, and that $\mathcal{P}_{0|_{\mathcal{H}_0^{s,p}(\mathbb{R}_+^n)}} = \mathbf{I}$. The boundedness properties, as claimed, follow from Proposition 2.9 and Proposition 3.1. ■

As well as the extension operator given by higher order reflection principle, the projection operator on "0-boundary condition" homogeneous Sobolev spaces satisfies homogeneous estimates on intersection spaces. The proof is a direct consequence of Proposition 3.3 and its formula introduced in the proof of Lemma 3.6.

**Corollary 3.7** Let $p_j \in (1, +\infty)$, $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, $m \in \mathbb{N}$, such that $(C_{s_0, p_0})$ is satisfied and $s_j < m + 1 + \frac{1}{p_j}$, and consider the projection operator $\mathcal{P}_0$ given by Lemma 3.6.

Then for all $u \in \dot{H}^{s_0, p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}^n)$, we have $\mathcal{P}_0 u \in \dot{H}_0^{s_j, p_j}(\mathbb{R}_+^n)$, $j \in \{0, 1\}$, with the estimate

$$ \|\mathcal{P}_0 u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}^n)} \lesssim_{s_j, m, p, n} \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}^n)}. $$

We still have Sobolev embeddings by definition of function spaces by restriction.

**Proposition 3.8** Let $p, q \in (1, +\infty)$, $s \in [0, n)$, such that

$$ \frac{1}{q} = \frac{1}{p} - \frac{s}{n}. $$

We have the estimates,

$$ \|u\|_{L^q(\mathbb{R}_+^n)} \lesssim_{n, s, p, q} \|u\|_{\dot{H}^{s, p}(\mathbb{R}_+^n)}, \quad \forall u \in \dot{H}^{s, p}(\mathbb{R}_+^n), \qquad (3.9) $$

$$ \|u\|_{\dot{H}_0^{-s, q}(\mathbb{R}_+^n)} \lesssim_{n, s, p, q} \|u\|_{L^p(\mathbb{R}_+^n)}, \quad \forall u \in L^p(\mathbb{R}_+^n), \qquad (3.10) $$

for which each underlying embedding is dense.

*Proof.* — First, let us recall the Hardy-Littlewood-Sobolev inequality from (i) in Proposition 2.2, which says that

$$ \|u\|_{\dot{H}^{-s, q}(\mathbb{R}^n)} \lesssim_{n, s, p, q} \|u\|_{L^p(\mathbb{R}^n)}, \quad \forall u \in L^p(\mathbb{R}^n). $$

Hence, the embedding (3.10) is a direct consequence of plugging $\tilde{u}$ the extension to the whole $\mathbb{R}^n$ of $u \in L^p(\mathbb{R}_+^n)$.

The embedding (3.9) is a direct consequence of (i) in Proposition 2.2 and function spaces defined by restriction. Indeed, for $u \in \dot{H}^{s, p}(\mathbb{R}_+^n)$, we have for any extension $U \in \dot{H}^{s, p}(\mathbb{R}^n) \subset L^q(\mathbb{R}^n)$ such that $u = U|_{\mathbb{R}_+^n} \in L^q(\mathbb{R}_+^n)$ the estimate

$$ \|u\|_{L^q(\mathbb{R}_+^n)} \lesssim \|U\|_{L^q(\mathbb{R}^n)} \lesssim_{s, p, q, n} \|U\|_{\dot{H}^{s, p}(\mathbb{R}^n)}. $$

Looking at the infimum on all such $U$ gives the result.

The density for the first embedding follows from the fact that $S_0(\mathbb{R}_+^n) \subset \dot{H}^{s, p}(\mathbb{R}_+^n)$ is dense in $L^q(\mathbb{R}_+^n)$. The density in the second case, follows from the canonical embedding,

$$ L^p(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{-s, q}(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{-s, q}(\mathbb{R}_+^n), $$

which turn, by duality into embeddings,

$$ H^{s, q'}(\mathbb{R}_+^n) \hookrightarrow (\dot{H}_0^{-s, q}(\mathbb{R}_+^n))' \hookrightarrow L^{p'}(\mathbb{R}_+^n). $$

In particular, the following is a dense embedding

$$ (\dot{H}_0^{-s, q}(\mathbb{R}_+^n))' \hookrightarrow L^{p'}(\mathbb{R}_+^n) $$

hence by reflexivity, the one below also is

$$ L^p(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{-s, q}(\mathbb{R}_+^n). $$

Now, all the ingredients are there in order to build the main usual density result for our 0-boundary conditions homogeneous Sobolev spaces. ■
**Lemma 3.6** Let $p \in (1, +\infty)$, $s \in \mathbb{R}$, $m \in \mathbb{N}$, such that $-1 + \frac{1}{p} < s < m + 1 + \frac{1}{p}$, then there exists a bounded projection operator $\mathcal{P}_0$, depending on $m$, such that it maps $H^{s,p}(\mathbb{R}^n)$ to $H_0^{s,p}(\mathbb{R}_+^n)$, which also satisfies that:

If either one of the following conditions holds

* $s \ge 0$ and $u \in H^{s,p}(\mathbb{R}^n)$;

* $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ and $u \in \dot{H}^{s,p}(\mathbb{R}^n)$;

then we have the estimate

$$ \|\mathcal{P}_0 u\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,m,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

In particular, $\mathcal{P}_0$ extends uniquely to a bounded projection from $\dot{H}^{s,p}(\mathbb{R}^n)$ to $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$ whenever $(C_{s,p})$ is satisfied.

*Proof.* — Let $p \in (1, +\infty)$, $s > -1 + \frac{1}{p}$, $m \in \mathbb{N}$, such that $s < m + 1 + \frac{1}{p}$. Then we consider the operator $E$ given by Proposition 3.1, but we modify it into an operator $E^-$, for any measurable function $u : \mathbb{R}^n \to \mathbb{C}$, we set for almost every $x \in \mathbb{R}^n$

$$ E^{-}u(x) := \begin{cases} u(x) & , \text{ if } x \in \mathbb{R}_{-}^{n}, \\ \sum_{j=0}^{m} \alpha_{j} u(x', -\frac{x_{n}}{j+1}) & , \text{ if } x \in \mathbb{R}^{n} \setminus \mathbb{R}_{-}^{n}. \end{cases} $$

Hence, for any measurable function $u : \mathbb{R}^n \to \mathbb{C}$, we set for almost every $x \in \mathbb{R}^n$,

$$ \mathcal{P}_0 u := u - E^{-}[1_{\mathbb{R}^n} u]. $$

The fact that $\mathcal{P}_0^2 = \mathcal{P}_0$ is clear by definition, and we have $\mathcal{P}_0 H^{s,p}(\mathbb{R}^n) \subset H_0^{s,p}(\mathbb{R}_+^n)$, and that $\mathcal{P}_{0|_{H_0^{s,p}(\mathbb{R}_+^n)}} = \mathbf{I}$. The boundedness properties, as claimed, follow from Proposition 2.9 and Proposition 3.1. ■

As well as the extension operator given by higher order reflection principle, the projection operator on homogeneous Sobolev spaces including the “0-boundary condition” satisfies homogeneous estimates on intersection spaces. The proof is a direct consequence of Proposition 3.3 and its formula introduced in the proof of Lemma 3.6.

**Corollary 3.7** Let $p_j \in (1, +\infty)$, $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, $m \in \mathbb{N}$, such that $(C_{s_0, p_0})$ is satisfied and $s_j < m + 1 + \frac{1}{p_j}$, and consider the projection operator $\mathcal{P}_0$ given by Lemma 3.6.

Then for all $u \in \dot{H}^{s_0, p_0}(\mathbb{R}^n) \cap \dot{H}^{s_1, p_1}(\mathbb{R}^n)$, we have $\mathcal{P}_0 u \in \dot{H}_0^{s_j, p_j}(\mathbb{R}_+^n)$, $j \in \{0, 1\}$, with the estimate

$$ \|\mathcal{P}_0 u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}^n)} \lesssim_{s_j, m, p, n} \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}^n)}. $$

We still have Sobolev embeddings by definition of function spaces by restriction.

**Proposition 3.8** Let $p, q \in (1, +\infty)$, $s \in [0, n)$, such that

$$ \frac{1}{q} = \frac{1}{p} - \frac{s}{n}. $$

We have the estimates,

$$ \|u\|_{L^q(\mathbb{R}_+^n)} \lesssim_{n, s, p, q} \|u\|_{\dot{H}^{s, p}(\mathbb{R}_+^n)}, \quad \forall u \in \dot{H}^{s, p}(\mathbb{R}_+^n), \qquad (3.9) $$

$$ \|u\|_{\dot{H}_0^{-s, q}(\mathbb{R}_+^n)} \lesssim_{n, s, p, q} \|u\|_{L^p(\mathbb{R}_+^n)}, \quad \forall u \in L^p(\mathbb{R}_+^n), \qquad (3.10) $$

for which each underlying embedding is dense.

*Proof.* — First, let us recall the Hardy-Littlewood-Sobolev inequality from (i) in Proposition 2.2, which says that

$$ \|u\|_{\dot{H}^{-s, q}(\mathbb{R}^n)} \lesssim_{n, s, p, q} \|u\|_{L^p(\mathbb{R}^n)}, \quad \forall u \in L^p(\mathbb{R}^n). $$
**Proposition 3.9** The space $C_c^\infty(\mathbb{R}_+^n)$ is dense in the spaces

(i) $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$, when $p \in (1, +\infty)$, $s \in (-\frac{n}{p'}, \frac{n}{p})$,

(ii) $\dot{H}_0^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1, p_1}(\mathbb{R}_+^n)$, when $p_j \in (1, +\infty)$, $s_j \ge 0$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0})$ is satisfied.

*Proof.* — **Step 1:** Point (i) with $s \in [0, n)$. Let $p \in (1, +\infty)$, and assume that $(C_{s,p})$ is true, and consider $u \in \dot{H}_0^{s,p}(\mathbb{R}_+^n)$.

In particular, we have $u \in \dot{H}^{s,p}(\mathbb{R}^n)$. Hence, there exists $(u_k)_{k \in \mathbb{N}} \subset \dot{H}^{s,p}(\mathbb{R}^n)$ such that

$$u_k \xrightarrow{k \to +\infty} u \text{ in } \dot{H}^{s,p}(\mathbb{R}^n).$$

Thus, it follows from Lemma 3.6, that $(P_0 u_k)_{k \in \mathbb{N}} \subset \dot{H}_0^{s,p}(\mathbb{R}_+^n) \subset \dot{H}_0^{s,p}(\mathbb{R}_+^n)$ converge to $P_0 u = u$ in $\dot{H}^{s,p}(\mathbb{R}^n)$. For $\varepsilon > 0$, there exists some $k_0$, such that for all $k \ge k_0$, we have

$$\|u - P_0 u_k\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} < \varepsilon.$$

Now, we use density of $C_c^\infty(\mathbb{R}_+^n)$ in $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$ to assert that there exists $w \in C_c^\infty(\mathbb{R}_+^n)$ so that,

$$\|P_0 u_k - w\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} \le \|P_0 u_k - w\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} < \varepsilon.$$

This proves the density of $C_c^\infty(\mathbb{R}_+^n)$ in the space $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$, since

$$\|u - w\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} \le \|u - P_0 u_k\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} + \|P_0 u_k - w\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} < 2\varepsilon.$$

**Step 2:** Now for the second part of (i), let us consider $s \in (0, \frac{n}{p'})$. For $u \in \dot{H}_0^{-s,p}(\mathbb{R}_+^n)$, applying Proposition 3.8, for $\varepsilon > 0$ there exists a function $v \in L^q(\mathbb{R}_+^n)$, (with $\frac{1}{p} = \frac{1}{q} - \frac{s}{n}$) such that,

$$\|u - v\|_{\dot{H}_0^{-s,p}(\mathbb{R}_+^n)} < \varepsilon.$$

But recalling that $C_c^\infty(\mathbb{R}_+^n)$ is dense in $L^q(\mathbb{R}_+^n)$, there exists $w \in C_c^\infty(\mathbb{R}_+^n)$ such that

$$\|v - w\|_{\dot{H}_0^{-s,p}(\mathbb{R}_+^n)} \lesssim_{n,s,p,q} \|v - w\|_{L^q(\mathbb{R}_+^n)} \lesssim_{n,s,p,q} \varepsilon,$$

so the triangle inequality gives

$$\|u - w\|_{\dot{H}_0^{-s,p}(\mathbb{R}_+^n)} \lesssim_{n,s,p,q} \varepsilon,$$

which conclude the proof since $w \in C_c^\infty(\mathbb{R}_+^n)$.

**Step 3:** For the density in the intersection spaces, it suffices to reproduce the above **Step 1** by means of Corollary 3.7. ■

**Corollary 3.10** For all $p \in (1, +\infty)$, $s \in (-1 + \frac{1}{p'}, \frac{1}{p})$,

$$\dot{H}_0^{s,p}(\mathbb{R}_+^n) = \dot{H}^{s,p}(\mathbb{R}_+^n).$$

In particular, $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{H}^{s,p}(\mathbb{R}_+^n)$ for same range of indices.

*Proof.* — This is a direct consequence of the definition of restriction spaces and Proposition 2.9, the density result follows from Proposition 3.9. ■

**Proposition 3.11** Let $p \in (1, +\infty)$, $s \in (-\frac{n}{p'}, \frac{n}{p})$, we have

$$(\dot{H}^{s,p}(\mathbb{R}_+^n))' = \dot{H}_0^{-s,p'}(\mathbb{R}_+^n) \text{ and } (\dot{H}_0^{s,p}(\mathbb{R}_+^n))' = \dot{H}^{-s,p'}(\mathbb{R}_+^n).$$

*Proof.* — First, consider $s \in (-\frac{n}{p'}, \frac{n}{p})$, let $\Phi \in \dot{H}_0^{-s, p'}(\mathbb{R}_+^n) \subset \dot{H}^{-s, p'}(\mathbb{R}^n)$, then using definition of restriction spaces, the following map defines a linear functional on $\dot{H}^{s, p}(\mathbb{R}_+^n)$,

$$u \mapsto \langle \Phi, \tilde{u} \rangle_{\mathbb{R}^n},$$

where $\tilde{u}$ is any extension of $u$, and notice that the action of $\Phi$ does not depend on the choice of such extension of $u$. Indeed, if $U \in \dot{H}^{s,p}(\mathbb{R}^n)$ is another extension of $u$, we obtain that $w := U - \tilde{u} \in$
Hence, the embedding (3.10) is a direct consequence of plugging $\tilde{u}$ the extension to the whole $\mathbb{R}^n$ of $u \in L^p(\mathbb{R}_+^n)$.

The embedding (3.9) is a direct consequence of (i) in Proposition 2.2 and function spaces defined by restriction. Indeed, for $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$, we have for any extension $U \in \dot{H}^{s,p}(\mathbb{R}^n) \subset L^q(\mathbb{R}^n)$ such that $u = U|_{\mathbb{R}_+^n} \in L^q(\mathbb{R}_+^n)$ the estimate

$$ \|u\|_{L^q(\mathbb{R}_+^n)} \le \|U\|_{L^q(\mathbb{R}^n)} \lesssim_{s,p,q,n} \|U\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

Looking at the infimum on all such $U$ gives the result.

The density for the first embedding follows from the fact that $S_0(\overline{\mathbb{R}_+^n}) \subset \dot{H}^{s,p}(\mathbb{R}_+^n)$ is dense in $L^q(\mathbb{R}_+^n)$. The density in the second case, follows from the canonical embedding,

$$ L^p(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{-s,q}(\mathbb{R}_+^n) \hookrightarrow H_0^{-s,q}(\mathbb{R}_+^n), $$

which turn, by duality into embeddings,

$$ H^{s,q'}(\mathbb{R}_+^n) \hookrightarrow (\dot{H}_0^{-s,q}(\mathbb{R}_+^n))' \hookrightarrow L^{p'}(\mathbb{R}_+^n). $$

In particular, the following is a dense embedding

$$ (\dot{H}_0^{-s,q}(\mathbb{R}_+^n))' \hookrightarrow L^{p'}(\mathbb{R}_+^n) $$

Hence, by reflexivity, the one below also is

$$ L^p(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{-s,q}(\mathbb{R}_+^n). $$

<figure><img src="image_7.png" /></figure>

Now, all the ingredients are there in order to build the main usual density result for our 0-boundary conditions homogeneous Sobolev spaces.

**Proposition 3.9** The space $C_c^\infty(\mathbb{R}_+^n)$ is dense in the spaces

$$ (i) \quad \dot{H}_0^{s,p}(\mathbb{R}_+^n), \text{ when } p \in (1, +\infty), s \in (-\frac{n}{p'}, \frac{n}{p}), $$

$$ (ii) \quad \dot{H}_0^{s_0, p_0}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1, p_1}(\mathbb{R}_+^n), \text{ when } p_j \in (1, +\infty), s_j \ge 0, j \in \{0, 1\}, \text{ such that } (C_{s_0, p_0}) \text{ is satisfied.} $$

*Proof.* — **Step 1:** Point (i) with $s \ge 0$. Let $p \in (1, +\infty)$, and assume that $(C_{s,p})$ is true, and consider $u \in \dot{H}_0^{s,p}(\mathbb{R}_+^n)$.

In particular, we have $u \in \dot{H}^{s,p}(\mathbb{R}^n)$. Hence, there exists $(u_k)_{k \in \mathbb{N}} \subset H^{s,p}(\mathbb{R}^n)$ such that

$$ u_k \xrightarrow{k \to +\infty} u \text{ in } \dot{H}^{s,p}(\mathbb{R}^n). $$

Thus, it follows from Lemma 3.6, that $(\mathcal{P}_0 u_k)_{k \in \mathbb{N}} \subset H_0^{s,p}(\mathbb{R}_+^n) \subset \dot{H}_0^{s,p}(\mathbb{R}_+^n)$ converge to $\mathcal{P}_0 u = u$ in $\dot{H}^{s,p}(\mathbb{R}^n)$. For $\varepsilon > 0$, there exists some $k_0$, such that for all $k \ge k_0$, we have

$$ \|u - \mathcal{P}_0 u_k\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} < \varepsilon. $$

Now, we use density of $C_c^\infty(\mathbb{R}_+^n)$ in $H_0^{s,p}(\mathbb{R}_+^n)$, [JK95, Remark 2.7], to assert that there exists $w \in C_c^\infty(\mathbb{R}_+^n)$ so that,

$$ \|\mathcal{P}_0 u_k - w\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} \le \|\mathcal{P}_0 u_k - w\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} < \varepsilon. $$

This proves the density of $C_c^\infty(\mathbb{R}_+^n)$ in the space $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$, since

$$ \|u - w\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} \le \|u - \mathcal{P}_0 u_k\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} + \|\mathcal{P}_0 u_k - w\|_{\dot{H}_0^{s,p}(\mathbb{R}_+^n)} < 2\varepsilon. $$

**Step 2:** Now for the second part of (i), let us consider $s \in (0, \frac{n}{p'})$. For $u \in \dot{H}_0^{-s,p}(\mathbb{R}_+^n)$, applying Proposition 3.8, for $\varepsilon > 0$ there exists a function $v \in L^q(\mathbb{R}_+^n)$, (with $\frac{1}{p} = \frac{1}{q} - \frac{s}{n}$) such that,

$$ \|u - v\|_{\dot{H}_0^{-s,p}(\mathbb{R}_+^n)} < \varepsilon. $$

But recalling that $C_c^\infty(\mathbb{R}_+^n)$ is dense in $L^q(\mathbb{R}_+^n)$, there exists $w \in C_c^\infty(\mathbb{R}_+^n)$ such that

$$ \|v - w\|_{\dot{H}_0^{-s,p}(\mathbb{R}_+^n)} \lesssim_{n,s,p,q} \|v - w\|_{L^q(\mathbb{R}_+^n)} \lesssim_{n,s,p,q} \varepsilon, $$
$\dot{H}_0^{s,p}(\overline{\mathbb{R}}_+^n)^c$. It follows from Proposition 3.9 that $w$ is a strong limit in $\dot{H}_0^{s,p}(\overline{\mathbb{R}}_+^n)^c$ of a sequence of functions $(w_k)_{k \in \mathbb{N}} \subset C_c^\infty(\overline{\mathbb{R}}_+^n)$ so that, passing to the limit, in the duality bracket, we obtain

$$ \langle \Phi, U \rangle_{\mathbb{R}^n} - \langle \Phi, \tilde{u} \rangle_{\mathbb{R}^n} = \langle \Phi, w \rangle_{\mathbb{R}^n} = 0. $$

This gives a well defined continuous injective map

$$ \left\{ \begin{array}{lcl} \dot{H}_0^{-s,p'}(\mathbb{R}_+^n) & \to & (\dot{H}^{s,p}(\mathbb{R}_+^n))' \\ \Phi & \mapsto & \langle \Phi, \tilde{\gamma} \rangle_{\mathbb{R}^n} \end{array} \right. . \qquad (3.11) $$

Now, let $\Psi \in (\dot{H}^{s,p}(\mathbb{R}_+^n))'$, for all $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$, since $1_{\mathbb{R}_+^n} u = u$, we may write,

$$ \langle \Psi, u \rangle = \langle \Psi, 1_{\mathbb{R}_+^n} \tilde{u} \rangle, $$

for any extension $\tilde{u} \in \dot{H}^{s,p}(\mathbb{R}^n)$ of $u$, hence as a direct consequence of the definition of restriction space $1_{\mathbb{R}_+^n} \Psi \in (\dot{H}^{s,p'}(\mathbb{R}^n))' = \dot{H}^{-s,p'}(\mathbb{R}^n)$, so $1_{\mathbb{R}_+^n} \Psi \in \dot{H}_0^{-s,p'}(\mathbb{R}_+^n)$. The following map is well defined continuous and injective

$$ \left\{ \begin{array}{lcl} (\dot{H}^{s,p}(\mathbb{R}_+^n))' & \to & \dot{H}_0^{-s,p'}(\mathbb{R}_+^n) \\ \Psi & \mapsto & 1_{\mathbb{R}_+^n} \Psi \end{array} \right. . \qquad (3.12) $$

Both maps (3.11) and (3.12) are even isometric and we obtain,

$$ (\dot{H}^{s,p}(\mathbb{R}_+^n))' = \dot{H}_0^{-s,p'}(\mathbb{R}_+^n), $$

which was the first statement. The second statement follows from duality and convexity exchanging roles of involved exponents. ■

The next result aims to carry over density in intersection spaces to transfer itself as a density result in their real interpolation spaces.

**Corollary 3.12** Let $p \in (1, +\infty)$, $-n/p' < s_0 < s_1 < n/p$, i.e. such that $(C_{-s_0, p'})$ and $(C_{s_1, p})$ are both satisfied.

The space $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{H}_0^{s_0, p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1, p}(\mathbb{R}_+^n)$.

*Proof.* — Let $p \in (1, +\infty)$, $-n/p' < s_0 < s_1 < n/p$. There are three subcases, $0 \le s_0 < s_1$, $s_0 < 0 < s_1$, and $s_0 < s_1 \le 0$.

The case $0 \le s_0 < s_1$ follows the lines of Proposition 3.9 thanks to Corollary 3.7.

The case $s_0 < 0 < s_1$, can be done via duality argument as in Proposition 3.9 for the negative index of regularity. Let us consider $\frac{1}{q} = \frac{1}{p} - \frac{s_0}{n}$, the following embeddings are true

$$ H_0^{s_1-s_0,q}(\mathbb{R}_+^n) \hookrightarrow L^q(\mathbb{R}_+^n) \cap H_0^{s_1,p}(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1,p}(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{s_1,p}(\mathbb{R}_+^n). $$

One may dualize it to deduce

$$ \dot{H}^{-s_1, p'}(\mathbb{R}_+^n) \hookrightarrow (\dot{H}_0^{s_0, p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1, p}(\mathbb{R}_+^n))' \hookrightarrow H^{s_0-s_1, q'}(\mathbb{R}_+^n). $$

We deduce that the last embedding is dense, since $(\dot{H}_0^{s_0, p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1, p}(\mathbb{R}_+^n))'$ contains $\dot{H}^{-s_1, p'}(\mathbb{R}_+^n)$ via canonical embedding, so that by duality and convexity of all involved spaces, the following embedding is dense:

$$ H_0^{s_1-s_0,q}(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1,p}(\mathbb{R}_+^n). $$

Since $C_c^\infty(\mathbb{R}_+^n) \hookrightarrow H_0^{s_1-s_0,q}(\mathbb{R}_+^n)$ is dense, the result follows.

We end the proof claiming that the third case $s_0 < s_1 \le 0$ can be done similarly via duality and convexity arguments. ■

### 3.2.2 Homogeneous Besov spaces, Interpolation

We are done with properties of homogeneous Sobolev spaces. We continue with a real interpolation embedding lemma, that will allow us to transfer all nice properties, like boundedness of extension and projection operators, from homogeneous Sobolev spaces to homogeneous Besov spaces.
so the triangle inequality gives

$$ \|u - w\|_{\dot{H}_0^{-s,p}(\mathbb{R}_+^n)} \lesssim_{n,s,p,q} \varepsilon, $$

which concludes the proof since $w \in C_c^\infty(\mathbb{R}_+^n)$.

**Step 3:** For the density in the intersection spaces, it suffices to reproduce the above **Step 1** by means of Corollary 3.7. ■

**Corollary 3.10** For all $p \in (1, +\infty)$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$,

$$ \dot{H}_0^{s,p}(\mathbb{R}_+^n) = \dot{H}^{s,p}(\mathbb{R}_+^n). $$

In particular, $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{H}^{s,p}(\mathbb{R}_+^n)$ for same range of indices.

*Proof.* — This is a direct consequence of the definition of restriction spaces and Proposition 2.9, the density result follows from Proposition 3.9. ■

**Proposition 3.11** Let $p \in (1, +\infty)$, $s \in (-\frac{n}{p'}, \frac{n}{p})$, we have

$$ (\dot{H}^{s,p}(\mathbb{R}_+^n))' = \dot{H}_0^{-s,p'}(\mathbb{R}_+^n) \quad \text{and} \quad (\dot{H}_0^{s,p}(\mathbb{R}_+^n))' = \dot{H}^{-s,p'}(\mathbb{R}_+^n). $$

*Proof.* — First, consider $s \in (-\frac{n}{p'}, \frac{n}{p})$, let $\Phi \in \dot{H}_0^{-s,p'}(\mathbb{R}_+^n) \subset \dot{H}^{-s,p'}(\mathbb{R}^n)$, then using definition of restriction spaces, the following map defines a linear functional on $\dot{H}^{s,p}(\mathbb{R}_+^n)$,

$$ u \mapsto \langle \Phi, \tilde{u} \rangle_{\mathbb{R}^n}, $$

where $\tilde{u}$ is any extension of $u$, and notice that the action of $\Phi$ does not depend on the choice of such extension of $u$. Indeed, if $U \in \dot{H}^{s,p}(\mathbb{R}^n)$ is another extension of $u$, we obtain that $w := U - \tilde{u} \in \dot{H}_0^{s,p}(\mathbb{R}_+^n)$. It follows from Proposition 3.9 that $w$ is a strong limit in $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$ of a sequence of functions $(w_k)_{k \in \mathbb{N}} \subset C_c^\infty(\mathbb{R}_+^n)$ so that, passing to the limit, in the duality bracket, we obtain

$$ \langle \Phi, U \rangle_{\mathbb{R}^n} - \langle \Phi, \tilde{u} \rangle_{\mathbb{R}^n} = \langle \Phi, w \rangle_{\mathbb{R}^n} = 0. $$

This gives a well-defined continuous injective map

$$ \left\{ \begin{array}{l} \dot{H}_0^{-s,p'}(\mathbb{R}_+^n) \longrightarrow (\dot{H}^{s,p}(\mathbb{R}_+^n))' \\ \Phi \longmapsto \langle \Phi, \tilde{\gamma} \rangle_{\mathbb{R}^n} \end{array} \right. . \qquad (3.11) $$

Now, let $\Psi \in (\dot{H}^{s,p}(\mathbb{R}_+^n))'$, for all $u \in \dot{H}^{s,p}(\mathbb{R}_+^n)$, since $1_{\mathbb{R}_+^n}u = u$, we may write,

$$ \langle \Psi, u \rangle = \langle \Psi, 1_{\mathbb{R}_+^n} \tilde{u} \rangle, $$

for any extension $\tilde{u} \in \dot{H}^{s,p}(\mathbb{R}^n)$ of $u$, hence, as a direct consequence of the definition of restriction space $1_{\mathbb{R}_+^n}\Psi \in (\dot{H}^{s,p}(\mathbb{R}^n))' = \dot{H}^{-s,p'}(\mathbb{R}^n)$, so $1_{\mathbb{R}_+^n}\Psi \in \dot{H}_0^{-s,p'}(\mathbb{R}_+^n)$. The following map is well-defined continuous and injective

$$ \left\{ \begin{array}{l} (\dot{H}^{s,p}(\mathbb{R}_+^n))' \longrightarrow \dot{H}_0^{-s,p'}(\mathbb{R}_+^n) \\ \Psi \longmapsto 1_{\mathbb{R}_+^n} \Psi \end{array} \right. . \qquad (3.12) $$

Both maps (3.11) and (3.12) are even isometric, and we obtain

$$ (\dot{H}^{s,p}(\mathbb{R}_+^n))' = \dot{H}_0^{-s,p'}(\mathbb{R}_+^n), $$

which was the first statement. The second statement follows from duality and convexity exchanging roles of involved exponents. ■

One would also like to carry over density results in intersection spaces in order to deduce its counterpart in their real interpolation spaces.

**Corollary 3.12** Let $p \in (1, +\infty)$, $-n/p' < s_0 < s_1 < n/p$, i.e. such that $(C_{-s_0,p'})$ and $(C_{s_1,p})$ are both satisfied.

The space $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{H}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1,p}(\mathbb{R}_+^n)$.
**Lemma 3.13** Let $(p, q, q_0, q_1) \in (1, +\infty) \times [1, +\infty]^3$, $s_0, s_1 \in \mathbb{R}$, such that $s_0 < s_1$, and set
$$s := (1 - \theta)s_0 + \theta s_1.$$

If $(C_{s_0,p})$ is satisfied we have,

$$ \dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{H}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{H}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q}, \quad (3.13) $$

$$ \dot{B}_{p,q,0}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{H}_{0}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{H}_{0}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q}. \quad (3.14) $$

Similarly if $(C_{s_0,p,q_0})$ is satisfied, we also have

$$ \dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{B}_{p,q_0}^{s_0}(\mathbb{R}_{+}^{n}), \dot{B}_{p,q_1}^{s_1}(\mathbb{R}_{+}^{n}))_{\theta,q}, \quad (3.15) $$

$$ \dot{B}_{p,q,0}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{B}_{p,q_0,0}^{s_0}(\mathbb{R}_{+}^{n}), \dot{B}_{p,q_1,0}^{s_1}(\mathbb{R}_{+}^{n}))_{\theta,q}. \quad (3.16) $$

*Proof.* — For embeddings (3.13) and (3.15), one may follow the first part of the proof of [DHMT21, Proposition 3.22].

The third embedding (3.14), (the fourth one (3.16) can be treated similarly) is straightforward since,

$$ (\dot{H}_{0}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{H}_{0}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q} \hookrightarrow (\dot{H}^{s_0,p}(\mathbb{R}^{n}), \dot{H}^{s_1,p}(\mathbb{R}^{n}))_{\theta,q} = \dot{B}_{p,q}^{s}(\mathbb{R}^{n}). $$

By definition, $f \in (\dot{H}_{0}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{H}_{0}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q} \subset \dot{H}_{0}^{s_0,p}(\mathbb{R}_{+}^{n}) + \dot{H}_{0}^{s_1,p}(\mathbb{R}_{+}^{n})$, hence $\text{supp } f \subset \overline{\mathbb{R}}_{+}^{n}$ and $f \in \dot{B}_{p,q,0}^{s}(\mathbb{R}_{+}^{n})$. ■

As we mentioned, the above lemma can be used to prove the boundedness of some operators on a sufficiently large range of indices on Besov spaces via some sort of interpolation method, without the exact description of the interpolation space; see below.

**Corollary 3.14** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s > -1 + \frac{1}{p}$, $m \in \mathbb{N}$, such that $s < m + 1 + \frac{1}{p}$. Let us consider the extension operator $E$ (resp. $\mathcal{P}_0$) given by Proposition 3.1 (resp. Lemma 3.6).
If either

* $s > 0$ and $u \in B_{p,q}^s(\mathbb{R}_+^n)$ (resp. $u \in B_{p,q}^s(\mathbb{R}^n)$);

* $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ and $u \in \dot{B}_{p,q}^s(\mathbb{R}_+^n)$ (resp. $u \in \dot{B}_{p,q}^s(\mathbb{R}^n)$);

we have the estimate

$$ \|Eu\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{s,m,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} \cdot (\text{resp. } \|\mathcal{P}_0u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{s,m,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} \cdot ) $$

In particular, $E$ (resp. $\mathcal{P}_0$) is a bounded operator from $\dot{B}_{p,q}^s(\mathbb{R}_+^n)$ to $\dot{B}_{p,q}^s(\mathbb{R}^n)$ (resp. from $\dot{B}_{p,q}^s(\mathbb{R}^n)$ to $\dot{B}_{p,q,0}^s(\mathbb{R}_+^n)$) whenever $(C_{s,p,q})$ is satisfied.

*Proof.* — Let $p \in (1, +\infty)$, $q \in [1, +\infty)$, $s > -1 + \frac{1}{p}$, $m \in \mathbb{N}$, such that $s < m + 1 + \frac{1}{p}$. Without loss of generality, it suffices to prove the result for the operator $E$, since we have the identity $\mathcal{P}_0 = I - E^{-}[1_{\mathbb{R}_+^n}]$, as written in the proof of Lemma 3.6.

The boundedness of $E$ on $\dot{B}_{p,q}^s(\mathbb{R}^n)$ for $(p, q) \in (1, +\infty) \times [1, +\infty]$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ is again a direct consequence of Proposition 2.9.

It remains to prove boundedness for $s \ge \frac{1}{p}$. To do so, we proceed via a manual real interpolation scheme.

Let $u \in B_{p,q}^s(\mathbb{R}_+^n)$, $\theta \in (0, 1)$ such that $\theta s_1 = s$, where $s_1 \in (s, m + 1 + \frac{1}{p})$. One has

$$ u \in (L^p(\mathbb{R}_+^n), H^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \hookrightarrow (L^p(\mathbb{R}_+^n), \dot{H}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \subset L^p(\mathbb{R}_+^n) + \dot{H}^{s_1,p}(\mathbb{R}_+^n). $$

Hence, for $a \in L^p(\mathbb{R}_+^n)$, $b \in \dot{H}^{s_1,p}(\mathbb{R}_+^n)$ such that $f = a + b$, we can deduce that

$$ b = u - a \in B_{p,q}^s(\mathbb{R}_+^n) + L^p(\mathbb{R}_+^n) \subset L^p(\mathbb{R}_+^n), $$

so that $b \in L^p(\mathbb{R}_+^n) \cap \dot{H}^{s_1,p}(\mathbb{R}_+^n) = H^{s_1,p}(\mathbb{R}_+^n)$ thanks to Proposition 3.3. Hence, $Eu = Ea + Eb$, with $Ea \in L^p(\mathbb{R}_+^n)$, $Eb \in H^{s_1,p}(\mathbb{R}_+^n)$, with the homogeneous estimates provided by Proposition 3.1. Then
Proof. — Let $p \in (1, +\infty)$, $-n/p' < s_0 < s_1 < n/p$. There are three subcases, $0 \le s_0 < s_1$, $s_0 < 0 < s_1$, and $s_0 < s_1 \le 0$.

The case $0 \le s_0 < s_1$ follows the lines of Proposition 3.9 thanks to Corollary 3.7.

The case $s_0 < 0 < s_1$, can be done via duality argument as in Proposition 3.9 for the negative index of regularity. Let us consider $\frac{1}{q} = \frac{1}{p} - \frac{s_0}{n}$, the following embeddings are true

$$H_0^{s_1-s_0,q}(\mathbb{R}_+^n) \hookrightarrow L^q(\mathbb{R}_+^n) \cap H_0^{s_1,p}(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1,p}(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{s_1,p}(\mathbb{R}_+^n).$$

One may dualize it to deduce

$$\dot{H}^{-s_1, p'}(\mathbb{R}_+^n) \hookrightarrow (\dot{H}_0^{s_0, p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1, p}(\mathbb{R}_+^n))' \hookrightarrow H^{s_0-s_1, q'}(\mathbb{R}_+^n).$$

We deduce that the last embedding is dense, since $(\dot{H}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1,p}(\mathbb{R}_+^n))'$ contains $\dot{H}^{-s_1,p'}(\mathbb{R}_+^n)$ via canonical embedding, so that by duality and reflexivity of all involved spaces, the following embedding is dense:

$$H_0^{s_1-s_0,q}(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1,p}(\mathbb{R}_+^n).$$

Since $C_c^\infty(\mathbb{R}_+^n) \hookrightarrow H_0^{s_1-s_0,q}(\mathbb{R}_+^n)$ is dense, [JK95, Remark 2.7], the result follows.

We end the proof claiming that the third case $s_0 < s_1 \le 0$ can be done similarly via duality and reflexivity arguments. ■

### 3.2.2 Homogeneous Besov spaces, Interpolation

We are done with properties of homogeneous Sobolev spaces. We continue with a real interpolation embedding lemma, that allows us to transfer all nice properties, like boundedness of extension and projection operators, from homogeneous Sobolev spaces to homogeneous Besov spaces.

**Lemma 3.13** Let $(p, q, q_0, q_1) \in (1, +\infty) \times [1, +\infty]^3$, $s_0, s_1 \in \mathbb{R}$, such that $s_0 < s_1$, and set

$$s := (1 - \theta)s_0 + \theta s_1.$$

If $(C_{s_0,p})$ is satisfied we have,

$$\dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{H}_{0}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{H}_{0}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q}, \qquad (3.13)$$

$$\dot{B}_{p,q,0}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{H}_{0}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{H}_{0}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q}. \qquad (3.14)$$

Similarly if $(C_{s_0,p,q_0})$ is satisfied, we also have

$$\dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{B}_{p,q_0}^{s_0}(\mathbb{R}_{+}^{n}), \dot{B}_{p,q_1}^{s_1}(\mathbb{R}_{+}^{n}))_{\theta,q}, \qquad (3.15)$$

$$\dot{B}_{p,q,0}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{B}_{p,q_0,0}^{s_0}(\mathbb{R}_{+}^{n}), \dot{B}_{p,q_1,0}^{s_1}(\mathbb{R}_{+}^{n}))_{\theta,q}. \qquad (3.16)$$

Proof. — For embeddings (3.13) and (3.15), one may follow the first part of the proof of [DHMT21, Proposition 3.22].

The third embedding (3.14), (the fourth one (3.16) can be treated similarly) is straightforward since,

$$(\dot{H}_0^{s_0,p}(\mathbb{R}_+^n), \dot{H}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \hookrightarrow (\dot{H}^{s_0,p}(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n))_{\theta,q} = \dot{B}_{p,q}^s(\mathbb{R}^n).$$

By definition, $f \in (\dot{H}_0^{s_0,p}(\mathbb{R}_+^n), \dot{H}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \subset \dot{H}_0^{s_0,p}(\mathbb{R}_+^n) + \dot{H}_0^{s_1,p}(\mathbb{R}_+^n)$, hence $\text{supp } f \subset \overline{\mathbb{R}_+^n}$ and $f \in \dot{B}_{p,q,0}^s(\mathbb{R}_+^n)$. ■

As we mentioned, the above lemma can be used to prove the boundedness of some operators on a sufficiently wide range of indices on Besov spaces via some sort of interpolation method, without the exact description of the interpolation space; see below.

**Corollary 3.14** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s > -1 + \frac{1}{p}$, $m \in \mathbb{N}$, such that $s < m + 1 + \frac{1}{p}$. Let us consider the extension operator $E$ (resp. $P_0$) given by Proposition 3.1 (resp. Lemma 3.6).

If either one of the following conditions holds

$$\bullet \ s > 0 \text{ and } u \in B_{p,q}^s(\mathbb{R}_+^n) \text{ (resp. } u \in B_{p,q}^s(\mathbb{R}^n)) ;$$
$Eu|_{\mathbb{R}_+^n} = u$, and we have the estimates

$$K(t, Eu, L^p(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n)) \le \|Ea\|_{L^p(\mathbb{R}^n)} + t\|Eb\|_{\dot{H}^{s_1,p}(\mathbb{R}^n)} \lesssim_{p,m,n} \|a\|_{L^p(\mathbb{R}_+^n)} + t\|b\|_{\dot{H}^{s_1,p}(\mathbb{R}_+^n)}.$$

Hence, taking infimum on all such functions $a$ and $b$, and multiplying by $t^{-\theta}$ leads to

$$t^{-\theta} K(t, Eu, L^p(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n)) \lesssim_{p,s,s_1,n} t^{-\theta} K(t, u, L^p(\mathbb{R}_+^n), \dot{H}^{s_1,p}(\mathbb{R}_+^n)),$$

so one may take the $L_*^q$-norm of above inequality and use (3.13) from Lemma 3.13 to deduce that

$$\|Eu\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{p,s,q,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}.$$

If $q < +\infty$, then $B_{p,q}^s(\mathbb{R}_+^n)$ is dense in $\dot{B}_{p,q}^s(\mathbb{R}^n)$, so that the conclusion holds by density whenever $(C_{s,p,q})$ is satisfied.

If $q = +\infty$, and $(C_{s,p,q})$ is satisfied, necessarily $s < \frac{n}{p}$. We introduce $\mathcal{E} := E[1_{\mathbb{R}_+^n}]$ which is bounded, thanks to the above step, seen as an operator

$$\mathcal{E} : \dot{B}_{p,q_j}^{s_j}(\mathbb{R}^n) \to \dot{B}_{p,q_j}^{s_j}(\mathbb{R}^n),$$

provided $s_0 < s < s_1 < \frac{n}{p}$, and $q_j \in [1, \infty)$, $j \in \{0,1\}$. Thus, by real interpolation argument, thanks to Theorem 2.6, for all $U \in \dot{B}_{p,\infty}^s(\mathbb{R}^n)$, we have

$$\|\mathcal{E}U\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)} \lesssim_{p,s,q,n} \|U\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)}.$$

In particular, for all $u \in \dot{B}_{p,\infty}^s(\mathbb{R}_+^n)$, and all $U \in \dot{B}_{p,\infty}^s(\mathbb{R}^n)$ such that $U|_{\mathbb{R}_+^n} = u$, we have

$$\|Eu\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)} \lesssim_{p,s,q,n} \|U\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)}.$$

Hence, taking the infimum on all such functions $U$ gives the result when $q = +\infty$ and $(C_{s,p,q})$ is satisfied. ■

**Proposition 3.15** Let $p, q \in [1, +\infty]$, $s \in (0, n)$, such that

$$\frac{1}{q} = \frac{1}{p} - \frac{s}{n}.$$

We have the following estimates,

$$\begin{align*}
\|u\|_{L^q(\mathbb{R}_+^n)} &\lesssim_{n,s,p,q,r} \|u\|_{\dot{B}_{p,r}^s(\mathbb{R}_+^n)}, \quad \forall u \in \dot{B}_{p,r}^s(\mathbb{R}_+^n), r \in [1, q] \\
\|u\|_{\dot{B}_{q,r,0}^{-s}(\mathbb{R}_+^n)} &\lesssim_{n,s,p,q,r} \|u\|_{L^p(\mathbb{R}_+^n)}, \quad \forall u \in L^p(\mathbb{R}_+^n), r \in [q, +\infty].
\end{align*}$$

Moreover, we also have $\dot{B}_{p,1}^n(\mathbb{R}_+^n) \hookrightarrow C_0^0(\overline{\mathbb{R}_+^n})$, whenever $p$ is finite.

*Proof.* — Each embedding is a direct consequence of the definition of each space and the corresponding ones on $\mathbb{R}^n$, see point (iv) of Proposition 2.1. ■

**Lemma 3.16** Let $p \in (1, +\infty)$, $q \in [1, +\infty)$ and $s > 0$. The function space $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{B}_{p,q,0}^s(\mathbb{R}_+^n)$ whenever $(C_{s,p,q})$ is satisfied.

*Proof.* — As in the proof of Proposition 3.9, in the case of non negative index: by a successive approximations scheme, we use density of $B_{p,q}^s(\mathbb{R}^n)$ in $\dot{B}_{p,q}^s(\mathbb{R}^n)$, to approximate functions in $\dot{B}_{p,q,0}^s(\mathbb{R}_+^n)$. Then the boundedness of $\mathcal{P}_0$ on $\dot{B}_{p,q}^s(\mathbb{R}_+^n)$, and the density of $C_c^\infty(\mathbb{R}_+^n)$ in $B_{p,q,0}^s(\mathbb{R}_+^n)$ yields the result. ■

**Proposition 3.17** Let $(p_0, p_1, p, q) \in (1, +\infty)^3 \times [1, +\infty]$, $s_0, s_1 \in \mathbb{R}$, such that $s_0 < s_1$, let $(\mathfrak{h}, \mathfrak{b}) \in \{(\mathfrak{H}, \mathfrak{B}), (\mathfrak{H}_0, \mathfrak{B}_{\cdot, \cdot, 0})\}$, and set

$$\left(s, \frac{1}{p_\theta}\right) := \left(1 - \theta\right) \left(s_0, \frac{1}{p_0}\right) + \theta \left(s_1, \frac{1}{p_1}\right).$$

If either one of following assertions is satisfied,

$$(i) \ q \in [1, +\infty), \ s_j > -1 + \frac{1}{p_j}, \ j \in \{0, 1\};$$