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SOME GEOMETRIC AND SPECTRAL ASPECTS OF
RESTRICTION PROBLEMS

Hajer Bahouri, V Éronique Fischer

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Hajer Bahouri, V Éronique Fischer. SOME GEOMETRIC AND SPECTRAL ASPECTS OF RESTRICTION PROBLEMS. 2025. hal-05127012

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# SOME GEOMETRIC AND SPECTRAL ASPECTS OF RESTRICTION PROBLEMS

HAJER BAHOURI AND VÉRONIQUE FISCHER

*In memory of Haïm Brezis*

**ABSTRACT.** This texts commemorates the memory of Haïm Brezis and explores some aspects of the restriction problem, particularly its connections to spectral and geometric analysis. Our choice of subject is motivated by Brezis' significant contributions to various domains related to this problem, including harmonic analysis, partial differential equations, spectral theory, representation theory, number theory, and many others.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. The first L<sup>2</sup>-restriction theorem: on the sphere</td><td>3</td></tr><tr><td>2.1. Statement</td><td>3</td></tr><tr><td>2.2. Proof of Theorem 2.1</td><td>4</td></tr><tr><td>2.3. Tomas' improvement</td><td>5</td></tr><tr><td>3. Tomas-Stein Restriction theorem on hyper-surfaces</td><td>5</td></tr><tr><td>3.1. The induced measure on a hyper-surface</td><td>5</td></tr><tr><td>3.2. Statement of the Tomas-Stein Restriction Theorem</td><td>7</td></tr><tr><td>4. Spectral viewpoint</td><td>8</td></tr><tr><td>4.1. Spectral description of the operator P</td><td>8</td></tr><tr><td>4.2. Spectral reformulation of the restriction property</td><td>10</td></tr><tr><td>4.3. Links with the cluster estimates</td><td>12</td></tr><tr><td>5. Restriction theorems on Lie groups and homogeneous domains</td><td>14</td></tr><tr><td>5.1. A few words on the analysis on Lie groups and homogeneous domains</td><td>14</td></tr><tr><td>5.2. Restriction theorem on the Heisenberg group</td><td>18</td></tr><tr><td>6. Related issues and open questions</td><td>19</td></tr><tr><td>6.1. Related questions in harmnoic analysis</td><td>19</td></tr><tr><td>6.2. Applications to evolution equations</td><td>20</td></tr><tr><td>6.3. Some generalisations of Tomas-Stein estimates</td><td>25</td></tr><tr><td>6.4. Discrete framework</td><td>28</td></tr><tr><td>6.5. Number theory connection</td><td>30</td></tr><tr><td>6.6. Open problems</td><td>31</td></tr><tr><td>Appendix A. The TT* argument</td><td>32</td></tr><tr><td>References</td><td>33</td></tr></table>

**Keywords:** Restriction problems, Fourier analysis, spectral theory, subelliptic operators.

**AMS Subject Classification (2020):** 43A30, 43A80, 53C17, 30C40.

## 1. INTRODUCTION

The original problem of Fourier restriction was first observed by E. Stein in the late 1960s, as reported in C. Fefferman's PhD thesis [44]: the restriction of the Fourier transform of

*Date:* June 24, 2025.
functions $f \in L^p(\mathbb{R}^n)$ to the unit sphere $\mathbb{S}^{n-1} \subset \mathbb{R}^n$ makes sense when $p$ is close enough to 1 and $n > 1$. We will recall the precise statement in Theorem 2.1 and sketch its original proof in Section 2.

Soon after this observation, E. Stein and P. Tomas in [107, 116] found the optimal range of indices $p$ for this restriction, and the sharpness of this result was proved with a counterexample due to A. Knapp, see [111]. In the meantime, A. Zygmund [122] proved a discrete analogue restriction property in the context of the two-dimensional flat torus.

Over the subsequent decades, restriction problems have experienced an incredible effer-
escence and excitement. They remain a topical issue to this day. Generalised to other
hyper-surfaces in Euclidean spaces and other settings, they are closely related to problems
in Harmonic Analysis and Partial Differential Equations (PDEs), such as the Bochner-
Riesz means, Strichartz estimates, and the Kakeya conjecture proved by R. Davies [35] in
the two dimensional case in 1971, and recently solved by H. Wang and J. Zahl [118] in
the three-dimensional case. They are also related to many questions at the interface of
Harmonic Analysis and Number Theory; an instance of this is the Vinogradov conjecture
(recalled in Section 6.5.1), now a theorem proved in 2015 independently by Bourgain-
Demeter-Guth [18] and T. Wooley [120, 121], using decoupling theory for the former and
for the latter the efficient congruencing. For an in-depth presentation of these topics, we
refer to the two milestone surveys on the subject by T. Tao [112] and L. Guth [63], as
well as the texts of C. Demeter [37, 38] about decoupling methods and Bourgain's work
in Fourier restriction.

Restriction problems are also connected to many phenomena in Spectral Theory, and our aim is to explore geometric and spectral aspects of them. In Section 4, we present the well-known interpretation of the original restriction problem on the sphere as a spectral property of the Laplace operator $\Delta_{\mathbb{R}^n}$ on $\mathbb{R}^n$. Replacing $-\Delta_{\mathbb{R}^n}$ with other positive operators allows us to formulate the problem of Fourier restriction into a spectral problem that makes sense in other settings. In contrast with T. Tao [112] and L. Guth [63]'s surveys, this text emphasises this reformulation which is especially relevant in contexts where the Euclidean Fourier transform is unavailable. These settings include the Riemannian manifolds with the Laplace-Beltrami operators, but more generally any setting with a Laplace-type operator, such as graphs with the graph Laplacian or manifolds equipped with positive sub-elliptic operators. We will naturally recall the connection with cluster estimates understood as $L^p - L^q$-bounds for the spectral projectors of a Laplace-type operator (or its square root) in a window $[\lambda, \lambda + 1]$ with $\lambda \gg 1$ large (see Section 4.3). The particular case of the Laplace-Beltrami operators on compact Riemannian manifolds has been extensively studied by C. Sogge [101, 102, 103].

A perhaps less understood question is to study cluster estimates or spectral restric-
tion problems for sub-elliptic operators, for instance, sub-Laplacians, that is, the sum of
squares of vector fields satisfying the Hörmander condition [69]. Exploring Stein's prob-
lem in sub-elliptic frameworks is motivated by the wide range of ramifications of this
field in several parts of mathematics and applied areas, such as crystallography, particle
physics, optimal control, image processing, etc. [77]. The prototype of sub-Laplacians is
the canonical ones on the Heisenberg groups; these groups can be defined via the canonical
commutation relations, known as CCR in quantum mechanics, and are at the confluence
of many mathematical and physical fields. Restriction properties for the canonical sub-
Laplacians on the Heisenberg groups were studied by D. Müller [92]. In this setting, there
are no non-trivial solutions in the $L^p$-spaces for $p > 1$, leading D. Müller to reformulate
the problem in anisotropic spaces, see Section 5.

This text is organised as follows. In Section 2, we discuss the first $L^2$-restriction theorem which was set on the unit sphere $\mathbb{S}^{n-1}$ of $\mathbb{R}^n$. We emphasise the crucial role of the Fourier transform of the canonical measure of $\mathbb{S}^{n-1}$ and its decay at infinity. In Section 3,
we present the extension of these questions to the setting of hyper-surfaces with non-vanishing Gaussian curvature in $\mathbb{R}^n$ with optimal range of indices and exponents. This is known as the famous Tomas-Stein Theorem. Again, the proofs rely fundamentally on the decay of the Fourier transform of the induced measures on the hyper-surfaces with non-vanishing curvature. Indeed, this decay is the same as for the unit sphere $\mathbb{S}^{n-1}$. In Section 4, we reformulate the restriction estimates on the sphere in spectral terms. We show that they are equivalent to cluster estimates. In Section 5, we discuss restriction and cluster estimates on Lie groups and homogeneous domains, where few cases have been studied with surprising results, such as M¨uller’s results for the canonical sub-Laplacian on the Heisenberg group. Section 6 is devoted to applications of restriction results and open problems. The first part of this section will be mainly concerned with Strichartz estimates which have become a powerful tool in the study of nonlinear evolution equations involved in physics, quantum mechanics and general relativity. In the second part of Section 6, we briefly present restriction problems in the challenging discrete framework, and its connections with Number Theory. Finally, in Appendix A, we recall the $TT^*$ argument.

We conclude this introduction with a comment on notation. In this paper, the letter $C$ will be used to denote universal constants which may vary from line to line. If we need the implied constant to depend on parameters, we shall indicate this by subscripts. We also use the notation $A \lesssim B$ to denote bound of the form $A \le CB$, and $A \lesssim_{\alpha} B$ for $A \le C_{\alpha}B$, where $C_{\alpha}$ depends only on $\alpha$.

**Acknowledgments.** The authors wish to thank Julien Sabin for enlightening discussions about the refined restriction estimates.

## 2. THE FIRST $L^2$-RESTRICTION THEOREM: ON THE SPHERE

Historically, the first $L^2$-restriction problem was set on the unit sphere

$$ \mathbb{S}^{n-1} = \{x = (x_1, \dots, x_n) \in \mathbb{R}^n : |x|^2 = x_1^2 + \dots + x_n^2 = 1\}, $$

equipped with its canonical measure $\sigma_{\mathbb{S}^{n-1}}$.

### 2.1. Statement.
As the Fourier transform of a Schwartz function is Schwartz, its restriction to the unit sphere makes sense. The first restriction theorem (Theorem 2.1 below) states that the Fourier transform of a function in $L^p(\mathbb{R}^n)$ restricts to an $L^2$-function on the unit sphere for $p \in [1, 4n/(3n+1))$. To keep the notation consistent with the case of more general settings, in what follows, we will distinguish $\mathbb{R}^n$ and its dual $\hat{\mathbb{R}}^n$, which is of course isomorphic to $\mathbb{R}^n$ itself, and write $\hat{\mathbb{S}}^{n-1}$ for the unit sphere of $\hat{\mathbb{R}}^n$. By a classical density argument, it suffices to establish the a priori estimate

$$ (2.1) \qquad \| \mathcal{F}(f) \|_{\hat{\mathbb{S}}^{n-1}} \| L^2(\hat{\sigma}_{\mathbb{S}^{n-1}}) \le C \|f\|_{L^p(\mathbb{R}^n)}, $$

for all $f$ in the Schwartz space $S(\mathbb{R}^n)$ and for a constant $C > 0$ independent of $f$. Above, $\mathcal{F}f$ denotes the Fourier transform of $f$ on $\mathbb{R}^n$:

$$ \mathcal{F}f(\xi) = \hat{f}(\xi) = \int_{\mathbb{R}^n} e^{-2\pi i x \cdot \xi} f(x) dx, \quad \xi \in \hat{\mathbb{R}}^n. $$

Indeed, (2.1) implies that the linear map $f \mapsto \mathcal{F}f|_{\hat{\mathbb{S}}^{n-1}}$ defined on $S(\mathbb{R}^n)$ extends uniquely continuously $L^p(\mathbb{R}^n) \to L^2(\hat{\mathbb{S}}^{n-1})$.

**Theorem 2.1 (First restriction theorem).** Let $n \ge 2$ and $p \in [1, 4n/(3n+1))$. The estimates in (2.1) hold for any $f \in S(\mathbb{R}^n)$.
Theorem 2.1 is far from being optimal. For instance, a modification of the argument below would allow us to reach the end-point $p = 4n/(3n + 1)$ using Hardy-Littlewood-Sobolev inequalities. However, the optimal end-point ($p_{TS} := (2n+2)/(n+3)$) was proved later on with the Tomas-Stein Theorem [116] (see Theorems 2.2 and 3.2 below).

2.2. **Proof of Theorem 2.1.** We reproduce here the well-known argument due to E. Stein and C. Fefferman [107, 44, 45].

*Proof of Theorem 2.1.* The proof relies on noticing the following equality valid for any $f \in S(\mathbb{R}^n)$

$$ \|\mathcal{F}f\|_{L^2(\hat{\sigma}_{\mathbb{S}^{n-1}})}^2 = \int_{\mathbb{R}^n} f(x) \overline{f * \kappa(x)} dx, \quad \text{where} \quad \kappa := \mathcal{F}^{-1}\hat{\sigma}_{\mathbb{S}^{n-1}}. $$

By Stein-Weiss [109] Section IV.3 (see also the monograph of Gel'fand-Shilov [55]), the convolution kernel $\kappa$ is explicitly given in terms of the Bessel function $J_\alpha$ of the first kind and of order $\alpha = (n-2)/2$ as

$$ (2.2) \qquad \kappa(x) = \int_{\hat{\mathbb{R}}^n} e^{2\pi i x \cdot \xi} d\sigma_{\hat{\mathbb{S}}^{n-1}}(\xi) = 2\pi \frac{J_{(n-2)/2}(2\pi|x|)}{|x|^{(n-2)/2}}. $$

It is well known that, for any $\alpha > -1$, the function $\frac{J_\alpha(z)}{(z/2)^\alpha}$ coincides on $\mathbb{R}$ with a power series in $z^2$, more precisely with ${}_0F_1(\alpha + 1, -z^2/4)/\Gamma(\alpha + 1)$ using generalised hyper-geometric series and the Gamma function. Moreover, it satisfies the asymptotic bound

$$ \exists C > 0 \mid \forall z \in \mathbb{R}, \quad |J_{\alpha}(z)| \le C(1 + |z|)^{-1/2}. $$

Hence, $\kappa$ is a smooth function on $\mathbb{R}^n$ satisfying

$$ (2.3) \qquad |\kappa(x)| \lesssim (1 + |x|)^{-(n-1)/2}, $$

so that $\kappa \in L^q(\mathbb{R}^n)$ if and only if $q > 2n/(n-1)$. By Hölder's inequality and then Young's convolution inequality, we have

$$ \|\mathcal{F}f\|_{L^2(\hat{\sigma}_{\mathbb{S}^{n-1}})}^2 \le \|f\|_{L^p(\mathbb{R}^n)} \|f * k\|_{L^{p'}(\mathbb{R}^n)} \le \|f\|_{L^p(\mathbb{R}^n)}^2 \|k\|_{L^q(\mathbb{R}^n)}, \quad \frac{1}{p} = 1 - \frac{1}{2q}. $$

The conclusion follows.

□

The crucial step in the proof above is to show that the operator $\mathcal{P}$ defined by

$$ (2.4) \qquad \mathcal{P}f = f * \kappa, \quad \text{where} \quad \kappa = \mathcal{F}^{-1}\hat{\sigma}_{\mathbb{S}^{n-1}}, $$

or equivalently via

$$ \widehat{\mathcal{P}f} = \mathcal{F}f \, d\sigma_{\widehat{\mathbb{S}}^{n-1}}, $$

is bounded $L^p(\mathbb{R}^n) \to L^{p'}(\mathbb{R}^n)$ whenever $p \in [1, 4n/(3n+1))$. This is in fact an early instance of the $TT^*$ argument (see Appendix A) as we may write

$$ \mathcal{P} = TT^*, \quad \text{i.e.} \quad \forall f \in S(\mathbb{R}), \qquad \mathcal{P}f = f * \kappa = TT^*f, $$

where $T^*$ is the actual restriction operator $f \mapsto \mathcal{F}|_{\widehat{\mathbb{S}}^{n-1}}$ whose boundedness $L^p \to L^2$ we wish to prove, and $T$ its formal dual given by $Tg = \mathcal{F}^{-1}(gd\sigma_{\widehat{\mathbb{S}}^{n-1}})$. By the $TT^*$ argument, the boundedness of one of the following operators

$$ \mathcal{P} : L^p(\mathbb{R}^n) \to L^{p'}(\mathbb{R}^n), \quad T : L^2(\widehat{\mathbb{S}}^{n-1}) \to L^{p'}(\mathbb{R}^n), \quad T^* : L^p(\mathbb{R}^n) \to L^2(\widehat{\mathbb{S}}^{n-1}) $$

implies the boundedness of the others. These boundedness properties may be expressed equivalently with the following a priori estimates valid for $f \in S(\mathbb{R}^n)$ and $g \in C^\infty(\widehat{\mathbb{S}}^{n-1})$:

$$ \begin{align*} \|\mathcal{P}f\|_{L^{p'}(\mathbb{R}^n)} &\le C_p \|f\|_{L^p(\mathbb{R}^n)}, \\ \|Tg\|_{L^{p'}(\mathbb{R}^n)} &= \|\mathcal{F}^{-1}(gd\sigma_{\widehat{\mathbb{S}}^{n-1}})\|_{L^{p'}(\mathbb{R}^n)} \le \sqrt{C_p} \|g\|_{L^2(\widehat{\mathbb{S}}^{n-1})}, \\ \|T^*f\|_{L^2(\widehat{\mathbb{S}}^{n-1})} &= \|\mathcal{F}f\|_{\widehat{\mathbb{S}}^{n-1}} \|f\|_{L^p(\mathbb{R}^n)}. \end{align*} $$
2.3. **Tomas' improvement.** In [116], P. Tomas improved the range of $p$ for which the estimates in (2.1) hold:

**Theorem 2.2 (Tomas).** Let $n \ge 2$ and $p \in [1, p_{TS})$ with

$$p_{TS} := \frac{2n + 2}{n + 3}.$$

The estimates in (2.1) hold for any $f \in S(\mathbb{R}^n)$.

Sketch of the proof of Theorem 2.2 for $p \in [1, p_{TS})$. The ideas in P. Tomas' proof [116] rely on a dyadic decomposition and interpolating between the boundedness $L^1 \to L^\infty$ and $L^2 \to L^2$ for each dyadic piece. These two types of boundedness are proved using (respectively) the estimates in (2.3) for $\kappa = \mathcal{F}^{-1}\sigma_{\hat{S}^{n-1}}$ and

$$(2.5) \qquad \sigma_{\hat{S}^{n-1}}(\hat{S}^{n-1} \cap B(\xi_0, r)) = \mathcal{H}^{n-1}(\hat{S}^{n-1} \cap B(\xi_0, r)) \sim r^{n-1};$$

above, $r > 0$ is small enough, $\xi_0 \in \hat{S}^{n-1}$, $B(\xi_0, r)$ denotes the Euclidean ball about $\xi_0$ with radius $r > 0$ and $\mathcal{H}^{n-1}$ the Hausdorff measure of dimension $n-1$. For more details, see Section 7 in [119]. □

In [107], E. Stein shows that the index $p_{TS}$ is achieved, using complex interpolation methods.

Let us also emphasise that a counterexample attributed to Knapp (see [111], Lemma 3) shows that the estimates in (2.1) do not hold for any $p > p_{TS}$. It is given as follows: consider $g_\delta$ ($\delta > 0$ very small) the characteristic function of a spherical cap

$$\hat{C}_\delta := \{x \in \hat{S}^{n-1} : |x \cdot e_n| < \delta\}.$$

One can prove that, as $\delta \to 0$,

$$\|g_\delta\|_{L^2(\sigma_{\hat{S}^{n-1}})} \sim \delta^{(n-1)/2}, \qquad \|\mathcal{F}^{-1}(g_\delta)\|_{L^{p'}(\mathbb{R}^n)} \ge C\delta^{n-1}\delta^{-(n+1)/p'}$$

hence the estimate for some constant $C > 0$

$$\|\mathcal{F}^{-1}(g_\delta)\|_{L^{p'}(\mathbb{R}^n)} \le C\|g_\delta\|_{L^2(\sigma_{\hat{S}^{n-1}})},$$

can hold only if $p' \ge (2n+2)/(n-1)$, that is to say if $p \le p_{TS}$. By the $TT^*$ argument, the estimates in (2.1) can hold only for $p \le p_{TS}$.

### 3. TOMAS-STEIN RESTRICTION THEOREM ON HYPER-SURFACES

Given a hyper-surface $\hat{S} \subset \mathbb{R}^n$ endowed with a smooth measure $\sigma$, the restriction problem, that has been introduced by E.-M. Stein in the seventies, asks for which pairs $(p, q)$ an inequality of the form

$$(3.1) \qquad \exists C > 0 \quad / \quad \forall f \in S(\mathbb{R}^n), \qquad \|\mathcal{F}(f)|_{\hat{S}}\|_{L^q(\hat{S}, \sigma)} \le C\|f\|_{L^p(\mathbb{R}^n)},$$

holds. The classical Tomas-Stein theorem focuses on the case $q = 2$. The measure $\sigma$ is any measure having a smooth and compactly supported density with respect to the induced measure $\sigma_{\hat{S}}$ on $\hat{S}$. Before stating the famous result, let us recall the definition of $\sigma_{\hat{S}}$ and its fundamental properties used in the proof of restriction theorems.

#### 3.1. The induced measure on a hyper-surface.
3.1.1. Definition. We consider a hyper-surface $S$ of $\mathbb{R}^n$ with $n \ge 2$. Equipping $\mathbb{R}^n$ with its Euclidean structure, $S$ inherits a Riemannian structure, the metric being obtained by restricting the Euclidean metric to the tangent space of $S$. By definition, the induced measure $\sigma_S$ is the volume measure on $S$ associated with this Riemannian metric. This generalises the notion of ‘surface area’ measures for surfaces in $\mathbb{R}^3$.

**Example 1.** Naturally, in the case of the unit sphere $\mathbb{S}^{n-1}$, the induced measure $\sigma_{\mathbb{S}^{n-1}}$ coincides with its canonical measure.

**Example 2.** If the hyper-surface is given by the graph of a smooth function $\phi : \mathbb{R}^{n-1} \to \mathbb{R}$

$$S = \{(x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R} \sim \mathbb{R}^n : x_n = \phi(x')\},$$

then the induced measure is given by:

$$\int_S f d\sigma_S = \int_{\mathbb{R}^{n-1}} f(x', \phi(x')) \sqrt{1 + |\nabla_{x'} \phi(x')|^2} dx',$$

for any function $f : \mathbb{R}^n \to \mathbb{C}$ in the space $C_c(\mathbb{R}^n)$ of continuous functions with compact support.

As a hyper-surface may be described locally as in Example 2, it follows that the estimates in (2.5) set in the context of the unit sphere generalise into

$$(3.2) \qquad \sigma_S(S \cap B(x_0, r)) \sim r^{n-1},$$

for any $x_0 \in S$ and $r > 0$ small enough.

For the subsequent applications of the restriction problems to PDE's (see Section 6), the most relevant examples are the following particular cases of Example 2:

**Example 3.** For the parabola

$$S_{par} := \{(x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R} \sim \mathbb{R}^n : x_n = |x'|^2\},$$

the induced measure is given by

$$\forall f \in C_c(\mathbb{R}^n), \qquad \int_{S_{par}} f d\sigma_{S_{par}} = \int_{\mathbb{R}^{n-1}} f(x', |x'|^2) \sqrt{1 + |x'|^2} dx.$$

Although the cone,

$$S_{cone} := \{(x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R} \sim \mathbb{R}^n : x_n = |x'|\}$$

has a singularity at 0, it is possible to define its induced measure as being given by

$$\forall f \in C_c(\mathbb{R}^n), \qquad \int_{S_{cone}} f d\sigma_{S_{cone}} = \sqrt{2} \int_{\mathbb{R}^{n-1}} f(x', |x'|) dx.$$

3.1.2. Estimates for the Fourier transform of the induced measure. A fundamental result in the analysis of restriction problems is the following:

**Theorem 3.1** ([107]). Let $S$ be a smooth hyper-surface in $\mathbb{R}^n$ with $n \ge 2$. We denote by $\sigma_S$ the induced measure on $S$. We fix $\psi \in C_c^\infty(S)$ and consider the measure given by

$$d\sigma(x) := \psi(x) d\sigma_S(x).$$

The Fourier transform of $\sigma$ is a smooth function $\hat{\sigma} \in C^\infty(\mathbb{R}^n)$. Moreover, if the Gaussian curvature of $S$ does not vanish at every point, then it satisfies the following estimate:

$$\exists C > 0 / \quad \forall \xi \in \mathbb{R}^n, \qquad |\hat{\sigma}(\xi)| \le C(1 + |\xi|)^{-\frac{n-1}{2}}.$$

In the case of $S = \mathbb{S}^{n-1}$ and $\psi = 1$, we recognise the estimate in (2.3) used in the proof of the first $L^2$-restriction theorem (Theorem 2.1).
Sketch of the proof of Theorem 3.1. As $\sigma$ is a measure with compact support, its Fourier transform $\hat{\sigma}$ is a smooth function on $\mathbb{R}^n$. The estimate is obtained by applications of stationary phase methods that we now sketch.

We may assume that the support of $\psi$ is included in a chart for $S$. Moreover, after an orthogonal change of coordinates in $\mathbb{R}^n$, we may assume that on the support of $\psi$, $S$ is described by the graph of a function as in Example 2. We then obtain

$$ \hat{\sigma}(\xi) = \int_{\mathbb{R}^{n-1}} e^{-i\phi_{\xi}(x')} \psi_1(x') dx', \quad \text{where } \psi_1(x') := \psi(x', \phi(x')) \sqrt{1 + |\nabla \phi|^2(x')}, $$

with phase given by

$$ \phi_{\xi}(x') = (x', \phi(x')) \cdot \xi = x_1\xi_1 + \dots + x_{n-1}\xi_{n-1} + \phi(x')\xi_n. $$

We write $\xi = t\tilde{\xi}$ with $t \ge 0$ and $\tilde{\xi} \in \mathbb{S}^{n-1}$, and consider $t$ large. If the direction $\tilde{\xi}$ is away from the South and North poles $(0, \dots, 0, \pm 1)$, then the non-stationary phase yields readily $|\hat{\sigma}(\xi)| \lesssim_N t^{-N} = |\xi|^{-N}$, for any $N \in \mathbb{N}$. If $\tilde{\xi}$ is near the South and North poles, then, since the Gaussian curvature of $S$ does not vanish, we may apply the stationary phase to obtain $|\hat{\sigma}(\xi)| \lesssim t^{-(n-1)/2} = |\xi|^{-(n-1)/2}$. $\square$

3.1.3. Measure theoretic description. The definition in Section 3.1.1 from differential geometry may also be described in terms of objects in measure theory, see the monographs of H. Federer [43] or P. Mattila [85]. Indeed, the induced measure $\sigma_S$ coincides with the restriction to $S$ of the Hausdorff measure $\mathcal{H}^{n-1}$ of dimension $n-1$:

$$ \sigma_S(A) = \mathcal{H}^{n-1}(A), $$

for any Borelian subset $A$ of $S$. From elementary tools in measure theory and differential geometry, in particular Example 2, it follows that, when $S$ is bounded, the restriction of $\mathcal{H}^{n-1}$ to $S$ may also be described as the weak limit of the measure

$$ \mathcal{H}^{n-1}(B \cap S) = \lim_{\epsilon \to 0} \frac{\mathrm{vol} B \cap S_{\epsilon}}{\mathrm{vol} S_{\epsilon}}, $$

for any Borelian subset $B \subset \mathbb{R}^n$; above, vol denotes the Lebesgue measure of $\mathbb{R}^n$, and $S_\epsilon$ is the $\epsilon$-tubular neighbourhood of $S$, that is, the set of points in $\mathbb{R}^n$ at distance less than $\epsilon$ to $S$. Equivalently,

$$ \sigma_S(A) = \mathcal{H}^{n-1}(A) = \lim_{\epsilon \to 0} \frac{\mathrm{vol} A_1 \cap S_{\epsilon}}{\mathrm{vol} S_{\epsilon}}, $$

for any Borelian subset $A \subset S$, having denoted by $A_1$ its $\epsilon'$-tubular neighbourhood with $\epsilon' = 1$. In the particular case where $S$ is compact with $\mathcal{H}^{n-1}(S) = 1$, the above formula simplifies as:

$$ (3.3) \qquad \sigma_S(A) = \mathcal{H}^{n-1}(A) = \lim_{\epsilon \to 0} \frac{\mathrm{vol} A_1 \cap S_{\epsilon}}{2\epsilon}. $$

We will see in Section 4.1.1 that the Fourier restriction problem on the sphere may be reformulated into a spectral problem, with an operator defined as the spectral analogue to (3.3).

3.2. Statement of the Tomas-Stein Restriction Theorem. Generalising the results obtained in the case of the unit sphere (see Theorem 2.1) and with optimal range including the Tomas-Stein index (see Section 2.3), Stein [107] proved the following result:

**Theorem 3.2** ([116, 107]). Let $\hat{S}$ be a smooth hyper-surface in $\hat{\mathbb{R}}^n$ with $n \ge 2$. We denote by $\sigma_{\hat{S}}$ the induced measure on $\hat{S}$. We fix $\psi \in C_c^\infty(\hat{S})$ and consider the measure given by

$$ d\sigma(\xi) := \psi(\xi) d\sigma_{\hat{S}}(\xi). $$
If the Gaussian curvature of $\hat{S}$ does not vanish, then the estimates in (3.1) hold for $q = 2$ and every $1 \le p \le p_{TS} = (2n+2)/(n+3)$. Consequently, the linear map $f \mapsto \mathcal{F}f|_{\hat{S}}$ defined on $S(\mathbb{R}^n)$ extends uniquely continuously $L^p(\mathbb{R}^n) \to L^2(\hat{S})$.

The methods of proofs in the case of the sphere (see Section 2.3) generalise to hyper-surfaces with non-vanishing Gaussian curvature. In particular, they rely on the estimates for the Fourier transform inverse of $\sigma$ in Theorem 3.1.

Arguing along the same lines as in Section 2, we deduce that the following estimates hold true, for any hyper-surface $\hat{S}$ satisfying the hypothesis of Theorem 3.2:

$$ (3.4) \qquad \|\mathcal{P}f\|_{L^{p'}(\mathbb{R}^n)} \le C_p \|f\|_{L^p(\mathbb{R}^n)} $$

$$ (3.5) \qquad \|Tg\|_{L^{p'}(\mathbb{R}^n)} = \|\mathcal{F}^{-1}(gd\sigma_{\hat{S}})\|_{L^{p'}(\mathbb{R}^n)} \le \sqrt{C_p}\|g\|_{L^2(\hat{S})} $$

$$ (3.6) \qquad \|T^*f\|_{L^2(\hat{S})} = \|\mathcal{F}f|_{\hat{S}}\|_{L^2(\hat{S})} \le \sqrt{C_p}\|f\|_{L^p(\mathbb{R}^n)}. $$

Naturally, the canonical example of a surface satisfying the hypotheses of Theorem 3.2 is the unit sphere $\hat{S}^{n-1}$ of $\mathbb{R}^n$, see Section 2. Other examples of hyper-surfaces such as the paraboloid whose Gaussian curvature does not vanish (see Example 3) have been the subject of a number of works. The hypothesis of curvature in Theorem 3.2 can be weakened: similar results are possible for hyper-surfaces that are not flat but with vanishing Gaussian curvature, such as the cone (see Example 3) which differs from the sphere and the paraboloid, since it has a vanishing principal curvature. In this case, the range of $p$ is smaller depending on the order of tangency of the surface to its tangent space. However, the hypothesis of non-flatness of the hyper-surface is mandatory. Indeed, consider the function $f: \mathbb{R}^n \to \mathbb{R}$ defined by

$$ (3.7) \qquad f(x) = \frac{e^{-|x'|^2}}{1 + |x_1|}, \quad x = (x_1, x') \in \mathbb{R}^n. $$

Then, one can check that $f$ belongs to $L^p(\mathbb{R}^n)$, for all $p > 1$, but its Fourier transform does not admit a restriction on the hyper-plane $\hat{S}$ of $\mathbb{R}^n$ defined by $\hat{S} = \{\xi \in \mathbb{R}^n / \xi_1 = 0\}$.

## 4. SPECTRAL VIEWPOINT

In this section, we reformulate the Fourier restriction problem on the unit sphere in terms of the spectral decomposition of the Laplace operator. This allows us to avoid relying on the Fourier transform and to make the link with the cluster estimates.

### 4.1. Spectral description of the operator $\mathcal{P}$.
In this section, we go back to the first $L^2$-restriction theorem on the unit sphere (Section 2) in order to start reformulating the problem spectrally.

We recall that the canonical measure $\sigma_{S^{n-1}}$ on $S^{n-1}$ coincides with the measure used in the polar change of coordinates:

$$ (4.1) \qquad \forall f \in S(\mathbb{R}^n), \quad \int_{\mathbb{R}^n} f(x)dx = \int_{r=0}^{+\infty} \int_{S^{n-1}} f(rx)d\sigma_{S^{n-1}}r^{n-1}dr. $$

#### 4.1.1. The operator $\mathcal{P}$.
The operator $\mathcal{P}$ defined in (2.4) may be understood in terms of the spectral decomposition of the (self-adjoint extension of the) rescaled Laplace operator

$$ (4.2) \qquad -(2\pi)^{-2}\Delta = \int_{0}^{\infty} \lambda dE_{\lambda} $$

of $\mathbb{R}^n$; for spectral decomposition see for instance [98] or the book of M. Lewin [79] and the references therein. The factor $2\pi$ is irrelevant in our analysis. It appears so that the spectral decomposition $E_\lambda$ interacts well with our (choice of normalisation for the) Fourier
transform on $\mathbb{R}^n$. The spectral projectors $E_\lambda$ are explicitly given via the Fourier transform as

$$ (4.3) \qquad E[a, b]f = \mathcal{F}^{-1}(\mathbb{1}_{[a,b]}(|\cdot|^2)\mathcal{F}(f)), \quad a < b, \ f \in L^2(\mathbb{R}^n). $$

In this particular case, spectral multipliers in $-(2\pi)^{-2}\Delta$, that is,

$$ m(-(2\pi)^{-2}\Delta) = \int_{0}^{\infty} m(\lambda) dE_{\lambda}, \quad m \in L^{\infty}(\mathbb{R}), $$

coincide with radial Fourier multipliers:

$$ (4.4) \qquad m(-(2\pi)^{-2}\Delta)f = \mathcal{F}^{-1}(m(|\cdot|^2)\mathcal{F}(f)). $$

We observe:

**Lemma 4.1.** The operator $\mathcal{P}$ defined via (2.4) coincides with $2\lim_{\varepsilon\to 0} \frac{1}{\varepsilon}E[1, 1+\varepsilon]$ in the sense that

$$ \forall f, g \in \mathcal{S}(\mathbb{R}^n), \quad (\mathcal{P}f, g)_{L^2(\mathbb{R}^n)} = 2 \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} (E_{[1,1+\varepsilon]}f, g)_{L^2(\mathbb{R}^n)}. $$

We may write

$$ \mathcal{P} = 2 \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} E[1, 1 + \varepsilon] = 2 \partial_{\lambda=1} E[0, \lambda], $$

since $E[0, \lambda_2] - E[0, \lambda_1] = E[\lambda_1, \lambda_2]$, for any $0 \le \lambda_1 \le \lambda_2$.

*Proof of Lemma 4.1.* Using the description of the spectral projector $E[1, 1+\varepsilon]$ in terms of the Fourier transform and a polar change of coordinates (see (4.3) and (4.1) respectively), we obtain:

$$ \begin{aligned} \frac{1}{\varepsilon} (E[1, 1 + \varepsilon]f, g)_{L^2(\mathbb{R}^n)} &= \frac{1}{\varepsilon} (\mathbb{1}_{[1,1+\varepsilon]}(|\cdot|^2)\mathcal{F}f, \mathcal{F}g)_{L^2(\mathbb{R}^n)} \\ &= \frac{1}{\varepsilon} \int_{\{\xi: 1 \le |\xi|^2 \le 1+\varepsilon\}} \mathcal{F}f(\xi) \overline{\mathcal{F}g}(\xi) \, d\xi \\ &= \frac{1}{\varepsilon} \int_1^{\sqrt{1+\varepsilon}} \left( \int_{\hat{\mathbb{S}}^{n-1}} \mathcal{F}f(r\omega) \overline{\mathcal{F}g}(r\omega) d\sigma_{\hat{\mathbb{S}}^{n-1}}(\omega) \right) r^{n-1} dr \end{aligned} $$

whose limit as $\varepsilon \to 0$ is

$$ \frac{1}{2} \int_{\hat{\mathbb{S}}^{n-1}} \mathcal{F}f(\omega) \overline{\mathcal{F}g}(\omega) d\sigma_{\hat{\mathbb{S}}^{n-1}}(\omega) = \frac{1}{2} (f * \mathcal{F}^{-1}\sigma_{\hat{\mathbb{S}}^{n-1}}, g)_{L^2(\mathbb{R}^n)} = \frac{1}{2} (\mathcal{P}f, g)_{L^2(\mathbb{R}^n)}. $$

□

We may equivalently define $\mathcal{P}$ as

$$ \mathcal{P} = -2 \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} E_{[1-\varepsilon, 1]}, \quad \text{or as} \quad \mathcal{P} = \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} E[1-\varepsilon, 1+\varepsilon]. $$

With the latter expression, we see that the construction for $\mathcal{P}$ is the spectral analogue of the geometric construction of the surface measure, see Section 3.1, especially (3.3).

We may similarly define

$$ \mathcal{P}_{\lambda_0} := 2\partial_{\lambda=\lambda_0} E[0, \lambda] = 2 \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} E[\lambda_0, \lambda_0 + \varepsilon] = \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} E[\lambda_0 - \varepsilon, \lambda_0 + \varepsilon], $$

for any $\lambda_0 \in \mathbb{R}$, with $\mathcal{P}_1 = \mathcal{P}$ for the case of $\lambda_0 = 1$.
4.1.2. Dilations properties. The properties of $\mathcal{P}_{\lambda_0}$ may be deduced from the properties of $\mathcal{P}$ by exploiting the 2-homogeneity of the differential operator $\Delta$,

$$ \text{i.e.} \quad \forall f \in C_c^\infty(\mathbb{R}^n), \quad \forall r > 0, \quad \Delta(f \circ \delta_r) = r^2(\Delta f) \circ \delta_r, $$

for the dilations $\delta_r: x \mapsto rx, r > 0$ on $\mathbb{R}^n$. Indeed, the 2-homogeneity of $\Delta$ implies for any $m \in L^\infty(\mathbb{R})$

$$ (4.5) \quad \forall r > 0, \quad \forall f \in L^2(\mathbb{R}^n), \quad m(-(2\pi)^{-2}\Delta)(f \circ \delta_r) = (m(-r^2(2\pi)^{-2}\Delta)f) \circ \delta_r, $$

and in particular for $m = 1_{[a,b]}$,

$$ E[a, b] = 1_{[a,b]}(-(2\pi)^{-2}\Delta) \quad \text{and} \quad E[r^{-2}a, r^{-2}b] = 1_{[a,b]}(-r^2(2\pi)^{-2}\Delta). $$

Hence, we obtain for any $\lambda_0 > 0$:

$$ \mathcal{P}_{\lambda_0} f = \frac{1}{\lambda_0} (\mathcal{P}(f \circ \delta_{\sqrt{\lambda_0}}^{-1})) \circ \delta_{\sqrt{\lambda_0}}. $$

Indeed in view of (4.5) and in the sense of Lemma 4.1, we have

$$ \begin{aligned} \mathcal{P}(f \circ \delta_{\sqrt{\lambda_0}}^{-1}) &= 2 \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} (E[\lambda_0, \lambda_0 + \varepsilon\lambda_0]f) \circ \delta_{\sqrt{\lambda_0}}^{-1} \\ &= 2 \lim_{\varepsilon' \to 0} \frac{\lambda_0}{\varepsilon'} (E[\lambda_0, \lambda_0 + \varepsilon']f) \circ \delta_{\sqrt{\lambda_0}}^{-1} \\ &= \lambda_0 (\mathcal{P}_{\lambda_0} f) \circ \delta_{\sqrt{\lambda_0}}^{-1}. \end{aligned} $$

Using the properties of the dilations on $L^p$-norms, that is,

$$ \forall f \in S(\mathbb{R}^n), \quad \forall r > 0, \quad \forall p \in [1, \infty], \quad \|f \circ \delta_r\|_{L^p(\mathbb{R}^n)} = r^{-\frac{n}{p}} \|f\|_{L^p(\mathbb{R}^n)}, $$

we obtain for any $p, q \in [1, \infty]$,

$$ (4.6) \quad \|\mathcal{P}_{\lambda_0}\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))} = \lambda_0^{-1+\frac{n}{2}(\frac{1}{p}-\frac{1}{q})} \|\mathcal{P}\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))}. $$

Recalling that with $\lambda = \varepsilon^{-1}$

$$ E[\lambda, \lambda + 1]f = (E[1, 1 + \varepsilon](f \circ \delta_{\sqrt{\lambda}}^{-1})) \circ \delta_{\sqrt{\lambda}}, $$

we also obtain (for $\lambda = \varepsilon^{-1}$)

$$ (4.7) \quad \|E[1, 1 + \varepsilon]\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))} = \lambda^{-\frac{n}{2}(\frac{1}{p}-\frac{1}{q})} \|E[\lambda, \lambda + 1]\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))}. $$

4.2. Spectral reformulation of the restriction property. The properties above imply readily that the $L^p-L^q$-boundedness of $\mathcal{P}$ is in fact equivalent to certain spectral estimates:

**Lemma 4.2.** Let $p, q \in [1, \infty]$. The operator $\mathcal{P}$ is bounded $L^p(\mathbb{R}^n) \to L^q(\mathbb{R}^n)$ if and only if

$$ (4.8) \quad \exists C > 0 / \quad \forall \lambda \ge 1, \quad \|E[\lambda, \lambda + 1]\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))} \le C\lambda^{-1+\frac{n}{2}(\frac{1}{p}-\frac{1}{q})}. $$

*Proof.* Since $\mathcal{P}_{\lambda_0} = 2\partial_{\lambda=\lambda_0}E[0, \lambda]$, we have, at least formally,

$$ E[\lambda, \lambda + 1] = \frac{1}{2} \int_{\lambda}^{\lambda+1} \mathcal{P}_{\lambda_0} d\lambda_0. $$

Therefore, if $\mathcal{P}$ is bounded $L^p(\mathbb{R}^n) \to L^q(\mathbb{R}^n)$, then we have, for all $\lambda \ge 1$,

$$ \begin{aligned} \|E[\lambda, \lambda + 1]\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))} &\le \frac{1}{2} \int_{\lambda}^{\lambda+1} \|\mathcal{P}_{\lambda_0}\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))} d\lambda_0 \\ &\le \frac{1}{2} \|\mathcal{P}\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))} \int_{\lambda}^{\lambda+1} \lambda_0^{-1+\frac{n}{2}(\frac{1}{p}-\frac{1}{q})} d\lambda_0 \asymp \lambda^{-1+\frac{n}{2}(\frac{1}{p}-\frac{1}{q})}, \end{aligned} $$
having used (4.6). Conversely, combining Lemma 4.1 with (4.7),

$$
\begin{align*}
\|\mathcal{P}\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))} &= \lim_{\varepsilon \to 0} \frac{2}{\varepsilon} \|E[1, 1 + \varepsilon]\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))} \\
&= 2 \lim_{\lambda \to +\infty} \lambda^{1-\frac{n}{2}(\frac{1}{p}-\frac{1}{q})} \|E[\lambda, \lambda+1]\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^q(\mathbb{R}^n))}
\end{align*}
$$

is bounded when (4.8) holds.

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

4.2.1. Equivalences to the restriction property. Let us summarise the equivalences to the restriction property for the sphere that we have obtained in the following statement:

**Corollary 4.3.** Let $p \in [1, +\infty)$. The following are equivalent:

(1) The restriction operator $f \mapsto \mathcal{F}f|_{\hat{S}^{n-1}}$ is bounded $L^p(\mathbb{R}^n) \to L^2(\hat{S}^{n-1})$.

(2) The operator $g \mapsto \mathcal{F}^{-1}(gd\sigma_{\hat{S}^{n-1}})$ is bounded $L^2(\hat{S}^{n-1}) \to L^p(\mathbb{R}^n)$.

(3) The operator $\mathcal{P}$ is bounded $L^p(\mathbb{R}^n) \to L^{p'}(\mathbb{R}^n)$.

(4) There exists $C > 0$ such that for any $\lambda \ge 1$, we have:

$$
\|E[\lambda, \lambda + 1]\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^{p'}(\mathbb{R}^n))} \le C\lambda^{-1+n(\frac{1}{p}-\frac{1}{2})},
$$

or equivalently

$$
\|E[\lambda, \lambda + 1]\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^2(\mathbb{R}^n))} \le \sqrt{C}\lambda^{-\frac{1}{2}+\frac{n}{2}(\frac{1}{p}-\frac{1}{2})},
$$

or equivalently

$$
\|E[\lambda, \lambda + 1]\|_{\mathcal{L}(L^2(\mathbb{R}^n), L^{p'}(\mathbb{R}^n))} \le \sqrt{C}\lambda^{-\frac{1}{2}+\frac{n}{2}(\frac{1}{p}-\frac{1}{2})}.
$$

Moreover, by the Tomas-Stein theorem (Theorem 3.2), the sharp range for Part (1) above
is $1 \le p \le p_{TS} = (2n + 2)/(n + 3)$.

*Proof of Corollary 4.3.* The equivalence follows readily from Lemma 4.2 together with the comments on the proof of Theorem 2.1 at the end of Section 2.2 and the application of the *TT\** argument to $E[\lambda, \lambda + 1] = E[\lambda, \lambda + 1]^2$. $\square$

Instead of considering $-\Delta$, we can obtain similar estimates as in Part (4), but for $\sqrt{-\Delta}$ instead of $-\Delta$ by modifying the arguments above (a more direct reasoning using the Fourier transform is explained at the beginning of Chapter 5 in [103]):

**Corollary 4.4.** For any $1 \le p \le p_{TS}$, there exists $C > 0$ such that for any $\lambda \ge 1$, we have

$$
\| \mathbb{1}_{[\lambda, \lambda+1]}(\sqrt{-\Delta}) \|_{\mathcal{L}(L^p(\mathbb{R}^n), L^2(\mathbb{R}^n))} \le C\lambda^{\delta(p)}, \quad \delta(p) := n\left(\frac{1}{p} - \frac{1}{2}\right) - \frac{1}{2}.
$$

We have equivalently,

$$
\| \mathbb{1}_{[\lambda, \lambda+1]}(\sqrt{-\Delta}) \|_{\mathcal{L}(L^2(\mathbb{R}^n), L^{p'}(\mathbb{R}^n))} \le C \lambda^{\delta(p)},
$$

or

$$
\| \mathbb{1}_{[\lambda, \lambda+1]}(\sqrt{-\Delta}) \|_{\mathcal{L}(L^p(\mathbb{R}^n), L^{p'}(\mathbb{R}^n))} \le C^2 \lambda^{2\delta(p)}.
$$

*Proof of Corollary 4.4.* The equivalence comes from the *TT\** argument. Hence, it suffices to prove the *L*^p - *L*^{p\'*'}-inequality. We write

$$
\begin{align*}
\mathbb{1}_{[\lambda, \lambda+1]}(\sqrt{-\Delta}) &= \mathbb{1}_{[\lambda^2, (\lambda+1)^2]}(-\Delta) = E[\lambda^2, (\lambda+1)^2] \\
&= \int_{\lambda^2}^{(\lambda+1)^2} \partial_{\lambda_0} E[0, \lambda_0] d\lambda_0 = \frac{1}{2} \int_{\lambda^2}^{(\lambda+1)^2} \mathcal{P}_{\lambda_0} d\lambda_0,
\end{align*}
$$
so using (4.6)

$$
\begin{align*}
& \|1_{[\lambda, \lambda+1]}(\sqrt{-\Delta})\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^{p'}(\mathbb{R}^n))} \\
& \le \frac{1}{2} \int_{\lambda^2}^{(\lambda+1)^2} \|\mathcal{P}_{\lambda_0}\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^{p'}(\mathbb{R}^n))} d\lambda_0 \\
& = \frac{1}{2} \|\mathcal{P}\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^2(\mathbb{R}^{p'}))} \int_{\lambda^2}^{(\lambda+1)^2} \lambda_0^{-1+\frac{n}{2}(\frac{1}{p}-\frac{1}{p'})} d\lambda_0 \\
& \asymp \|\mathcal{P}\|_{\mathcal{L}(L^p(\mathbb{R}^n), L^{p'}(\mathbb{R}^2))} \lambda^{n(\frac{1}{p}-\frac{1}{p'})-1}.
\end{align*}
$$

We recognise $n(\frac{1}{p} - \frac{1}{p'}) - 1 = 2\delta(p)$. We conclude with $\mathcal{P}$ being $L^p \to L^{p'}$-bounded for $1 \le p \le p_{TS}$ (see Corollary 4.3). $\square$

Proceeding as for the proof of Corollary 4.3, we can show that any of the equivalent estimates in Corollary 4.4 is equivalent to the Tomas-Stein restriction theorem in Theorem 3.2. Moreover, the range in $p$ is sharp.

4.3. **Links with the cluster estimates.** We can consider similar estimates as in Part (4) of Corollary 4.3 or in Corollary 4.4 in other contexts, that is, the following estimates

$$
(4.9) \quad \exists C > 0 \ / \ \forall \lambda \ge 1, \quad \|1_{[\lambda, \lambda+1]}(\mathcal{L})\|_{\mathcal{L}(L^p(M), L^q(M))} \le C \lambda^{\gamma_{p,q}},
$$

for some indices $p, q \in [1, \infty]$ and exponents $\gamma_{p,q} \in \mathbb{R}$; expressing these estimates necessi-
tates a non-negative self-adjoint operator $\mathcal{L}$ on $L^2(M)$ and a measurable space $M$ equipped
with a sigma-finite positive measure. This may be studied in settings different from $\mathbb{R}^n$,
and no meaning needs to be attached to the operator $\mathcal{P}$ or to the restriction operator. In
particular, the existence of a Fourier transform is not required. Furthermore, the space $M$
need not be even a manifold but e.g. a tree or a graph with $\mathcal{L}$ being the graph Laplacian.

Roughly speaking, the estimates in (4.9) measure the density or the distribution of $\mathcal{L}$-eigenvalues in a window of length 1 in the $\mathcal{L}(L^p, L^q)$-norm. When the spectrum is discrete, taking $q=2$, they yield $L^p$-estimates of $\mathcal{L}$-eigenfunctions, and furthermore, estimates for the $L^p - L^2$-boundedness of the corresponding spectral projectors, see (4.15) below in the case of a compact Riemannian manifold. Variations of the estimates in (4.9) may be considered, for instance for other spaces than $L^r(M)$, $r=p,q$, or different windows from $[\lambda, \lambda+1]$, see e.g. the review of P. Germain [58]. These types of estimates are often called cluster estimates.

The most notable example of cluster estimates are set on a compact Riemannian mani-
fold M equipped with the volume element. In this case, the Laplace-Beltrami operator L
on a compact Riemannian manifold has compact resolvent, so its spectrum is discrete
and each eigenvalue has finite multiplicity. The spectral decomposition of Lsqrt; may be
described very concretely in terms of a chosen orthonormal basis of L-eigenfunctions. The
corresponding cluster estimates have been proved by C. Sogge on the sphere and then on
any compact Riemannian manifold [103]:

**Theorem 4.5.** Let $(M, g)$ be a compact Riemannian manifold of dimension $n \ge 2$. We keep the same notation for the Laplace-Beltrami operator $\mathcal{L}$ and its self-adjoint extension on $L^2(M)$. The operator $\sqrt{\mathcal{L}}$ satisfies the cluster estimates as in (4.9) with $q = 2$, $p \in [1, 2]$ and

$$
\gamma_{p,2} := \begin{cases} n \left(\frac{1}{p} - \frac{1}{2}\right) - \frac{1}{2} & \text{if } 1 \le p \le p_{TS} = \frac{2n+2}{n+3}, \\ \frac{n-1}{2} \left(\frac{1}{p} - \frac{1}{2}\right) & \text{if } p_{TS} \le p \le 2. \end{cases}
$$

Furthermore, these estimates are sharp in the sense that, for each $p \in [1, 2]$, there exist
compact Riemannian manifolds for which the exponent $\gamma_{p,2}$ is sharp.
Theorem 4.5 means that for any $p \le 2$, there exists a constant $C > 0$ such that

$$ (4.10) \quad \forall \lambda \ge 1, \quad \forall f \in L^p(M), \quad \|1_{[\lambda, \lambda+1]}(\sqrt{\mathcal{L}})f\|_{L^2(M)} \le C\lambda^{\gamma_{p,2}}\|f\|_{L^p(M)}, $$

with exponents $\gamma_{p,2}$ given in the statement. In particular, the critical index is again $p_{TS} = \frac{2n+2}{n+3}$. The regimes $p \in [1, p_{TS}]$ and $p \in [p_{TS}, 2]$ are sometimes called *point-focusing* and *geodesic-focusing* respectively. The exponents in the point-focusing regime, that is,

$$ \gamma_{p,2} = n \left( \frac{1}{p} - \frac{1}{2} \right) - \frac{1}{2} = \delta(p), \quad 1 \le p \le p_{TS}, $$

are the same as for the cluster estimates on $\mathbb{R}^n$ for $\sqrt{-\Delta}$, see Corollary 4.4.

Before sketching elements of the proof, we observe that, by the $TT^*$ argument (see Appendix A), the cluster estimates in (4.10) are equivalent to

$$ (4.11) \quad \forall g \in L^2(M), \quad \|1_{[\lambda, \lambda+1]}(\sqrt{\mathcal{L}})g\|_{L^{p'}(M)} \le C\lambda^{\gamma_{p,2}}\|g\|_{L^2(M)} $$

$$ (4.12) \quad \forall f \in L^p(M), \quad \|1_{[\lambda, \lambda+1]}(\sqrt{\mathcal{L}})f\|_{L^{p'}(M)} \le C^2 \lambda^{2\gamma_{p,2}} \|f\|_{L^p(M)}. $$

*Elements of the proof of Theorem 4.5.* The first idea of the proof consists in establishing (4.10) for $p = 1, 2$, using functional analysis. Indeed, we readily have that

$$ (4.13) \quad \|1_{[\lambda, \lambda+1]}(\sqrt{\mathcal{L}})\|_{\mathcal{L}(L^2(M))} \le 1, $$

and combining the spectral theory for unbounded self-adjoint operators on compact Riemannian manifold and the point-wise sharp Weyl law, it follows that

$$ (4.14) \quad \|1_{[\lambda, \lambda+1]}(\sqrt{\mathcal{L}})\|_{\mathcal{L}(L^1(M), L^2(M))} \lesssim \lambda^{\frac{n-1}{2}}. $$

Let us give a few more details for the proof of (4.14): for any function $\psi: \mathbb{R} \to \mathbb{C}$, we have

$$ \|\psi(\sqrt{\mathcal{L}})\|_{\mathcal{L}(L^1(M), L^2(M))} \le \sup_{x \in M} |K_{\psi}(x, x)|, $$

using functional properties of the integral kernel $K_{\psi}$ of the operator $\psi(\sqrt{\mathcal{L}})$ and the spectral decomposition of $\sqrt{\mathcal{L}}$. The estimate for the kernel for $\psi = 1_{[\lambda, \lambda+1]} = 1_{[0, \lambda+1]} - 1_{[0, \lambda]}$ may be found in [103, Lemma 4.2.4] or can be deduced from the point-wise sharp Weyl law in compact manifolds [104]. This shows (4.14).

The second idea which will complete the proof of the theorem, is to interpolate the two estimates (4.13)-(4.14) corresponding respectively to $p = 2$ and $p = 1$ with the estimate for the case $p = p_{TS}$. However in this setting, the study of the case $p = p_{TS}$ requires to first replace $1_{[\lambda, \lambda+1]}(\sqrt{\mathcal{L}}) = 1_{[0,1]}(\sqrt{\mathcal{L}} - \lambda)$ with a smooth approximant $\chi(\sqrt{\mathcal{L}} - \lambda)$ for some $\chi \in S(\mathbb{R})$, and then to perform a deep study of certain oscillatory integral operators. See [103] for the complete proof. □

A consequence of the cluster estimates in the form (4.11) is that if $g$ is an eigenfunction for $\mathcal{L}$ with eigenvalue $\mu$, then $g$ is also an eigenfunction for $\sqrt{\mathcal{L}}$ but with eigenvalue $\lambda = \sqrt{\mu}$, and we have

$$ \|1_{[\lambda, \lambda+1]}(\sqrt{\mathcal{L}})g\|_{L^{p'}(M)} = \|g\|_{L^{p'}(M)} \le C\lambda^{\gamma_{p,2}}\|g\|_{L^2(M)}. $$

Consequently, the orthogonal projector $\Pi_{\mu} := 1_{\{\mu\}}(\mathcal{L}) = 1_{\{\sqrt{\mu}\}}(\sqrt{\mathcal{L}})$ onto the $\mu$-eigenspace for $\mathcal{L}$ is bounded $L^2 \to L^{p'}$ with operator norm

$$ \|\Pi_{\mu}\|_{\mathcal{L}(L^2(M), L^{p'}(M))} \le C\mu^{\gamma_{p,2}/2}. $$

By duality, we also have:

$$ (4.15) \quad \|\Pi_{\mu}\|_{\mathcal{L}(L^p(M), L^2(M))} \le C\mu^{\gamma_{p,2}/2}. $$
5. RESTRICTION THEOREMS ON LIE GROUPS AND HOMOGENEOUS DOMAINS

Here, we discuss the restriction and cluster estimates in the setting of Lie groups and homogeneous domains. In the compact case, the case of a Laplace operator invariant under the action of the group is covered by C. Sogge's results on cluster estimates on compact manifold (see Theorem 4.5). However, more can be said in the case of the torus (see Section 6.4) and a different question may be asked for instance for nilpotent Lie groups equipped with sub-Laplacians.

5.1. **A few words on the analysis on Lie groups and homogeneous domains.**
Here, we consider (real, finite dimensional) Lie groups and their homogeneous domains.
Recall that by definition, homogeneous domains are quotients of Lie groups by closed
subgroups.

5.1.1. *First examples of Lie groups and homogeneous domains.* The Euclidean space $\mathbb{R}^n$ or the torus $\mathbb{T}^n$ are naturally commutative Lie groups. The torus may also be viewed as a homogeneous domain since it is the quotient $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$. Intuitively, homogeneous domains are manifolds that are highly symmetrical. Mathematically, this means that a large group acts on them. Examples include the unit sphere

$$
\mathbb{S}^{n-1} = \{ x = (x_1, \dots, x_n) \in \mathbb{R}^n : |x|^2 = x_1^2 + \dots + x_{n-1}^2 + x_n^2 = 1 \}, n \ge 2,
$$

and the hyperbola

$$
S_{\text{hyperbol}} = \{ x = (x_1, \dots, x_n) \in \mathbb{R}^n : x_1^2 + \dots + x_{n-1}^2 - x_n^2 = 1 \}, n \ge 2.
$$

The natural groups acting on them are, respectively, the groups $O(n)$ and $O(n-1, 1)$ of orthogonal and Lorentz transformations. Recall that orthogonal and Lorentz transformations are, by definition, the linear transformations preserving the quadratic forms $|x|^2 = x_1^2 + \dots + x_{n-1}^2 + x_n^2$ and $x_1^2 + \dots + x_{n-1}^2 - x_n^2$ (respectively).

5.1.2. *Definition and further examples.* By definition, a Lie group $G$ is a set equipped with compatible structures as a smooth manifold and as a group. In other words, the smooth manifold $G$ is also equipped with the group operations of multiplication $(x, y) \mapsto xy$ and inverses $x \mapsto x^{-1}$ which are assumed to be smooth ($G \times G \to G$ and $G \to G$ respectively).

**Example 4.** *The general linear group GL(n, R) of n × n invertible matrices over the reals is a Lie group. For n > 1, it is non-commutative.*

In fact, any Lie group is locally isomorphic to a matrix group, that is, a closed subgroup of GL($\mathbb{R}$, $n$) for $n$ large enough.

5.1.3. *Types of Lie groups by examples.* Matrix groups such as the orthogonal group $O(n)$, the unitary group, or the symplectic group together with their non-compact counterparts (e.g. $O(n-1,1)$) belong to the realm of *semisimple* or more generally *reductive* Lie groups [65, 67, 117]. They are crucial in describing natural homogeneous domains such as the sphere (viewed as the quotient $O(n)/O(n-1)$ or equivalently $SO(n)/SO(n-1)$) or the hyperbola $S_{\text{hyperbol}}$ (viewed as the quotient $O(n-1,1)/O(n-1)$). Moreover, their Lie algebras are fundamental objects in mathematical physics, especially for particle physics.

Another important class of Lie groups consists of the *solvable* Lie groups.

**Example 5.** *The group of affine transformations on $\mathbb{R}$ given by $x \mapsto ax+b$ with $a \in \mathbb{R} \setminus \{0\}$ and $b \in \mathbb{R}$ is a solvable Lie group that is not nilpotent. It is often called the AX+B group, and may be also realised as the matrix group:*

$$
\left\{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} : a \in \mathbb{R} \setminus \{0\}, b \in \mathbb{R} \right\}.
$$
A significant subclass among solvable Lie groups consists of the *nilpotent* Lie groups. The latter are often assumed connected and simply connected without a real loss of generality [34]. In this case, they can be described as the closed subgroups of the matrix group

$$ \begin{pmatrix} 1 & * & * \\ 0 & \ddots & * \\ 0 & 0 & 1 \end{pmatrix}. $$

Equivalently, they may be viewed as the group $\mathbb{R}^n$ equipped with a polynomial law [95].

**Example 6.** Perhaps the most famous examples of nilpotent Lie groups are the Heisenberg groups $\mathbb{H}_d$, with underlying manifold $\mathbb{R}^{2d+1}$. The three dimensional Heisenberg group may be realised as the matrix group

$$ (5.1) \qquad \mathbb{H}_1 \sim \left\{ \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix} : a, b, c \in \mathbb{R} \right\}. $$

Because of its relation to complex analysis, the Heisenberg group $\mathbb{H}_d$ is often realised as $\mathbb{R}^{2d+1}$ equipped with the following group law:

$$ (5.2) \qquad (Y, s)(Y', s') = (Y + Y', s + s' + 2(\langle y, \eta' \rangle - \langle y', \eta \rangle)), \\ \text{where } (Y, s) = (y, \eta, s) \text{ and } (Y', s') = (y', \eta', s') \text{ are in } \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R} \sim \mathbb{R}^{2d+1}. $$

5.1.4. Technical definitions. Let us briefly present the technical definitions of the notions sketched above:

**Definition 5.1.** A (connected) Lie group $G$ is reductive, semisimple, solvable or nilpotent when its Lie algebra $\mathfrak{g}$ is so.

(1) A Lie algebra is nilpotent if any finite number $s$ of iterated Lie brackets vanishes. In fact, there exists a smallest number $s$ valid for any nested Lie brackets; this is called the step of nilpotency of $\mathfrak{g}$.

Equivalently, a Lie algebra $\mathfrak{g}$ is nilpotent when its lower central series eventually reaches zero. The lower central series is defined with $\mathfrak{g}^1 = \mathfrak{g}$ and recursively $\mathfrak{g}^{k+1} = [\mathfrak{g}, \mathfrak{g}^k]$ being the linear span of elements $[V, W], V \in \mathfrak{g}$ and $W \in \mathfrak{g}^k$.

(2) A Lie algebra is solvable when its derived series eventually reaches zero. The derived series is defined with $\mathfrak{g}^{(0)} = \mathfrak{g}$ and recursively $\mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}]$ being the linear span of elements $[V, W], V, W \in \mathfrak{g}^{(k)}$.

(3) A Lie algebra is semi-simple when it has no-non zero abelian ideals.

(4) A Lie algebra is reductive when it can be written as the direct sum of its centre with a semi-simple Lie algebra.

The classifications of semi-simple Lie algebras and of the homogeneous domains of the semisimple Lie groups (symmetric domains) are well known [67]. This is a beautiful and elegant theory connected to the Killing form, Dynkin diagrams, root space decompositions, the Serre presentation, the theory of highest weight representations, the Weyl character formula for finite-dimensional representations, etc. By contrast, if it is possible to classify the nilpotent Lie algebra of dimension $\le 6$, in dimension 7, it is possible to identify families of non-isomorphic Lie algebras parametrised by real numbers [83].

Certain types of nilpotent Lie groups are relevant to control theory and sub-Riemannian geometry:

**Definition 5.2.** A Lie algebra is stratified when it can be decomposed as

$$ \mathfrak{g} = \mathfrak{g}_1 \oplus \dots \oplus \mathfrak{g}_s \quad \text{with} \quad [\mathfrak{g}_i, \mathfrak{g}_j] = \mathfrak{g}_{i+j}, $$

with the convention that $\mathfrak{g}_k = 0$ for $s > k$.
In this case, the Lie algebra is nilpotent, and the corresponding connected simply con-
nected Lie group is said to be stratified. For instance, the Heisenberg group is naturally
stratified. After a choice of an inner product or a basis on the first stratum $g_1$, a stratified
group is said to be *Carnot*.

5.1.5. *Solvmanifolds*. The homogeneous domains for solvable and nilpotent Lie groups are called solvmanifolds and nilmanifolds respectively.

**Example 7.** *The Möbius strip is a solvmanifold as it is the quotient of the group $AX+B$ (see Example 5) by the subgroup formed by the transformations corresponding to $a = \pm 1$ and $b = 0$.*

**Example 8.** *The canonical three dimensional Heisenberg nilmanifold is the matrix group in (5.1) quotiented by the matrix subgroup corresponding to integer coefficients. This may be described as an $S^1$-bundle over $T^2$.*

Not every nilpotent Lie group may be quotiented into a compact nilmanifold (see [34, Chapitre 5] and references therein). However, it is possible on the Heisenberg group and more generally nilpotent Lie groups with two-step of nilpotency, such as in [48].

5.1.6. *Analysis on compact Lie groups*. Global and Harmonic analysis on compact (hence reductive) Lie groups and their homogeneous domains is now well-understood [67]. Moreover, many results initially obtained in these settings have proven to be generalisable to compact manifolds without an underlying group structure. A notable example of this phenomenon is for instance the cluster estimates (see Theorem 4.5): although it was first proved by C. Sogge for the Laplace-Beltrami operator on the sphere using techniques of harmonic analysis [102], Sogge has generalised the result to elliptic pseudo-differential operators [103]. Naturally, many sophisticated questions may still use the underlying structure of Lie groups or homogeneous domains, for instance in the case of the torus (see Section 6.4) or general compact Lie groups, see e.g. [49] or [99].

The global analysis of non-compact Lie groups is more intricate for reasons explained
in the next paragraph.

5.1.7. *Convolution operators*. A Lie group $G$ is equipped with positive measures that are invariant under translation, meaning multiplication on the left or on the right by any group element. These measures are known as the (left or right) Haar measures. Up to a constant, a left (resp. right) Haar measure is unique. This leads to the definition of $L^p$-spaces on $G$, as well as convolution functions and operators.

On $\mathbb{R}^n$, the analysis of convolution operators is well-understood through the theory of singular integral operators developed by A.-P. Calderón and A. Zygmund [27] in the decades around the 1950s, and also the early works of Elias Stein [105]. This theory extends readily to operators with integral kernels (not necessarily of convolution) and can be adapted to other settings [31], including Lie groups and their homogeneous domains. However, this adaptation depends on the behaviour of the Haar measure. The growth of the Haar measure of a (connected) Lie group is either polynomial or exponential - see [62] for a precise statement of this property. Examples of Lie groups with polynomial growth of the volume include compact and nilpotent Lie groups, while non-compact reductive Lie groups and certain solvable Lie groups such as the $AX+B$ group have exponential growth of the volume. The adaptation of the Calderón-Zygmund theory to Lie groups has become well understood in the polynomial growth case [2, 41], while the case of exponential growth is technically challenging [66].
5.1.8. *Invariant Laplace operators.* Other operators that are naturally considered on Lie groups are differential operators invariant under left or right translations. To fix the ideas, let us consider invariance under left translations. The Lie algebra $\mathfrak{g}$ of the Lie group $G$ may be viewed as the Lie algebra of left-invariant vector fields.

**Example 9.** The following vector fields on $\mathbb{R}^{2d+1}$

$$ \mathcal{X}_j := \partial_{y_j} + 2\eta_j \partial_s \quad \text{and} \quad \Xi_j := \partial_{\eta_j} - 2y_j \partial_s, \quad j = 1, \dots, d, $$

are invariant under the left translations of the Heisenberg group $\mathbb{H}_d$. Together with $\partial_s$, they form the canonical basis of the Lie algebra of $\mathbb{H}_d$. They satisfy the following Canonical Commutation Relations mentioned in the introduction:

$$ [\mathcal{X}_j, \Xi_{\Xi_j}] = -4\partial_s, \quad [\mathcal{X}_j, \Xi_k] = 0 \text{ for } j \neq k. $$

(The non-zero factor $-4$ is irrelevant.)

The underlying vector space of the Lie algebra $\mathfrak{g}$ of $G$ is naturally isomorphic to the tangent space of $G$ at its neutral element. Any left-invariant differential operator may be written as a non-commutative polynomial in a basis of $\mathfrak{g}$, in a unique way once an ordering of the basis is fixed [67].

In the case of a compact Lie group $G$, it is natural to consider the left-invariant Laplace operator defined as follows: we equip the Lie algebra $\mathfrak{g}$ with a $G$-invariant inner product, yielding a left-invariant Riemannian metric on $G$ as well as a left-invariant Laplace-Beltrami operator given by $-(X_1^2 + \dots + X_n^2)$ where $X_1, \dots, X_n$ are left-invariant vector fields that form an orthonormal basis of $\mathfrak{g}$. A similar construction is often considered on symmetric spaces (i.e. the suitable compact and non-compact quotients of semisimple Lie groups, see [67]) such as the sphere $S^{n-1}$ and the hyperbola $S_{\text{hyperbol}}$.

5.1.9. *Sub-Laplacians.* An important class of differential operators related to the analysis on Lie groups are sub-Laplacians. By sub-Laplacians, here, we mean the sum of squares $X_1^2 + \dots + X_n^2$ of vector fields $X_1, \dots, X_n$ of a manifold $M$ that generate the entire tangent space at every point of $M$ by linear combinations of iterated nested commutators. This hypothesis on the family of vector fields is often referred to as the Hörmander condition, as L. Hörmander [69] showed in 1967 that it implies the hypoellipticity of sub-Laplacians, see also [110]. This was the motivation behind the research programme led by Folland and Stein and their collaborators around the 1980s, see e.g. [51, 97], on subelliptic operators modelled via nilpotent Lie groups, in particular on Carnot groups (see Section 5.1.4) which are naturally equipped with a canonical sub-Laplacian. This is related to non-holonomic geometries, often called *sub-Riemannian* since the 1990's, with applications in optimal control, image processing and biology (eg human vision), see for instance the book by A. Agrachev, D. Barilari, and U. Boscain [1]. It has rich motivations and ramifications in several parts of mathematics: the analysis of hypoelliptic PDEs and stochastic (Malliavin) calculus but also geometric measure theory, metric group theory, etc., see e.g. [77].

5.1.10. *$L^2$-theory.* The starting point of many questions in analysis is an $L^2$-decomposition, be it as a hypothesis in singular integral theory or from Fourier analysis or as a consequence of the self-adjointness of an operator.

On $\mathbb{R}^n$ and $\mathbb{T}^n$, $L^2$-decompositions are obtained readily according to the Plancherel formula via the Euclidean Fourier transform and the Fourier series. The latter are strongly related to their group structure and can be generalised on many topological groups. The case of compact groups was proved by F. Peter and H. Weyl [94] in 1927 (see also [106]), and provides a decomposition of the $L^2$-space analogous to the one provided by Fourier series: the sum is over all the irreducible representation modulo equivalence. In the 1960's, this was further generalised by J. Dixmier [40] to a very large class of topological
groups (more technically: locally compact, unimodular, type I) and provides a Fourier transform in terms of the (unitary irreducible) representations of the group. It provides not only a decomposition of the $L^2$-space but also an understanding of Fourier multipliers, or equivalently, the operators that are invariant under (e.g. left) translations and bounded on $L^2$. Although J. Dixmier's results are very general and abstract, they can be made very concrete on groups whose representation theory is very explicit, such as semisimple Lie groups via weight theory, and nilpotent Lie groups with the orbit method [76].

Considering operators that are self-adjoint will also provide an $L^2$-decomposition via their spectral decomposition $E(\lambda)$, see e.g. [98]. This leads to the definition of spectral multipliers, and is of particular interest for sub-Laplacians (see e.g. [2]).

## 5.2. Restriction theorem on the Heisenberg group.

### 5.2.1. The setting.
Realising the Heisenberg group $\mathbb{H}_d$ as in Example 6 with group law given in (5.2), we define the following differential operator:

$$ (5.3) \qquad \Delta_{\mathbb{H}_d} u := \sum_{j=1}^{d} (\chi_j^2 u + \Xi_j^2 u), $$

where the vector fields $\chi_j$ and $\Xi_j$ were defined in Example 9. Note that this choice of vector fields on the first stratum of the Heisenberg Lie algebra equipped $\mathbb{H}_d$ with its natural structure of Carnot group (see Section 5.1.4) for which $\Delta_{\mathbb{H}_d}$ is its canonical sub-Laplacian. The operator $\Delta_{\mathbb{H}_d}$ is homogeneous of order two for the anisotropic dilations

$$ (5.4) \qquad \delta_r(Y, s) = (rY, r^2s). $$

These dilations are automorphisms of the group $\mathbb{H}_d$ - unlike the isotropic ones.

### 5.2.2. Müller's result.
The operator $-\Delta_{\mathbb{H}_d}$ is non-negative and essentially self-adjoint on $C_c^\infty(\mathbb{H}_d) \subseteq L^2(\mathbb{H}_d)$, and it is also invariant under left-translation. From the $L^2$-theory viewpoint (see Section 5.1.10), this implies that its spectral decomposition $E(\lambda)$ may be expressed in terms of the Fourier decomposition of the Heisenberg group, see [50] and [8, 9]. Consequently, $E[\alpha, \beta]$ may be described with special functions connected to the representation theory of $\mathbb{H}_d$. D. Müller has given explicit formulae for these in [92]. In the same paper, he studies the restriction theorem on the sphere of the Heisenberg group: as emphasised in Section 4.1.1, this amounts in studying the analogue of the operator $2\mathcal{P}$ formally defined as

$$ 2\mathcal{P} = \partial_{\lambda=1} E[0, \lambda]. $$

In particular, he shows that this operator is a convolution operator whose convolution kernel is a tempered distribution formally given by

$$ (5.5) \qquad G(Y, s) = \frac{2^d}{\pi^{d+1}} \sum_{m \in \mathbb{N}^d} \frac{1}{(2|m| + d)^{d+1}} \cos\left(\frac{s}{2|m| + d}\right) \mathcal{W}\left(m, m, 1, \frac{Y}{\sqrt{2|m| + d}}\right), $$

with $Y = (y, \eta) \in \mathbb{R}^d \times \mathbb{R}^d$ and where $\mathcal{W}$ denotes the Wigner transform of the (renormalized) Hermite functions

$$ (5.6) \qquad \mathcal{W}(m, m, \lambda, Y) := \int_{\mathbb{R}^d} e^{2i\lambda\langle\eta, z\rangle} H_{m,\lambda}(y+z) H_{m,\lambda}(-y+z) dz. $$

Here $H_{m,\lambda}$ stands for the renormalized Hermite function on $\mathbb{R}^d$, namely (for further details, see [55, 84])

$$ H_{m,\lambda}(x) := |\lambda|^{\frac{d}{4}} H_m(|\lambda|^{\frac{1}{2}}x), $$

with $(H_m)_{m \in \mathbb{N}^d}$ the Hermite orthonormal basis of $L^2(\mathbb{R}^d)$ given by the eigenfunctions of the harmonic oscillator:

$$ -(\Delta - |x|^2)H_m = (2|m| + d)H_m, $$
specifically

$$ (5.7) \qquad H_m := \left( \frac{1}{2^{|m|}m!} \right)^{\frac{1}{2}} \prod_{j=1}^{d} \left( -\partial_j H_0 + x_j H_0 \right)^{m_j}, $$

with $H_0(x) := \pi^{-d/4}e^{-|x|^2/2}$, $m! := m_1! \cdots m_d!$ and $|m| := m_1 + \cdots + m_d$.

The formula in (5.5) should be compared with its analogue $\kappa$ on $\mathbb{R}^n$ given in (2.2). From (5.5), we observe that the behaviour in $s$ is almost transport-like. In fact, D. Müller in the proof of [92, Proposition 3.1] constructs a Schwartz function $f \in S(\mathbb{H}_d)$ such that

$$ 2\mathcal{P}f(Y, s) = e^{-\frac{|Y|^2}{d}} \cos \frac{s}{d}. $$

This is an obstruction to any $L^p \to L^q$-boundedness for $\mathcal{P}$. It is also related to the Bahouri-Gérard-Xu counter-example to the Schrödinger propagation for $\Delta_{\mathbb{H}_d}$, see page 96 in [9]. However, D. Müller proves also the boundedness of $\mathcal{P}$ for the anisotropic Lebesgue spaces

$$ L_Y^p L_s^q = L^p(\mathbb{R}^{2d} : L^q(\mathbb{R})), $$

namely the boundedness $L_Y^p L_s^1 \to L_Y^{p'} L_s^{\infty}$:

$$ \|\mathcal{P}f\|_{L_Y^{p'} L_s^{\infty}} \le C \|f\|_{L_Y^p L_s^1}, \quad 1 \le p < 2. $$

This has been re-interpreted in terms of a Fourier restriction theorem for the group Fourier transform of the Heisenberg group [5], then adapted to suitable hyper-surfaces in the setting of $\mathbb{R} \times \mathbb{H}_d$, see further discussion in Section 6.2.1.

Note that restriction issues has been also investigated in a few other sub-elliptic frameworks, see for instance V. Casarino-P. Ciatti [28], H. Liu-M. Song [81, 82] and D. Barilari-S. Flynn [13].

## 6. RELATED ISSUES AND OPEN QUESTIONS

Restriction problems are at the intersection of many areas of mathematics such as Harmonic Analysis, Spectral and Geometry Theories, with a broad range of applications covering PDEs, Number Theory, Probability Theory, etc. This subject is strongly linked with many questions that are still largely opened, hence giving a complete survey to all these connexions is beyond the scope of the present text. However, we will present in the first part of this section some applications of Tomas-Stein inequalities (in well-known frameworks) in the fields of PDEs and Spectral Theory. Then, we will close the text by highlighting the interplay between restriction problems and Number Theory.

**6.1. Related questions in harmonic analysis.** Fourier restriction problems are deeply connected to two other conjectures central to Euclidean harmonic analysis, and known as Kakeya and Bochner-Riesz. The Kakeya conjecture states that each Kakeya set in $\mathbb{R}^n$, that is to say a set containing a unit line in every direction, has Minkowski and Hausdorff dimension equal to $n$. This was proved by R. Davies [35] in the two dimensional case in 1971, and very recently by H. Wang and J. Zahl [118] in the three dimensional case; for insights about the Kakeya conjecture, we refer the reader to [37, 112, 118].

The Bochner-Riesz conjecture (recalled below) is a significant problem in harmonic analysis, focusing on the convergence and boundedness in $L^p$ of the Bochner-Riesz means. It has been proved by L. Carleson and P. Sjölin [29] in the two dimensional case, but remains open in higher dimensions. As regards to its link with restriction problems, we refer the interested reader to the survey of T. Tao [112] and the references therein.

The Bochner-Riesz conjecture arose in the study of the ball multipliers at frequency $R$

$$ \mathbb{1}_{B(0,R)}(D)f := \mathcal{F}^{-1}(\mathbb{1}_{B(0,R)}\hat{f}), \quad f \in S(\mathbb{R}^n). $$
In other words,

$$1_{B(0,R)}(D)f(x) = \int_{B(0,R)} e^{2\pi i x \cdot \xi} \hat{f}(\xi) d\xi, \quad x \in \mathbb{R}^n.$$

According to Plancherel's formula, the family $1_{B(0,R)}(D)$ converges to $f$ in $L^2(\mathbb{R}^n)$ as $R$ tends to infinity. By contrast, the famous result of C. Fefferman [46] implies its unbound- edness in $L^p(\mathbb{R}^n)$ as soon as $n \ge 2$ and $p \ne 2$. The failure of $L^p$ convergence of the ball multipliers has raised the same question with the ball multipliers being replaced by the following family of smoother Fourier multiplier operators, known as the Bochner-Riesz multipliers:

$$S_R^\delta f := \mathcal{F}^{-1}(m_{\delta,R}(|\cdot|^2)\hat{f}), \quad f \in \mathcal{S}(\mathbb{R}^n),$$

where

$$m_{\delta,R}(\lambda) := \left(1 - \frac{\lambda}{R^2}\right)_+^\delta,$$

with $x_+ := \max(x, 0)$ denoting the positive part of $x$. In other words,

$$S_R^\delta f(x) = \int_{\mathbb{R}^n} \left(1 - \frac{|\xi|^2}{R^2}\right)_+^\delta \hat{f}(\xi) e^{2\pi i x \cdot \xi} d\xi, \quad x \in \mathbb{R}^n.$$

The Bochner-Riesz conjecture asks if $\delta > 0$ and $n|\frac{1}{2} - \frac{1}{p}| - \frac{1}{2} < \delta$, then $S_R^\delta f$ converges to $f$ in $L^p(\mathbb{R}^n)$, as $R$ tends to infinity? Note that such condition on $p$ is known to be necessary, according to a counter example where the symbol of $S_R^\delta$ is divided into a large number of Knapp's examples, see E.-M. Stein [108].

We observe that the Riesz means $S_R^\delta$ are radial Fourier multipliers, and can therefore be equivalently described as a spectral multiplier in $-(2\pi)^{-2}\Delta$, see (4.4):

$$S_R^\delta f = m_{\delta,R}(-(2\pi)^{-2}\Delta).$$

Hence, a similar question may be asked for any positive self-adjoint operator in other contexts. The most natural is certainly the case of the Laplace operator on the torus, yielding to the Bochner-Riesz means for Fourier series. The case of the canonical sub-Laplacians on the Heisenberg groups has also been considered [88, 91].

## 6.2. Applications to evolution equations.
In the present paragraph, we aim at briefly providing the different strategies used to establish Strichartz estimates, which are another face of the Tomas-Stein theorem.

Strichartz estimates date back to the 70s through the founding paper of R. Strichartz [111]. They have become an efficient tool for analysing many phenomena in physics, biology, fluid mechanics, general relativity, etc.

### 6.2.1. Origin of the Strichartz estimates.
Here, we consider the Schrödinger equation, that have been introduced in the context of quantum mechanics by E. Schrödinger in 1925,

$$(6.1) \qquad \begin{cases} i\partial_t u - \Delta u = 0 \\ u|_{t=0} = u_0 \in L^2(\mathbb{R}^n). \end{cases}$$

Based on standard arguments, one can easily check that the solution of the above Cauchy problem can be written as follows

$$(6.2) \qquad u(t, x) = \int_{\mathbb{R}^n} e^{i(x \cdot \xi + t|\xi|^2)} \hat{u}_0(\xi) d\xi$$

$$(6.3) \qquad = \frac{e^{i \frac{|t|^2}{4t}}}{(4\pi it)^{\frac{n}{2}}} \star u_0.$$
In [111], R. Strichartz pointed out that Formula (6.2) can be interpreted as the restriction of the Fourier transform on the paraboloid $\hat{S}$ in the space of frequencies $\hat{\mathbb{R}}^{n+1} = \hat{\mathbb{R}} \times \hat{\mathbb{R}}^n$, defined as

$$ \hat{S} := \{(\alpha, \xi) \in \hat{\mathbb{R}} \times \hat{\mathbb{R}}^n \mid \alpha = |\xi|^2\}. $$

Combining the $L^2 \to L^{p'}$-bound formulation of the restriction theorem (see (3.5)) with scaling arguments, he deduced the following bounds for space-time norms of the solution $u$ to the Schrödinger equation by means of the $L^2$-norm of the initial data $u_0$:

$$ (6.4) \qquad \|u\|_{L^{\frac{2n+4}{n}}(\mathbb{R}, L^{\frac{2n+4}{n}}(\mathbb{R}^n))} \le C \|u_0\|_{L^2(\mathbb{R}^n)}. $$

Since then, these type of estimates, which have appropriate analogue results for the wave equation and several systems involved in fluid mechanics, were christened with the name of Strichartz estimates.

### 6.2.2. Dispersion and Strichartz estimates.
In mathematics, a dispersive evolution corresponds to the phenomena that waves with different frequencies move at different velocities in time. For the Schrödinger equation in (6.1), this can be seen in the Fourier representation (6.2) of its solution. Applying Young's convolution inequality to (6.3) implies that this solution satisfies the so-called *dispersive estimate*:

$$ (6.5) \qquad \|u(t, \cdot)\|_{L^{\infty}(\mathbb{R}^n)} \le \frac{1}{(4\pi|t|)^{\frac{n}{2}}} \|u_0\|_{L^1(\mathbb{R}^n)}, \quad \forall t \ne 0. $$

Commonly, a dispersive estimate corresponds to a pointwise inequality in time decay of the solution $u$ of an evolution PDE by means of the $L^1$-norm of the data $u_0$, namely ($t \ne 0$)

$$ \|u(t, \cdot)\|_{L^{\infty}} \lesssim \frac{\|u_0\|_{L^1}}{|t|^r} $$

where (in general) the rate of decay $r > 0$ depends on the equation, the dimension and the setting.

In the late 1990's, the seminal work of J. Ginibre and G. Velo [59] (see also M. Keel and T. Tao [74] for the endpoint) resorted to dispersive estimates to allow for the derivation of Strichartz estimates for a larger range of indices, thanks to the $TT^*$ argument recalled in Appendix A. The research that has ensued in this subject has been marked by a whole series of dramatic results on dispersive and Strichartz estimates which are the key to proving well-posedness results for nonlinear evolution equations.

We illustrate this idea with the $L^p - L^q$-Strichartz estimates for the Schrödinger equation. This result contains, in particular, the first Strichartz estimates in (6.4).

**Theorem 6.1.** For any $u_0 \in L^2(\mathbb{R}^n)$, the solution $u$ to the Schrödinger equation in (6.1) satisfies the following family of Strichartz estimates:

$$ (6.6) \qquad \|u\|_{L^q(\mathbb{R}, L^p(\mathbb{R}^n))} \le C(p,q) \|u_0\|_{L^2(\mathbb{R}^n)}; $$

above $(p, q)$ satisfies the scaling admissibility condition

$$ (6.7) \qquad \frac{2}{q} + \frac{n}{p} = \frac{n}{2} \quad \text{with } q \ge 2 \quad \text{and} \quad (n, q, p) \ne (2, 2, \infty). $$

**Sketch of the proof of Theorem 6.1.** An argument of complex interpolation between the dispersive estimate in (6.5) with the conservation of the mass

$$ \|u(t, \cdot)\|_{L^2(\mathbb{R}^n)} = \|u_0\|_{L^2(\mathbb{R}^n)}, $$

implies, for any $2 \le p \le \infty$,

$$ (6.8) \qquad \|u(t, \cdot)\|_{L^p(\mathbb{R}^n)} \lesssim |t|^{-n(\frac{1}{2}-\frac{1}{p})} \|u_0\|_{L^{p'}(\mathbb{R}^n)}. $$
Invoking the $TT^*$ argument, this gives rise when $u_0 \in L^2(\mathbb{R}^n)$ to the following family of Strichartz estimates which contains the first Strichartz estimates in (6.4):

$$ (6.9) \qquad \|u\|_{L^q(\mathbb{R}, L^p(\mathbb{R}^n))} \le C(p,q) \|u_0\|_{L^2(\mathbb{R}^n)}; $$

above $(p,q)$ satisfies the scaling admissibility condition in (6.7) Indeed, if we denote $u(t, \cdot) = U(t)u_0$, then by definition of the $L_t^q(L_x^p)$-norm, we have

$$ \|U(t)u_0\|_{L^q(\mathbb{R}, L^p(\mathbb{R}^n))} = \sup_{\varphi \in \mathcal{B}_{q,p}} \left| \int_{\mathbb{R}} (U(t)u_0|\varphi(t))_{L^2(\mathbb{R}^n)} dt \right|, $$

where $\mathcal{B}_{q,p} = \{\phi \in \mathcal{D}(\mathbb{R}^n) / \|\phi\|_{L^{q'}(\mathbb{R}, L^{p'}(\mathbb{R}^n))} \le 1\}$. By the definition of the adjoint operator, we thus have

$$ \|U(t)u_0\|_{L^q(\mathbb{R}, L^p(\mathbb{R}^n))} \le \|u_0\|_{L^2(\mathbb{R}^n)} \sup_{\varphi \in \mathcal{B}_{q,p}} \left\| \int_{\mathbb{R}} U^*(t)\varphi(t, \cdot) dt \right\|_{L^2(\mathbb{R}^n)}. $$

However, we have by functional analysis and Hölder's inequality,

$$ \begin{aligned} \left\| \int_{\mathbb{R}} U^*(t) \varphi(t, \cdot) dt \right\|_{L^2(\mathbb{R}^n)}^2 &= \int_{\mathbb{R}^2} \langle U(t-t')\varphi(t', \cdot), \bar{\varphi}(t, \cdot) \rangle dt' dt \\ &\le \int_{\mathbb{R}^2} \|U(t-t')\varphi(t', \cdot)\|_{L^p(\mathbb{R}^n)} \|\varphi(t, \cdot)\|_{L^{p'}(\mathbb{R}^n)} dt' dt, \end{aligned} $$

which implies thanks to the dispersive estimate (6.8)

$$ \left\| \int_{\mathbb{R}} U^*(t) \varphi(t, \cdot) dt \right\|_{L^2(\mathbb{R}^n)}^2 \le \int_{\mathbb{R}^2} \frac{1}{|t-t'|^{n(\frac{1}{2}-\frac{1}{p})}} \|\varphi(t', \cdot)\|_{L^{p'}(\mathbb{R}^n)} \|\varphi(t, \cdot)\|_{L^{p'}(\mathbb{R}^n)} dt' dt. $$

This completes the proof of the result by Hardy-Littlewood-Sobolev inequality. $\square$

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

In fact, there is a plethora of Strichartz estimates expressed in Lebesgue spaces as well as in Sobolev spaces or more generally in Besov spaces, in homogeneous or inhomogeneous settings, whether for evolution equations with constant coefficients or rough variable coefficients. For a brief introduction to this topic, we refer the reader to the text [10] and the references therein.

In the particular case of the Schrödinger equation in (6.1), combined with a scaling argument, the method of proof outlined above provides a gain of one half derivative with respect to the Sobolev embedding $H^s(\mathbb{R}^n) \hookrightarrow L^\infty(\mathbb{R}^n)$, $s > n/2$, where $H^s(\mathbb{R}^n)$ stands for the inhomogeneous Sobolev space of regularity index $s$. As a result of this fact, nonlinear Schrödinger equations can be solved for initial data less regular than what is required by the energy method.

To address the quasilinear framework, Strichartz estimates for evolution equations with variable (rough) coefficients have been also extensively studied: it is the combination of geometrical optics and harmonic analysis that allows us to derive in that (more involved) framework Strichartz estimates, with some loss compared to the constant coefficient case. In this context, the heart of the matter consists in constructing a microlocal parametrix, that is to say an approximation of $u_q$, the part of the solution relating to frequencies of size $2^q$, which solves a regular evolution equation (thanks to paradifferential calculus, see [7]) but only on a small time interval whose size depends on the frequency. Families of dispersive estimates can then be established for small time, leading to microlocal Strichartz estimates by the $TT^*$ argument. The local (in time) Strichartz estimate are then obtained by gluing the microlocal estimates. We refer the interested reader to H. Bahouri and J.-Y. Chemin [6], N. Burq, P. Gérard and N. Tzvetkov [25], and D. Tataru [114].
6.2.3. Evolution with lack of dispersion. Dispersion may not hold or can be weak, such as for the wave equation on $\mathbb{H}_d$, where dispersive estimates have been established in [9] with an optimal rate of decay of order $|t|^{-\frac{1}{2}}$, regardless of the dimension $d$. This is the case for instance, on compact Riemannian manifolds and on some bounded domains. The Euclidean strategy described in Section 6.2.2 breaks down, and then obtaining Strichartz estimates is considered a very difficult problem. Other general approaches can be used (with a possible loss of derivatives), such as the estimates of resolvents (see for instance [70]), the microlocal analysis as explained above [6, 25, 114], or the use of adapted restriction results in the spirit of the pioneering work of R. Strichartz [111].

A model setting for totally non-dispersive evolution equations is the Heisenberg group $\mathbb{H}_d$ (see Example 6), and more precisely the Schrödinger equation for its canonical sub-Laplacian defined in (5.3):

$$ (6.10) \qquad \left\{ \begin{array}{l} i \partial_t u - \Delta_{\mathbb{H}_d} u = f \\ u|_{t=0} = u_0, \end{array} \right. $$

Indeed, in [9], the authors exhibited examples of Cauchy data $u_0$ for which the Heisenberg Schrödinger equation in (6.10) behaves as a transport equation along the variable $s$ (called the vertical variable), in the sense that

$$ u(t, Y, s) = u_0(Y, s + 4td). $$

More generally, it was emphasised in H. Bahouri, D. Barilari and I. Gallagher [5] that the Heisenberg Schrödinger equation in (6.10) can be interpreted as a superposition of transport equations $\pm T_{2\ell+d}$ along the vertical direction, with velocity $\pm 4(2\ell + d)$, $\ell \in \mathbb{N}$.

Although the Schrödinger equation on the Heisenberg group in (6.10) is considered totally non-dispersive, the authors in [5] obtained a restriction result in the setting of $\mathbb{R} \times \mathbb{H}_d$ for an adapted hyper-surface to the Schrödinger equation (6.10), that can be interpreted as the parabola $S_{par}$ in the setting of $\mathbb{R}^n$; they also obtained appropriate analogue results for the wave equation on $\mathbb{H}_d$. They were inspired by Müller's restriction result [92], and in particular took advantage of Formula (5.5). Following Strichartz' approach [111], they then established the following weak Strichartz estimate which shows the distinguished role of the vertical direction:

$$ (6.11) \qquad \|u\|_{L_s^\infty L_t^q L_Y^p} \lesssim_{p,q} \|u_0\|_{H^{\frac{2d+2}{2}-\frac{2}{q}-\frac{2d}{p}}(\mathbb{H}_d)}, $$

for all $(p, q) \in [2, \infty]^2$ such that $p \le q$ and $\frac{2}{q} + \frac{2d}{p} \le \frac{2d+2}{2}$. Here, $H^s(\mathbb{H}_d)$ denotes the Sobolev space on $\mathbb{H}_d$ of regularity index $s$; it can be defined by several ways, for instance via the functional calculus of $-\Delta_{\mathbb{H}_d}$ (see for instance [5, 97] and the references therein). In comparison with (6.9), one can easily check that the following weighted estimate holds

$$ (6.12) \qquad \| \langle s \rangle^{-\alpha} u \|_{L^2(\mathbb{R}, L^2(\mathbb{H}_d))} \lesssim_{\alpha} \| u_0 \|_{L^2(\mathbb{H}_d)}, $$

for all $\alpha \ge \frac{1}{2}$, where $\langle s \rangle := \sqrt{1 + |s|^2}$.

The restriction problem amounts to investigate an operator $\mathcal{P}$ defined for instance spectrally as in Lemma 4.1. This operator may be further described with the group Fourier transform (see Section 5.1.10) or/and via its convolution kernel in terms of special functions. To our knowledge, the first result in that direction was achieved by D. Müller [92] on the Heisenberg group $\mathbb{H}_d$ in 1990. Müller's result created a surprise, since while stressing that there are no non-trivial solutions in the $L^p$-spaces for $p > 1$, it provided a positive answer in anisotropic Lebesgue spaces, see Section 5.

Contrary to the Euclidean setting, knowing restriction's theorems on $\mathbb{H}_d$ does not translate straightforwardly into results of the same type in $\mathbb{R} \times \mathbb{H}_d$, which is of course not equal to $\mathbb{H}_{d'}$, for some $d'$. Hence, to get restriction results in the setting of $\mathbb{R} \times \mathbb{H}_d$, one needs to adapt the results pertaining to $\mathbb{H}_d$.
Let us end this section by pointing out that, based on the restriction result of V. Casarino and P. Ciatti [28], D. Barilari and S. Flynn [13] proved Strichartz estimates for the wave and Schrödinger equations for any H-type group, which are examples of nilpotent Lie groups with two-step of nilpotency.

### 6.2.4. Kato-smoothing effect.
Variations of the Tomas-Stein estimate (2.1), including refined restriction results or trace theorems, come also into play in the description of solutions of some evolution equations. These estimates, known as Kato-smoothing properties, have been discovered independently by A.-V. Faminskii and S.-N. Kruzhkov [42] and T. Kato [72] for the Korteweg-de Vries equation, then extended by P. Constantin and J.-C. Saut [32] to a large number of dispersive equations and systems on $\mathbb{R}^n$. The Kato-smoothing effect for the Schrödinger equation on $\mathbb{R}^n$ writes as follows:

**Theorem 6.2.** For any $u_0 \in L^2(\mathbb{R}^n)$, the solution $u$ to the Schrödinger equation in (6.1) satisfies the following Kato-smoothness estimate:

$$ (6.13) \qquad \| \langle x \rangle^{-1} \langle D_x \rangle^{\frac{1}{2}} u \|_{L^2(\mathbb{R}_t \times \mathbb{R}^n)} \lesssim \|u_0\|_{L^2(\mathbb{R}^n)}, \\ \text{where } \langle \cdot \rangle := \sqrt{1 + |\cdot|^2}. $$

The estimate in (6.13) tells us that the solution $u(t, \cdot)$ is, for almost $t$, locally $1/2$ derivative smoother than the initial data $u_0$. This regularization effect is different from the one displayed by Strichartz estimates, which as mentioned above can be only interpreted with respect to the Sobolev embedding: the estimate (6.13) showcases the effective gain of the one half derivative. Needless to say, the local character of Kato-smoothing effect is essential: indeed due to the conservation of the $H^s$-norms along the flow of the Schrödinger equation, namely

$$ (6.14) \qquad u_0 \in H^s(\mathbb{R}^n) \Leftrightarrow u(t, \cdot) = e^{-it\Delta}u_0 \in H^s(\mathbb{R}^n), \forall t \in \mathbb{R}, $$

a global smoothing effect is excluded in Sobolev spaces.

The strategy of the proof of estimates of type (6.13) depends on the setting: equations with constant or variable coefficients, on the whole space $\mathbb{R}^n$ or in exterior domains, etc. For further details, one can consult the monograph of L. Robbiano [96]. Here we sketch the idea of M. Ben-Artzi and S. Klainerman [15].

*Sketch of the proof of Theorem 6.2 following [15]*. The starting point is the following variant of Tomas-Stein restriction estimate

$$ (6.15) \qquad \| \mathcal{F}(f) \|_{\hat{S}^{n-1}(\mathbb{R})} \| L^2(\hat{\sigma}_{\hat{S}^{n-1}}(\mathbb{R})) \| \lesssim \min(r^{\frac{1}{2}}, 1) \| \langle x \rangle f \|_{L^2(\mathbb{R}^n)}, $$

where $\hat{S}^{n-1}(\mathbb{R})$ denotes the sphere centered at the origin and of radius $r$ in $\mathbb{R}^n$. Its proof is mainly based on classical trace theorems in the framework of Sobolev spaces: for $r \ge 1$, it is deduced straightaway from the basic trace theorem, while for $r \le 1$ it follows from a combination of scaling arguments with the Hardy inequality and the application of the following trace theorem on $\hat{S}^{n-1}$

$$ \| \mathcal{F}(f) \|_{\hat{S}^{n-1}} \| L^2(\hat{\sigma}_{\hat{S}^{n-1}}) \| \lesssim \| \mathcal{F}(f) \|_{H^1(\hat{B}(0,1))}^2 \lesssim \| \mathcal{F}(f) \|_{H^1(\mathbb{R}^n)}^2. $$

For further details, see the monograph of L. Robbiano [96].

With (6.15) at hand, the idea of M. Ben-Artzi and S. Klainerman [15] consists first in reducing the proof of the local smoothing property (6.13) by duality arguments to establish

$$ (6.16) \qquad \left| \left( (\mathrm{Id} - \Delta)^{\frac{1}{4}} u \right) |_{L^2(\mathbb{R}_t \times \mathbb{R}^n)} \right| \le C \|u_0\|_{L^2(\mathbb{R}^n)} \| \langle x \rangle v \|_{L^2(\mathbb{R}_t \times \mathbb{R}^n)}, $$
for any test function $v(t, x)$. Then, as the spectral decomposition of the (self-adjoint extension of the) rescaled Laplace operator given by (4.2) reads according to Fourier-Plancherel formula, for all $f, g$ in $S(\mathbb{R}^n)$,

$$ \begin{aligned} \left| \left(-2\pi\right)^{-2} \Delta f \right|_{L^2(\mathbb{R}^n)} &= \int_{\hat{\mathbb{R}}^n} |\xi|^2 \mathcal{F}f(\xi) \overline{\mathcal{F}g}(\xi) d\xi \\ &= \int_0^\infty r^2 \left( \int_{\hat{\mathbb{S}}^{n-1}} \mathcal{F}f(r\omega) \overline{\mathcal{F}g}(r\omega) d\sigma_{\hat{\mathbb{S}}^{n-1}}(\omega) \right) r^{n-1} dr. \end{aligned} $$

Therefore

$$ (6.17) \qquad \left| \left(-2\pi\right)^{-2} \Delta f \right|_{L^2(\mathbb{R}^n)} = \int_0^\infty \mu(A(\mu)f|g)_{L^2(\mathbb{R}^n)} d\mu, $$

with

$$ (6.18) \qquad (A(\mu)f|g)_{L^2(\mathbb{R}^n)} = \frac{1}{\sqrt{\mu}} (\mathcal{F}(f), \mathcal{F}(g))_{L^2(\sigma_{\hat{\mathbb{S}}^{n-1}}(\sqrt{\mu}))}. $$

Taking advantage of the basic properties of the spectral measure of $-\Delta$ on $\mathbb{R}^n$, one can then deduce that the following bound holds

$$ \left| \left( \mathrm{Id} - (2\pi)^{-2} \Delta \right)^{1/4} u|v \right|_{L^2(\mathbb{R}_t \times \mathbb{R}^n)} \le \|u_0\|_{L^2(\mathbb{R}^n)} \sqrt{\int_0^\infty \sqrt{1+\mu} (A(\mu)\tilde{v}(\mu, \cdot)|\tilde{v}(\mu, \cdot)) d\mu}. $$

With the restriction estimate (6.15) and Fourier-Plancherel formula with respect to time, the conclusion follows. $\square$

### 6.3. Some generalisations of Tomas-Stein estimates.
When functional inequalities are at hand, it is important to study their invariance by scaling, oscillations and translations, etc, as this often leads to their refinement. Functional inequalities were one of Brezis' predilection topics, where he made deep and influential contributions that have been at the origin of a large number of research projects, see for instance [19, 20, 22, 23, 24].

#### 6.3.1. Sobolev inequalities in Lebesgue spaces.
Among the most famous functional inequalities, we can mention the Sobolev inequalities in Lebesgue spaces:

$$ (6.19) \qquad \dot{H}^s(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n), $$

where $\dot{H}^s(\mathbb{R}^n)$ stands for the homogeneous Sobolev space of regularity index $s$, $0 \le s < n/2$ and $p = 2n/(n-2s)$. Those inequalities are invariant by translation and scaling, namely $u_\lambda := u(\lambda \cdot)$, but they are not invariant by oscillations, that is, by multiplication by oscillating functions, namely by those of the type $u_\epsilon(x) = e^{i\frac{(x|\omega)}{\epsilon}}\varphi(x)$, where $\omega$ is a unit vector of $\mathbb{R}^n$, and $\varphi$ is a function in $S(\mathbb{R}^n)$. In [57], the authors P. Gérard, Y. Meyer and F. Oru refined (6.19) as follows:

$$ (6.20) \qquad \|u\|_{L^p(\mathbb{R}^n)} \lesssim \|u\|_{\dot{B}^{\frac{s-n}{2}}_{\infty, \infty}(\mathbb{R}^n)}^{1-\frac{2}{p}} \|u\|_{\dot{H}^s(\mathbb{R}^n)}^{\frac{2}{p}}, $$

where $\dot{B}^{\frac{s-n}{2}}_{\infty, \infty}(\mathbb{R}^n)$ denotes the homogeneous Besov space of regularity index $s - \frac{n}{2}$. Hence, they have obtained a sharp inequality invariant under scaling, oscillations and translations. Recall that Besov spaces (which are in some sense generalisations of Sobolev spaces) can be defined in several ways (see for instance [10] for an overview of these spaces). In particular, for $\sigma > 0$ and $\psi \in S(\mathbb{R}^n)$ with compactly supported Fourier transform $\hat{\psi}$, identically equal to 1 near the origin, and taking its values in $[0, 1]$, one has (independently of $\psi$)

$$ \|u\|_{\dot{B}^{-\sigma}_{\infty, \infty}(\mathbb{R}^n)} := \sup_{A>0} A^{n-\sigma} \|\psi(A\cdot) \star u\|_{L^{\infty}(\mathbb{R}^n)} = \sup_{A>0} A^{-\sigma} \|\mathcal{F}^{-1}(\hat{\psi}((A^{-1}\cdot)\hat{u}))\|_{L^{\infty}(\mathbb{R}^n)}. $$

The refined estimate (6.20) is one of the key arguments in [56], where P. Gérard gave a characterisation of the defect of compactness of the critical Sobolev embedding (6.19) by means of the profile decompositions. This characterisation can be formulated in the
following terms: a bounded sequence $(u_n)_{n \ge 0}$ in $\dot{H}^s(\mathbb{R}^n)$ can be decomposed (up to a subsequence extraction) according to

$$ (6.21) \qquad u_n = \sum_{l=1}^{L} h_{l,n}^{s-n/p} \phi^l \left( \frac{\cdot -x_{l,n}}{h_{l,n}} \right) + r_{n,L}, \quad \lim_{L \to +\infty} \left( \limsup_{n \to +\infty} \|r_{n,L}\|_{L^p} \right) = 0, $$

where $(\phi^l)_{l \ge 1}$ is a family of functions in $\dot{H}^s(\mathbb{R}^n)$ and where the cores $(x_{l,n})_{n \in \mathbb{N}}$ satisfy, for all $k \ne l$,

$$ |\log(h_{l,n}/h_{k,n})| \to +\infty \quad \text{or} \quad |x_{l,n} - x_{k,n}|/h_{l,n} \to +\infty, \quad \text{as } n \to +\infty. $$

In short, the decomposition (6.21) shows that translational and scaling invariance are the only features responsible for the defect of compactness of the Sobolev embedding (6.19).

Nonlinear analysis progressed substantially in the last decades due to profile decomposition techniques that originated from the work of H. Brézis and J.-M. Coron [21] motivated by geometric problems. As can be seen from the extensive literature on the subject, these techniques are now essential in PDEs in the contexts of both elliptic and evolutionary equations. A vast literature on the subject has been growing: among others, let us mention H. Bahouri and G. Perelman [11], M. Del Pino, F. Mahmoudi and M. Musso [36], J. Jendrej and A. Lawrie [71], E. Lenzmann and M. Lewin [78], F. Merle and L. Vega [86], C.-E. Kenig and F. Merle [87], and the references therein. Let us also emphasise that from the profile decompositions (6.21), one can recover that the best Sobolev constant in (6.19) is reached, as well as the structure of minimising sequences (see the founding paper of P.-L. Lions [80]).

The question of optimal constants and the existence of extremisers in functional inequalities such as Sobolev embedding or Gagliardo-Nirenberg inequalities is a long standing problem that has been the subject of many papers for the sake of geometric problems, or physical studies in many settings, but remains a topical issue. This kind of analysis can be carried out by variational analysis, optimal transport methods (see for instance [33]) or concentration compactness arguments as mentioned above.

### 6.3.2. Case of the Heisenberg group.
In [14], J. Ben Ameur obtained, in the framework of the Heisenberg group, an appropriate analogue profile decomposition to (6.21). J. Ben Ameur's result characterises the defect of compactness of the Sobolev embedding:

$$ (6.22) \qquad \dot{H}^s(\mathbb{H}_d) \hookrightarrow L^p(\mathbb{H}_d), $$

where $\dot{H}^s(\mathbb{H}_d)$ denotes the homogeneous Sobolev space on $\mathbb{H}_d$ of regularity index $s$, with $0 \le s < Q/2$ and $p = 2Q/(Q-2s)$, and where $Q := 2d+2$ stands for the homogeneous dimension of $\mathbb{H}_d$ given by the anisotropic dilations (5.4). Based on this decomposition, L. Gassot [54] showed that the best Sobolev constant is attained in the setting of $\mathbb{H}_d$, and also constructed families of traveling waves for the cubic Schrödinger equation on the Heisenberg group.

### 6.3.3. Refined Tomas-Stein estimates.
Concerning the Tomas-Stein inequality, similar studies have been conducted. On the one hand, the literature includes several refinements of the functional inequality (2.1): among others, one can cite R.-L. Frank, E. Lieb and J. Sabin [53], R. Killip and M. Visan [75], A. Moyua, A. Vargas and L. Vega [89], D. Oliveira e Silva [93], S. Shao [100], T. Tao [113], etc. In particular, one has the following refined estimate to be compared with (6.20): there exists a positive constant $C$, such that for all $g \in L^2(\hat{\mathbb{S}}^{n-1}, d\sigma_{\hat{\mathbb{S}}^{n-1}})$,

$$ (6.23) \qquad \|\mathcal{F}^{-1}(gd\sigma_{\hat{\mathbb{S}}^{n-1}})\|_{L^{\frac{2n+2}{n-1}}(\mathbb{R}^n)} \lesssim \left( \sup_{Q \in D} |Q|^{-\frac{1}{2}} \|\mathcal{F}^{-1}(1_Q gd\sigma_{\hat{\mathbb{S}}^{n-1}})\|_{L^\infty(\mathbb{R}^n)} \right)^\theta \|g\|_{L^2(d\sigma_{\hat{\mathbb{S}}^{n-1}})}^{1-\theta}, $$
for some $0 < \theta < 1$, where $D$ denotes a family of dyadic cubes covering $\hat{S}^{n-1}$. Note that (6.23) implies the standard Tomas-Stein inequality:

$$ (6.24) \qquad \| \mathcal{F}^{-1}(gd\sigma_{\hat{S}^{n-1}}) \|_{L^{\frac{2n+2}{n-1}}(\mathbb{R}^n)} \lesssim \|g\|_{L^2(d\sigma_{\hat{S}^{n-1}})}. $$

Indeed, for any $Q \in D$, we have by Riemann-Lebesgue theorem

$$ \begin{aligned} |Q|^{-\frac{1}{2}} \| \mathcal{F}^{-1}(1_Q g d\sigma_{\hat{S}^{n-1}}) \|_{L^{\infty}(\mathbb{R}^n)} &\lesssim |Q|^{-\frac{1}{2}} \|1_Q g\|_{L^1(d\sigma_{\hat{S}^{n-1}})} \\ &\lesssim |Q|^{-\frac{1}{2}} \|1_Q\|_{L^2(d\sigma_{\hat{S}^{n-1}})} \|g\|_{L^2(d\sigma_{\hat{S}^{n-1}})} \\ &\lesssim \|g\|_{L^2(d\sigma_{\hat{S}^{n-1}})}. \end{aligned} $$

The proof of the various refined Tomas-Stein estimates are different from that of (6.20). While the proof of (6.20) results from a combination of Cavalieri principle with the classical method of decomposition of functions into low and high frequencies (see for instance Theorem 1.43 in [7]), that of the refined estimates of Tomas-Stein inequality relies rather on bilinear restriction results. In particular, the proof of (6.23) relies on a deep bilinear restriction theorem of Tao [113]. Recall that such bilinear estimates lead to a better gain of regularity for the product of two solutions of the Schrödinger equation in (6.1) compared with the product laws and the gain of one half derivative (with respect to the Sobolev embedding) with purely Strichartz methods.

On the other hand, along the same lines as in the proof of the profile decomposition (6.21) where (6.20) plays a key role, the refined estimate (6.23) involves in a sort of "Knapp" profile decomposition, that have been introduced by M. Christ and S. Shao [30]. Roughly speaking, the "Knapp" algorithm decomposition implemented in [30] is built on the fact that if a bounded sequence $(f_k)_{k \in \mathbb{N}}$ of $L^2(\hat{S}^{n-1})$ do not concentrate around the north pole (up the invariance of (6.24)), then (see Lemma 5.2 in [53])

$$ \sup_{Q \in D} |Q|^{-\frac{1}{2}} \| \mathcal{F}^{-1}(1_Q f_k d\sigma_{\hat{S}^{n-1}}) \|_{L^{\infty}(\mathbb{R}^n)} \xrightarrow{k \to \infty} 0. $$

Let us also underline that similarly to the case of the Sobolev embedding (6.19), the refinements of Tomas-Stein inequalities as well as the "Knapp" profile decompositions come to play in the study of best constants and extremisers involved in this setting, which is not completely settled in its general form and remains a topical issue. Indeed, works addressing the existence of extremisers for Tomas-Stein inequalities tend to be a tour de force, using a variety of tools and technics, such as the refined estimates, bilinear estimates, and the concentration compactness arguments mentioned above. For an overview about this topic, see for instance the recent paper Foschi-Oliveira e Silva [52].

### 6.3.4. Other frameworks.
We have seen in Section 5.2.2 that in the particular case of the Heisenberg group $\mathbb{H}^d$, an estimate of type (2.1) can only hold for $p = 1$, which is the trivial index given by Riemann-Lebesgue theorem! However, Müller [92] established an appropriate restriction estimate involved anisotropic Lebesgue spaces. This prompts us to generalise the question of Stein in the following way: given a Lie group $G$, for which norm $N_G$ on $G$, an inequality of the form

$$ (6.25) \qquad \| \mathcal{F}_G(f) |_{\hat{S}_G} \|_{L^2(d\sigma_{\hat{S}_G})} \leq C N_G(f), $$

holds for all $f$ in the Schwartz space $S(G)$, where $\mathcal{F}_G(f)$ denotes the Fourier transform of $f$ on the group $G$, and $\hat{S}_G$ stands for the dual unit sphere. The definition of the dual unit sphere $\hat{S}_G$ and its operator-valued measure $\sigma_{\hat{S}_G}$ is open to interpretation. In the case of the Heisenberg groups, the dual unit sphere and its operator-valued measure are defined in spectral terms using the relation of the spectral decomposition of the canonical sub-Laplacian and the group-Fourier transform as in Section 5.2.2. This is more generally
the case on Carnot groups (see Section 5.1.4 for a definition) as they are equipped with a canonical sub-Laplacian.

Let us end this section by stressing that numerous works related to restriction problems and its applications have been carried out in the framework of hyperbolic geometry, especially the hyperpolic space (one model of which is the hyperbola as in Section 5.1.1). Regarding restriction theorems, one can consult the recent work of S. Buschenhenke, D. Müller, A. Vargas, and the references therein. For cluster estimates, one can see the latter paper of J.-P. Anker, P. Germain and T. Léger [3], while for Strichartz estimates, one can look for instance to the studies by V. Banica [12] as well as that by J.-P. Anker and V. Pierfelice [4] concerning the Schrödinger equation. Note finally that the issue of extremals for Gagliardo-Niremberg inequalities in hyperbolic geometry has been investigated by M. Mukherjee in [90], whereas, as far as we know, the study of extremals of restriction's estimates in hyperbolic geometry is still an open question.

**6.4. Discrete framework.** In the discrete case, we have to deal with exponential sums, such as for instance

$$ (6.26) \qquad f_A(x) = \sum_{a \in A} c_a e^{2i\pi a \cdot x}, $$

with A a subset of $\mathbb{Z}^n$. Evaluating $f_A$ in $L^p([0, 1]^n)$ is a challenging issue: the involved terms have various phases in the complex plane, and it is difficult to tell what happens when we add them all up.

**6.4.1. The first $L^2$-restriction theorem on the flat Torus.** To our knowledge, the first restriction result on the flat Torus dates back to the seventies with the result of A. Zygmund [122] concerning the two dimensional case.

**Theorem 6.3 ([122]).** There exists a positive constant C such that, for any regular function $f$ on $Q := [0, 1]^2$ and any $r > 0$, we have

$$ (6.27) \qquad \left( \sum_{|\mu|=r} |c_\mu|^2 \right)^{\frac{1}{2}} \le C \|f\|_{L^{\frac{4}{3}}(Q)}, $$

where, for $\mu = (m, n) \in \mathbb{Z}^2$, $c_\mu$ denotes the Fourier coefficient of $f$ given by

$$ c_{\mu} := \int_{Q} e^{-2i\pi\mu \cdot x} f(x) dx. $$

*Sketch of the proof of Theorem 6.3.* Assume that $\hat{S}_r$ the subset of points $\mu = (m, n)$ in the lattice $\mathbb{Z}^2$ satisfying $|\mu| = r$, namely $|m| + |n| = r$, is non empty, and write

$$ \begin{aligned} \left( \sum_{|\mu|=r} |c_\mu|^2 \right)^{\frac{1}{2}} &= \sum_{|\mu|=r} c_\mu \gamma_\mu \\ &= \int_Q f(x) \left[ \sum_{|\mu|=r} \gamma_\mu e^{-2i\pi\mu \cdot x} \right] dx, \end{aligned} $$

with $\gamma_\mu := \frac{\bar{c}_\mu}{\|c_\mu\|_{\ell^2(\hat{S}_r)}}$.

Applying Hölder's inequality, and then Parseval's formula, we deduce that

$$ \begin{aligned} \left( \sum_{|\mu|=r} |c_\mu|^2 \right)^{\frac{1}{2}} &\le \|f\|_{L^{\frac{4}{3}}(Q)} \left\| \sum_{|\mu|=r} \gamma_\mu e^{-2i\pi\mu \cdot x} \right\|_{L^4(Q)} \\ &= \|f\|_{L^{\frac{4}{3}}(Q)} \|\Gamma_\rho\|_{\ell^2(\Lambda)}^2, \end{aligned} $$
where $\Lambda$ denotes the lattice of points $\rho \in \mathbb{Z}^2$ so that $\rho = \mu - \nu$, with $|\mu| = |\nu| = r$ and
(6.28) $$
$$

$$
\Gamma_{\rho} := \sum_{\substack{\mu-\nu=\rho \\ |\mu|=|\nu|=r}} \gamma_{\mu} \bar{\gamma}_{\nu}.
$$

By definition $\Gamma_0 = \|\gamma_\mu\|_{\ell^2(\hat{S}_r)}^2 = 1$ while, for $\rho \neq 0$ (belonging to the lattice $\Lambda$) $\Gamma_\rho$ includes at most two terms, which easily (according to the fact that $\|\gamma_\mu\|_{\ell^2(\hat{S}_r)} = 1$) allows us to conclude the proof of the result. $\square$

The proof of Zygmund’s theorem highlights the connexion between the restriction prob-
lem in the discrete case with Number Theory.

6.4.2. Schrödinger equations on flat tori. The Schrödinger equation makes sense on any Riemannian manifold, and in particular on flat tori. However, understanding the analogue of Strichartz estimates (6.4) on compact manifolds (M, g) is extremely difficult. The first examples where sharp Strichartz estimates were proved concern the case of S¹ by J. Bourgain [16] and S³ by N. Burq, P. Gérard and N. Tzvetkov [25]. Let us discuss briefly the main arguments for the above theorems.

In the case of the flat Torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$, using the Fourier series theory, one can write
any solution of the Schrödinger equation under the form:

$$
(6.29) \qquad u(t, x) = \sum_{k \in \mathbb{Z}^n} a_k e^{2i\pi(k \cdot x + |k|^2 t)},
$$

where $(a_k)_{k \in \mathbb{Z}^n}$ denote the Fourier coefficients of the Cauchy data $u_0 \in L^2(\mathbb{T}^n)$, and then of course $\|u_0\|_{L^2(\mathbb{T}^n)} = \|a_k\|_{\ell^2(\mathbb{Z}^n)}$.

In the one dimensional case, J. Bourgain in [16] established the following sharp Strichartz estimate

$$
(6.30) \qquad \left\| \sum_{n \in \mathbb{Z}} a_n e^{2i\pi(nx+n^2t)} \right\|_{L^4(\mathbb{T}^2)} \lesssim \|a_n\|_{\ell^2(\mathbb{Z})}.
$$

Its proof is in the same vein as the proof of the estimate (6.27) by A. Zygmund [122] out-
lined in Section 6.4.1: setting $f(t, x) = \sum_{k \in \mathbb{Z}} a_k e^{2i\pi(k \cdot x + k^2 t)}$ and observing that $\|f\|_{L^4(\mathbb{T}^2)}^2 = \|f \bar{f}\|_{L^2(\mathbb{T}^2)}$,
where

$$
(\overline{f\bar{f}})(t,x) = \sum_{k \in \mathbb{Z}} |a_k|^2 + \sum_{\substack{k_1 \neq k_2 \\ \sum_{k \in \mathbb{Z}} a_k \bar{a}_k e^{2i\pi((k_1-k_2)x + (k_1^2-k_2^2)t)}}
$$

one can easily infer that

$$
\|f\|_{L^4(\mathbb{T}^2)}^2 \le 2\|f\|_{L^2(\mathbb{T}^2)},
$$

which implies (6.30) by Parseval identity. Here again as for (6.27), we are dealing with an
easy example of arithmetic structure, and this allows us to end up easily with the result.

However, higher dimensions are more challenging: actually the sum (6.29) is large
near rational points of the form $(\frac{p_1}{q}, \dots, \frac{p_d}{q}, \frac{p_t}{q})$, and this makes the study of such sums
tricky. Optimal results in this setting have been obtained later on thanks to the decou-
pling method introduced by T. Wolff [119], and developed later on by several authors, in
particular by J. Bourgain and C. Demeter [17].

On compact Riemannian manifolds (M, g), a natural expansion of square integrable functions is provided by the spectral decomposition of the associated Laplace-Beltrami operator. However, the knowledge of the spectrum and of the eigenfunctions of such operators on arbitrary compact manifolds (M, g) is too poor to be able to adapt the method applied for flat Tori. The strategy adopted by the authors in [25] is rather the one described in Section 6.2.1, which relies on microlocal approximations.
Decoupling is a recent method in Fourier analysis that is helpful for estimating $||f||_{L^p}$, for $p \neq 2$, in terms of informations about the Fourier transform $\mathcal{F}(f)$. One impressive application of this method involves Strichartz estimates for the Schrödinger equation on the flat Torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$. When $u$ has frequencies at most $N$, one has the following result due to Bourgain-Demeter:

**Theorem 6.4 ([17]).** If the Fourier coefficients $(a_k)_{k \in \mathbb{Z}^n}$ are supported in

$$Q_N = \{(k_1, \dots, k_n) \in \mathbb{Z}^n : |k_j| \le N, j = 1, \dots, n\},$$

then the solution of the Schrödinger equation on $\mathbb{T}^n$ given by

$$u(t, x) = \sum_{n \in Q_N} a_n e^{2i\pi(n \cdot x + |n|^2 t)}$$

satisfies

$$(6.31) \qquad \|u\|_{L^{\frac{2(d+1)}{d}}(\mathbb{T}^n \times [0,1])} \lesssim \varepsilon \|a_n\|_{\ell^2}.$$

Decoupling is a recent development in Fourier analysis whose basis were developed within the framework of the theory of restriction. In abstract terms, if $\mathcal{F}(f)$ is supported in $\Omega \subset \mathbb{R}^n$, then we decompose

$$f = \sum_{\theta} f_{\theta},$$

where $\Omega = \cup \theta$ is a disjoint union of subsets $\theta$, and $\mathcal{F}(f_{\theta})$ is supported in $\theta$, namely

$$f_{\theta}(x) = \int_{\theta} \mathcal{F}(f)(\omega) e^{2i\pi\omega x} d\omega.$$

For each $p$, we define the decoupling constant $D_p := D_p(\Omega = \cup \theta)$ to be the smallest constant so that, for all $f$ with $\mathcal{F}(f)$ supported in $\Omega$,

$$\|f\|_{L^p(\mathbb{R}^n)}^2 \le D_p^2 \sum_{\theta} \|f_{\theta}\|_{L^p(\mathbb{R}^n)}^2.$$

For most applications, we are interested by sets $\theta$ which are partition of a thin neighborhood of a curved manifold, for instance a truncated parabola in Theorem 6.4. For an introduction to this method, one can consult the survey of L. Guth [63] and the references therein.

**6.5. Number theory connection.** Concerning connections to Number Theory, Additive Combinatorics and PDEs in discrete setting, one can consult [37, 38, 63] for an introduction to the decoupling methods whose foundations have grown inside the framework of restriction theory.

**6.5.1. Vinogradov's conjecture.** Let us illustrate the interplay between restriction's problem and Number Theory with Vinogradov's conjecture which concerns diophantine systems:

*For fixed positive integers $s, k, N$, let $J_{s,k}(N)$ be the number of integer solutions*

$$k_1^j + \cdots + k_s^j = k_{s+1}^j + \cdots + k_{2s}^j,$$

for all $1 \le j \le k$ with $(k_1, \dots, k_{2s}) \in \{1, \dots, N\}^{2s}$.

The Vinogradov's question, which dates back to the 1930's, focuses on the asymptotics of $J_{s,k}(N)$, for $N$ large enough.

Using analytic number theory, Vinogradov have proved good estimates for $J_{s,k}(N)$, for some values of $k$ and $s$. Later on in 2015, based on Fourier restriction ideas, and in particular on the decoupling method, J. Bourgain, C. Demeter and L. Guth [18] proved
the following asymptotic sharp bounds (see also the papers of T. Wooley [120, 121] using the efficient congruencing): for all $\epsilon > 0$,

$$ (6.32) \qquad J_{s,k}(N) \lesssim_{s,k,\epsilon} N^{\epsilon} (N^s + N^{2s - \frac{k(k+1)}{2}}), $$

for $N$ sufficiently large.

In particular, one has, in the case when $1 \le s \le \frac{k(k+1)}{2}$,

$$ (6.33) \qquad J_{s,k}(N) \lesssim_{s,k,\epsilon} N^{s+\epsilon}. $$

Observing that there are $N^s$ trivial solutions, those with $(k_1, \dots, k_s) = (k_{s+1}, \dots, k_{2s})$, the bound (6.33) shows that when $1 \le s \le \frac{k(k+1)}{2}$, there are not too many non trivial solutions.

At first glance, this conjecture linked to Number Theory seems to be very far from the restriction's problem! This is not the case, since it turns out that the number $J_{s,k}(N)$ admits the following analytic representation:

$$ (6.34) \qquad J_{s,k}(N) = \int_{[0,1]^k} \left| \sum_{j=1}^N e^{2i\pi(jx_1+j^2x_2+\dots+j^kx_k)} \right|^{2s} dx_1 \dots dx_k, $$

which by straightforward computations follows from the obvious fact that

$$ \int_{[0,1]} e^{2i\pi mx} dx = \begin{cases} 1 & \text{if } m = 0 \\ 0 & \text{if } m \neq 0. \end{cases} $$

This formulation can be seen as a discrete analogue of Tomas-Stein's estimate (3.6), where the measures $gd\sigma_{\hat{S}}$ are sums of exponentials with discrete frequencies located on manifolds such as the parabola when $k = 2$ or the momentum curve, for $k \ge 3$,

$$ \{(t, t^2, \dots, t^k) : t \in [0, 1]\}. $$

The example of Vinogradov's conjecture clearly illustrates the connection between Number and Restriction Theory. We will not comment further on this fact in this text, but we refer the interested reader to the survey of L. Guth [63] and the references therein.

### 6.5.2. Additive combinatorics.
Nilmanifolds appear in the study of arithmetic progressions and additive combinatorics, especially in works by B. Green and T. Tao, see e.g. [60, 61]. For instance, the Green-Tao theorem states that the primes contain arbitrarily long arithmetic progressions, and its proof relies on understanding the distribution of primes in 'nilsequences,' which are sequences arising from flows on nilmanifolds.

### 6.6. Open problems.
As is traditional in Brezis' papers, we conclude this text with a few open questions related to restriction Stein's problem.

From the above cited research a few inquiries arise:

(1) The first one concerns hyper-surfaces $\hat{S}$ of $\mathbb{R}^n$ whose Gaussian curvature vanishes on submanifolds of $\hat{S}$ with codimension $k > 1$ (and do not anywhere else). This is for instance the case of the revolution Torus in $\mathbb{R}^3$ which has for equation

$$ (\sqrt{x^2 + y^2 - R})^2 + z^2 = r^2, \quad 0 < r < R, $$

whose Gaussian curvature vanishes on the two circles at height $z = \pm r$, and is positive anywhere else. One wonders what should be the corresponding optimal estimate for Stein's problem.
(2) In Section 6.2.1, we raised the issue of quasilinear equations which requires more elaborate techniques than the constant coefficient case. In this setting, the heart of the matter consists to investigate approximate solutions under the form:

$$ \int e^{i\Phi(t,x,\xi)} \sigma(t,x,\xi) d\xi, $$

for some phase $\Phi$ and amplitude $\sigma$. In the articles [6, 25, 114] cited above, the involved oscillating integrals are investigated combining microlocal analysis and stationary phase theorem. It would be helpful to establish restriction theorems, under some hypothesis on the phase $\Phi$, for instance as some perturbation of the phase involved in (6.2). Note that L. Guth, J. Hickman and M. Iliopoulou provide in [64] a restriction result for a parametrix's class.

(3) Tomas-Stein estimates have appropriate analogues in the framework of hyperbolic geometry. However, to our knowledge, refined versions in the spirit of (6.23) have not been tackled. Although the roadmap of Section 6.3 is not fully resolved in the Euclidean setting, it would be interesting to refine restriction estimates in the hyperbolic geometry and to investigate adapted concentration compactness tools, with the aim of investigating the involved optimal constants and extremals.

(4) It is anticipated that cluster estimates for sub-Laplacians will not behave like their Riemannian counterparts in Theorem 4.5, especially when sharp. This intuition is based on the difference in propagation properties and Strichartz estimates for nilpotent Lie groups, especially the Heisenberg groups in sections 5.2 and 6.2.

A concrete open problem is to find cluster estimates (even non-sharp) on Heisenberg nilmanifolds for the canonical sub-Laplacians. The spectral theory of these sub-Laplacians are completely explicit [39], so cluster estimates in these settings may be viewed as related to analytic number theory. Other approaches (e.g. via Harmonic Analysis or Spectral Theory) may also be possible.

## APPENDIX A. THE $TT^*$ ARGUMENT

Here, we recall the $TT^*$ argument as presented in [59, Lemma 2.2].

If $Y$ and $Z$ are two vector spaces (without any assumption of norms or topology), we denote the space of linear maps $Y \to Z$ by $\mathcal{L}_a(Y, Z)$.

Let $\mathcal{H}$ be a Hilbert space. Let $X$ a Banach space; the Banach space dual to $X$ is denoted by $X^*$. Let $\mathcal{D}$ be a vector space densely contained in $X$.

If $A \in \mathcal{L}_a(\mathcal{D}, \mathcal{H})$, we define its algebraic adjoint $A^* \in \mathcal{L}(\mathcal{H}, \mathcal{D}_a^*)$ in the following way: the space $\mathcal{D}_a^* = \mathcal{L}(\mathcal{D}, \mathbb{C})$ is the algebraic dual of $\mathcal{D}$ equipped with the algebraic pairing $\langle \cdot, \cdot \rangle_{\mathcal{D}}$, and the operator $A^*$ is defined via

$$ \forall f \in \mathcal{D}, \quad \forall v \in \mathcal{H}, \quad \langle A^* v, f \rangle_{\mathcal{D}} = \langle v, Af \rangle_{\mathcal{H}}, $$

where the scalar product $\langle \cdot, \cdot \rangle_{\mathcal{H}}$ on $\mathcal{H}$ is conjugate linear in the first argument.

**Theorem A.1.** Under the hypotheses above, the following three conditions are equivalent:

(1) There exists $a \in [0, \infty)$ such that

$$ \forall f \in \mathcal{D}, \quad \|Af\|_{\mathcal{H}} \le a \|f\|_{X} $$

(2) The range of $A^*$ is included in $X^*$ and there exists $a \in [0, \infty)$ such that

$$ \forall v \in \mathcal{H}, \quad \|A^*v\|_{\mathbb{X}^*} \le a \|v\|_{\mathcal{H}} $$

(3) The range of $A^*A$ is included in $X^*$ and there exists $a \in [0, \infty)$ such that

$$ \forall f \in \mathcal{D}, \quad \|A^*Af\|_{\mathbb{X}^*} \le a^2 \|f\|_{X}. $$
If one of (all) those conditions is (are) satisfied, the operators $A$ and $A^*A$ extend by continuity to bounded operators $X \to \mathcal{H}$ and $X \to X^*$ respectively. In this case, we may take $a \in [0, +\infty)$ to be the same constant in all three parts with

$$a \ge \|A\|_{\mathcal{L}(X, \mathcal{H})} = \|A^*\|_{\mathcal{L}(\mathcal{H}, X^*)} = \sqrt{\|A^*A\|_{\mathcal{L}(\mathcal{H}, X^*)}}.$$

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(H. Bahouri) CNRS & SORBONNE UNIVERSITÉ, LABORATOIRE JACQUES-LOUIS LIONS (LJLL) UMR 7598, 4, PLACE JUSSIEU, 75005 PARIS, FRANCE.
*Email address: hajer.bahouri@sorbonne-universite.fr*

(V. Fischer) DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF BATH, BATH, BA2 7AY, UK
*Email address: v.c.m.fischer@bath.ac.uk*