Strichartz Estimates and Fourier Restriction Theorems on the Heisenberg Group

Abstract

This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group ${\mathop {\mathbb H}\nolimits }^d$ for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on ${\mathop {\mathbb H}\nolimits }^d$ is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. Our approach, inspired by the Fourier transform restriction method initiated in Tomas (Bull Am Math Soc 81: 477–478, 1975), is based on Fourier restriction theorems on ${\mathop {\mathbb H}\nolimits }^d$, using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.

1 Introduction

1.1 Strichartz estimates

In the past decades, Strichartz estimates for linear evolution equations such as the Schrödinger and wave equations, have been a central tool in the study of semilinear and quasilinear equations, which appear in numerous physical applications. In many cases and particularly in ${\mathop {\mathbb R}\nolimits }^n$, the proof of those inequalities, which involve space-time Lebesgue norms, is a combination of an abstract functional analysis argument known as the $TT^*$-argument (see [31]) and of a dispersive estimate. Concerning the Schrödinger equation on ${\mathop {\mathbb R}\nolimits }^{n}$

$$\begin{aligned} (S) \qquad \left\{ \begin{array}{c} i\partial _t u -\Delta u = 0\\ u_{|t=0} = u_0\,, \end{array} \right. \end{aligned}$$

the dispersive estimate writes (for $t \ne 0$)

$$\begin{aligned} \Vert u(t,\cdot )\Vert _{L^\infty ({\mathop {\mathbb R}\nolimits }^{n})} \le \frac{1}{(4\pi |t|)^{\frac{n}{2}}} \Vert u_0\Vert _{L^1({\mathop {\mathbb R}\nolimits }^{n})}\,, \end{aligned}$$

(1.1)

and can be easily derived from the explicit expression of the solution, which is based on Fourier analysis:

$$\begin{aligned} u(t,\cdot )= \frac{{\mathrm{e}}^{i \frac{|\cdot |^2}{4t} }}{(4 \pi i t)^\frac{n}{2}} \star u_0\, . \end{aligned}$$

The dispersive inequality (1.1) which expresses that waves with different frequencies move at different velocities, gives rise when $u_0$ is in $L^2({\mathop {\mathbb R}\nolimits }^{n})$ to the following Strichartz estimate (see for instance [15,16,17, 22, 37]) for the solution to the free Schrödinger equation

$$\begin{aligned} \Vert u\Vert _{L^q({\mathop {\mathbb R}\nolimits }, L^p({\mathop {\mathbb R}\nolimits }^{n}))} \le C(p,q) \Vert u_0\Vert _{L^2({\mathop {\mathbb R}\nolimits }^{n})}\,, \end{aligned}$$

(1.2)

where (p, q) satisfies the scaling admissibility condition

$$\begin{aligned} \frac{2}{q} + \frac{n}{p} = \frac{n}{2} \quad \hbox {with}\quad q \ge 2 \quad \hbox {and}\quad (n,q,p) \ne (2,2, \infty ) \, . \end{aligned}$$

(1.3)

It is worth noticing that the dispersive inequality (1.1) also yields the following Strichartz inequalities for the inhomogeneous Schrödinger equation, which have proven to be of paramount importance in the study of semilinear and quasilinear Schrödinger equations (one can for instance consult the monograph [6] and the references therein): if (p, q) and $(p_1,q_1)$ satisfy the admissibility condition (1.3), then

$$\begin{aligned} \Vert u\Vert _{L^q({\mathop {\mathbb R}\nolimits }, {L^{p}({\mathop {\mathbb R}\nolimits }^{n}))}} \le C(p,q, p_1,q_1) \Big ( \Vert u_0\Vert _{L^{2}({\mathop {\mathbb R}\nolimits }^{n})} + \Vert i\partial _t u -\Delta u\Vert _{L^{q'_1}({\mathop {\mathbb R}\nolimits }, {L^{{p'_1}}({\mathop {\mathbb R}\nolimits }^{n}))}}\Big )\,, \end{aligned}$$

denoting by $a'$ the dual exponent of any $a \in [1,\infty ]$¹.

In the case of the wave equation on ${\mathop {\mathbb R}\nolimits }^{n}$

$$\begin{aligned} (W) \qquad \left\{ \begin{array}{c} \partial _t^2 u -\Delta u = 0\\ (u,\partial _t u)_{|t=0} = (u_0,u_1)\,, \end{array} \right. \end{aligned}$$

the solution of which may be written by means of the Fourier transform $\mathcal F$ (in the case when $\mathcal F u_0$ and $\mathcal F u_1$ are supported in a ring) as $\displaystyle u(t) = \sum _{\pm } u_{\pm }(t)$ with

$$\begin{aligned} u_{\pm }(t) = {\mathcal F}^{-1} \big ( e^{\pm it|\xi |} \gamma _\pm (\xi )\big )\, , \quad \gamma _\pm (\xi )\buildrel \hbox { def}\over =\frac{1}{2} \big ( {\mathcal F} u_0(\xi ) \pm \frac{1}{i|\xi |}{\mathcal F} u_1(\xi ) \big ) \,, \end{aligned}$$

(1.4)

the dispersive estimate writes (for $t \ne 0$)

$$\begin{aligned} \Vert u(t,\cdot )\Vert _{L^\infty ({\mathop {\mathbb R}\nolimits }^{n})} \le \frac{C}{|t|^{\frac{n-1}{2}}} ( \Vert u_0\Vert _{L^1({\mathop {\mathbb R}\nolimits }^{n})}+ \Vert u_1\Vert _{L^1({\mathop {\mathbb R}\nolimits }^{n})})\,. \end{aligned}$$

(1.5)

Its proof requires more elaborate techniques involving oscillatory integrals and the application of a stationary phase theorem. This dispersive estimate leads to the following Strichartz estimate (see for instance [6, 31, 37] and the references therein)

$$\begin{aligned} \Vert u\Vert _{L^q({\mathop {\mathbb R}\nolimits }, L^p({\mathop {\mathbb R}\nolimits }^{n}))} \le C(p,q) \big ( \Vert \nabla u_0\Vert _{L^2({\mathop {\mathbb R}\nolimits }^{n})} + \Vert u_1\Vert _{L^2({\mathop {\mathbb R}\nolimits }^{n})} \big ) \,, \end{aligned}$$

(1.6)

where (p, q) satisfies the scaling admissibility condition

$$\begin{aligned} \frac{1}{q} + \frac{n}{p} = \frac{n}{2} -1\quad \hbox {with}\quad p,q \ge 2 \quad \hbox {and}\quad q<\infty \,. \end{aligned}$$

(1.7)

If (p, q) and $(p_1,q_1)$ satisfy (1.7), one can also infer

$$\begin{aligned} \Vert u\Vert _{L^q({\mathop {\mathbb R}\nolimits }, {L^{p}({\mathop {\mathbb R}\nolimits }^{n}))}}\le & {} C(p,q, p_1,q_1)\\&\times \Big ( \Vert \nabla u_0\Vert _{L^2({\mathop {\mathbb R}\nolimits }^{n})} + \Vert u_1\Vert _{L^2({\mathop {\mathbb R}\nolimits }^{n})} + \Vert \partial _t^2 u -\Delta u\Vert _{L^{q'_1}({\mathop {\mathbb R}\nolimits }, {L^{{p'_1}}({\mathop {\mathbb R}\nolimits }^{n}))}}\Big )\,. \end{aligned}$$

When some loss of dispersion occurs, as for instance in the case of compact Riemannian manifolds and of some bounded domains, or as was highlighted by Bahouri, Gérard and Xu in [12] in the case of the Schrödinger operator on ${\mathop {\mathbb H}\nolimits }^d$ (where it is shown that there is no dispersion at all), the Euclidean strategy referred to above fails and the problem of obtaining Strichartz estimates is considered as very difficult. Strichartz estimates in the setting of compact Riemannian manifolds and bounded domains (with a possible loss of derivatives) have been obtained in a number of works (see for instance Bourgain [18], Burq, Gérard and Tzvetkov [19], Ivanovici, Lebeau and Planchon [34] and the references therein). The case of the hyperbolic space (noncompact and negatively curved) is also considered in [2].

Even if the study of PDEs associated with sublaplacians on nilpotent groups is nowadays classical (see for instance the pioneering works [30, 33, 40]), obtaining Strichartz estimates for the Schrödinger operator on the Heisenberg group is still open and has to our knowledge never been tackled. Note that the Heisenberg group is one of the simplest examples of a noncommutative Lie group, whence our interest in proving those estimates in this setting. We are confident that our methods should apply to more general nilpotent Lie groups, provided some harmonic and Fourier analysis tools (that will be introduced in the setting of ${\mathop {\mathbb H}\nolimits }^d$ in Sect. 3) are extended from the Heisenberg framework to the context of these groups. This is for instance the case of H-type groups or more generally of step 2 stratified Lie groups: in [25] and [9] the lack of dispersion for the associated Schrödinger operators is indeed proved. We also refer to [13, 14] for a discussion about the link between dispersion, restriction estimates and the heat semigroup and to [29] for a study of the cubic Schrödinger equation on the Heisenberg group.

In this paper our main goal is thus to establish Strichartz estimates for the solutions to the linear Schrödinger equation on the Heisenberg group ${\mathop {\mathbb H}\nolimits }^d$, involving the sublaplacian, as well as for the wave equation. As already mentioned, in [12] the authors show the absence of dispersion – they actually prove that the Schrödinger equation on ${\mathop {\mathbb H}\nolimits }^d$ behaves as a transport equation with respect to one direction, known as the vertical direction (i.e., along the orbits of the Reeb vector field). But as will be clear later, a salutary fact is that the Schrödinger operator on ${\mathop {\mathbb H}\nolimits }^d$ behaves rather well in the complement to that vertical direction. This enables us to derive anisotropic Strichartz estimates for the Schrödinger operator on ${\mathop {\mathbb H}\nolimits }^d$, by adapting the Fourier transform restriction analysis initiated in [42] and [45] in the Euclidean case (see also [28]); this also leads to new, anisotropic Strichartz estimates for the wave equation, at least for the radial case. The approach we set up here is somewhat more challenging than in the Euclidean case because the Fourier analysis on the Heisenberg group is an intricate tool.

1.2 Basic facts about the Heisenberg group

Let us start by recalling that the d-dimensional Heisenberg group ${\mathop {\mathbb H}\nolimits }^d$ can be defined as $T^\star {\mathop {\mathbb R}\nolimits }^d \times {\mathop {\mathbb R}\nolimits }$ where $T^\star {\mathop {\mathbb R}\nolimits }^d$ is the cotangent bundle, endowed with the noncommutative product law

$$\begin{aligned} (Y,s)\cdot (Y',s') \buildrel \hbox { def}\over =\big (Y+Y', s+s'+2 \sigma (Y,Y')\big )\,, \end{aligned}$$

(1.8)

where²$w=(Y,s)=(y,\eta ,s)$ and $w'=(Y',s')=(y',\eta ',s')$ are elements of ${\mathop {\mathbb H}\nolimits }^d$, while $\sigma$ denotes the canonical symplectic form on $T^\star {\mathop {\mathbb R}\nolimits }^d$ defined by

$$\begin{aligned} \sigma (Y,Y') \buildrel \hbox { def}\over =\langle \eta ,y'\rangle -\langle \eta ',y\rangle \quad \hbox {for all }\ (Y,Y')\in T^\star {\mathop {\mathbb R}\nolimits }^d \times T^\star {\mathop {\mathbb R}\nolimits }^d\, , \end{aligned}$$

(1.9)

with $\langle \eta ,y\rangle$ the value of the one-form $\eta$ at y.

With this point of view, the Haar measure on ${\mathop {\mathbb H}\nolimits }^d$ is simply the Lebesgue measure on the space $T^\star {\mathop {\mathbb R}\nolimits }^d \times {\mathop {\mathbb R}\nolimits }$. In particular, one can define the following (noncommutative) convolution product for any two integrable functions f and g:

$$\begin{aligned} f \star g ( w ) \buildrel \hbox { def}\over =\int _{{\mathop {\mathbb H}\nolimits }^d} f ( w \cdot v^{-1} ) g( v)\, dv = \int _{{\mathop {\mathbb H}\nolimits }^d} f ( v ) g( v^{-1} \cdot w)\, dv\, . \end{aligned}$$

(1.10)

Even though the convolution on the Heisenberg group is noncommutative, if one defines the Lebesgue spaces $L^p({\mathop {\mathbb H}\nolimits }^d)$ to be simply $L^p(T^\star {\mathop {\mathbb R}\nolimits }^d \times {\mathop {\mathbb R}\nolimits }),$ then one still obtains Hölder and Young inequalities, in their classical and weak versions. In order to distinguish the vertical coordinate from the others, we shall also be using, for any two real numbers $1\le p,r \le \infty$, the anistropic Lebesgue spaces $L^{p}_{Y}L^{r}_{s}({\mathop {\mathbb H}\nolimits }^d)$ and $L^{r}_{s}L^{p}_{Y}({\mathop {\mathbb H}\nolimits }^d)$ endowed with the mixed norms

$$\begin{aligned}&\Vert f\Vert _{L^{p}_{Y}L^{r}_{s}}\buildrel \hbox { def}\over =\left( \int \left( \int |f(Y,s)| ^r \, ds\right) ^\frac{p}{r} \, dY\right) ^\frac{1}{p},\\&\quad \Vert f\Vert _{L^{r}_{s}L^{p}_{Y}}\buildrel \hbox { def}\over =\left( \int \left( \int |f(Y,s)| ^p \, dY\right) ^\frac{r}{p} \, ds\right) ^\frac{1}{r}\, . \end{aligned}$$

In the framework of the Heisenberg group, the scale invariance is investigated through the family of dilation operators $(\delta _a)_{a>0}$ (which are compatible with the product law (1.8)) defined by

$$\begin{aligned} \delta _a ( Y, s ) \buildrel \hbox { def}\over =( a Y, a^2s )\, .\end{aligned}$$

(1.11)

As the determinant of $\delta _a$ is $a^{2d+2}$, it is natural to define the homogeneous dimension of ${\mathop {\mathbb H}\nolimits }^d$ to be $Q\buildrel \hbox { def}\over =2d+2$.

The Schwartz class ${\mathcal S}({\mathop {\mathbb H}\nolimits }^d)$ coincides with ${\mathcal S}({\mathop {\mathbb R}\nolimits }^{2d+1})$, and can be characterized by the action of the sublaplacian

$$\begin{aligned} \Delta _{{\mathop {\mathbb H}\nolimits }}u \buildrel \hbox { def}\over =\sum _{j=1} ^d ({\mathcal X}_j^2u+\Xi _j^2u) \, , \end{aligned}$$

where the horizontal vector fields ${\mathcal X}_j$ and $\Xi _j$ are defined for $j\in \{1,\dots ,d\}$ by

$$\begin{aligned} {\mathcal X}_j\buildrel \hbox { def}\over =\partial _{y_j} +2\eta _j\partial _s\quad \hbox {and}\quad \Xi _j\buildrel \hbox { def}\over =\partial _{\eta _j} -2y_j\partial _s \, . \end{aligned}$$

(1.12)

We also define the horizontal gradient

$$\begin{aligned} \nabla _{{\mathop {\mathbb H}\nolimits }}u \buildrel \hbox { def}\over =({\mathcal X}_1 u,\ldots ,{\mathcal X}_d u,\Xi _1 u,\ldots ,\Xi _d u) \, . \end{aligned}$$

Note that Sobolev spaces can be defined thanks to the sublaplacian, for instance for any real number s

$$\begin{aligned} H^s({\mathop {\mathbb H}\nolimits }^d)\buildrel \hbox { def}\over =\Big \{u \in L^2({\mathop {\mathbb H}\nolimits }^d) \, , (-\Delta _{\mathop {\mathbb H}\nolimits })^\frac{s}{2} u \in L^2({\mathop {\mathbb H}\nolimits }^d) \Big \}\, , \end{aligned}$$

where non integer powers of $-\Delta _{\mathop {\mathbb H}\nolimits }$ can be defined via functional calculus. The purpose of this paper is to establish Strichartz estimates for the linear Schrödinger and wave equations on ${\mathop {\mathbb H}\nolimits }^d$ associated with the sublaplacian

$$\begin{aligned} (S_{\mathop {\mathbb H}\nolimits })\quad \left\{ \begin{array}{c} i\partial _t u -\Delta _{\mathop {\mathbb H}\nolimits }u = f\\ u_{|t=0} = u_0\,, \end{array} \right. \qquad \quad (W_{\mathop {\mathbb H}\nolimits })\quad \left\{ \begin{array}{c} \partial _t^2 u -\Delta _{\mathop {\mathbb H}\nolimits }u = f\\ (u,\partial _t u)_{|t=0} = (u_0,u_1)\,. \end{array} \right. \end{aligned}$$

As in the Euclidean case, among the most notable achievements of Fourier analysis on the Heisenberg group that we review in Sect. 3, one can mention that one can explicitly solve those equations by means of the Fourier transform. However as shown by the following proposition established in [12], $(S_{\mathop {\mathbb H}\nolimits })$ is a model for totally non-dispersive evolution equations.

Proposition 1.1

([12]) There exists a function $u_0$ in the Schwartz class ${\mathcal S}({\mathop {\mathbb H}\nolimits }^{d})$ such that the solution to the free Schrödinger equation $(S_{\mathop {\mathbb H}\nolimits })$ (with $f\equiv 0)$ satisfies

$$\begin{aligned} u(t,Y,s) = u_0(Y,s+4td)\, . \end{aligned}$$

Remark 1.2

Since the translation $(Y,s)\mapsto (Y,s+s_0)$ leaves the Lebesgue measure invariant for all $s_0 \in {\mathop {\mathbb R}\nolimits }$, the solution constructed in Proposition 1.1 satisfies

$$\begin{aligned} \forall p \in [1,\infty ]\,,\ \Vert u(t,\cdot )\Vert _{L^p({\mathop {\mathbb H}\nolimits }^{d})} = \Vert u_0\Vert _{L^p({\mathop {\mathbb H}\nolimits }^{d})} \end{aligned}$$

which shows that one cannot hope for a dispersion phenomenon of the type (1.1).

Proof

In order to establish Proposition 1.1, let us introduce a family of functions on ${\mathop {\mathbb H}\nolimits }^{d}$ which are the analogues of the solutions associated with plane waves in the classical Euclidean case, namely

$$\begin{aligned} (t,x)\in {\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb R}\nolimits }^n\longmapsto e^{i|\xi |^2 t +i\langle \xi ,x\rangle }\in {\mathbb S}^1 \end{aligned}$$

which of course satisfy

$$\begin{aligned} (i\partial _t -\Delta ) e^{i|\xi |^2 t +i\langle \xi ,x\rangle } =0\,. \end{aligned}$$

Similarly, consider the family of functions

$$\begin{aligned} \Theta _\lambda : (Y,s)\in {\mathop {\mathbb H}\nolimits }^d \longmapsto e^{is\lambda } e^{-\lambda |Y|^2}\in \mathop {\mathbb C}\nolimits \,. \end{aligned}$$

(1.13)

One can readily check that

$$\begin{aligned} -\Delta _{{\mathop {\mathbb H}\nolimits }} \Theta _\lambda = 4\lambda d \Theta _\lambda \,, \end{aligned}$$

(1.14)

therefore the functions

$$\begin{aligned} (t,Y,s)\in {\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb H}\nolimits }^d \longmapsto \Theta _\lambda (Y,s+4td)\in \mathop {\mathbb C}\nolimits \end{aligned}$$

satisfy

$$\begin{aligned} (i\partial _t -\Delta _{\mathop {\mathbb H}\nolimits }) \big ( \Theta _\lambda (Y, s+4td)\big ) =0\,. \end{aligned}$$

Now let g be a function in ${\mathcal D}(]0,\infty [)$, and define

$$\begin{aligned} u(t,Y,s) \buildrel \hbox { def}\over =\int _{{\mathop {\mathbb R}\nolimits }} \Theta _\lambda (Y,s+4td) g(\lambda ) \, |\lambda |^d d\lambda \,. \end{aligned}$$

It stems from the Lebesgue derivation theorem that u solves the Cauchy problem $(S_{\mathop {\mathbb H}\nolimits })$ with $f \equiv 0$ and initial data

$$\begin{aligned} u_0(Y,s) = \int _{{\mathop {\mathbb R}\nolimits }} \Theta _\lambda (Y,s) g(\lambda ) \, |\lambda |^d d\lambda \,, \end{aligned}$$

(1.15)

which easily ends the proof of the proposition. $\square$

Actually, as we shall see in Sect. 3 page 13, there is a family of functions $(\Theta ^{(\ell )}_\lambda )_{\ell \in \mathop {\mathbb N}\nolimits }$ on ${\mathop {\mathbb H}\nolimits }^d$ such that³

$$\begin{aligned} (i\partial _t -\Delta _{\mathop {\mathbb H}\nolimits }) \big ( \Theta ^{(\ell )}_\lambda (Y, s+4t(2\ell +d))\big ) =0\,. \end{aligned}$$

(1.16)

This readily ensures that the solution to the free Schrödinger equation $(S_{\mathop {\mathbb H}\nolimits })$ associated to the Cauchy data

$$\begin{aligned} u^{(\ell )}_0(Y,s) \buildrel \hbox { def}\over =\int _{{\mathop {\mathbb R}\nolimits }} \Theta ^{(\ell )}_\lambda (Y, s) g(\lambda ) \, |\lambda |^d d\lambda \,, \end{aligned}$$

with $g \in {\mathcal D}(]0,\infty [)$, behaves as a transport equation, with velocity depending on $\ell$. More precisely, we have

$$\begin{aligned} (i\partial _t -\Delta _{\mathop {\mathbb H}\nolimits }) u^{(\ell )}_0= i (\partial _t - 4 (2\ell +d) \partial _s) u^{(\ell )}_0\,, \end{aligned}$$

(1.17)

which again highlights the fact that one cannot hope for a dispersion phenomenon of the type (1.1).

Remark 1.3

In [12] the authors also prove that every solution to the wave equation on ${\mathop {\mathbb H}\nolimits }^{d}$ satisfies the dimension-independent dispersive estimate

$$\begin{aligned} \Vert u(t,\cdot )\Vert _{L^\infty ({\mathop {\mathbb H}\nolimits }^{d})} \le \frac{C}{|t|^{\frac{1}{2}}} ( \Vert u_0\Vert _{L^1({\mathop {\mathbb H}\nolimits }^{d})}+ \Vert u_1\Vert _{L^1({\mathop {\mathbb H}\nolimits }^{d})})\,,\end{aligned}$$

(1.18)

and show by an example similar to the ones above that this estimate is optimal. The rate of decay in (1.18) regardless to the dimension is due to the fact that only the center is involved in the dispersive effect. Note also that compared with the Euclidean framework, there is an exchange in the rates of decay between the wave and the Schrödinger equations on ${\mathop {\mathbb H}\nolimits }^{d}$. It is also proved in [12] that the dispersive estimate (1.18) gives rise to a Strichartz estimate

$$\begin{aligned} \Vert u\Vert _{L^q_t L^p_{Y,s}} \le C_{p, q,p_1,q_1} \Big ( \Vert \nabla _{{\mathop {\mathbb H}\nolimits }^d} u_0\Vert _{L^2({\mathop {\mathbb H}\nolimits }^{d})} + \Vert u_1\Vert _{L^2({\mathop {\mathbb H}\nolimits }^{d})} + \Vert f\Vert _{ L^{q_1'}_t L^{p'_1}_{Y,s}}\Big ) \end{aligned}$$

with $\displaystyle \frac{1}{q}+\frac{Q}{p} = \frac{Q}{2} -1$ and $q \ge 2Q-1$.

1.3 Statements of the results

Our first goal in this paper is to establish the following Strichartz estimates for the Schrödinger equation on ${\mathop {\mathbb H}\nolimits }^{d}$ for radial data — note that the Fourier transform in the radial setting is much easier to handle, and the geometry of sets on the Fourier side is also much easier to describe in the radial case (see for example (4.1) in Sect. 4.1 for the sphere), so we restrict our attention to that framework in this article. A function f on  ${\mathop {\mathbb H}\nolimits }^{d}$ is said to be radial if it is invariant under the action of the unitary group U(d) of $T^\star {\mathop {\mathbb R}\nolimits }^d$, which implies that f can be written under the form $f(Y,s)= f(|Y|, s)$.

Theorem 1

Given (p, q) belonging to the admissible set

$$\begin{aligned} {\mathcal A}^{{\text{ S }}}\buildrel \hbox { def}\over =\Big \{(p,q)\in [2,\infty ]^2\, /\,p\le q \quad \text{ and } \quad \frac{2}{q}+\frac{2d}{p}\le \frac{Q}{2}\Big \} \,, \end{aligned}$$

there is a constant $C_{p, q}$ such that the solution to the Schrödinger equation $(S_{\mathop {\mathbb H}\nolimits })$ associated with radial data satisfies the following Strichartz estimate (denoting by $a'$ the dual exponent of any $a \in [1,\infty ])$

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L^{q}_t L^{p}_{Y}} \le C_{p, q } \Big (\Vert u_0\Vert _{H^{ \frac{Q}{2} - \frac{2}{q}-\frac{2d}{p}}({\mathop {\mathbb H}\nolimits }^{d})} + \Vert f\Vert _{L^1({\mathop {\mathbb R}\nolimits }, H^{ \frac{Q}{2} - \frac{2}{q}-\frac{2d}{p}}({\mathop {\mathbb H}\nolimits }^{d}))} \Big ) \,. \end{aligned}$$

(1.19)

Remark 1.4

The above theorem deserves some comments:

• First notice that the limit case when $\displaystyle \frac{2}{q}+\frac{2d}{p}= \frac{Q}{2}$ provides (when $f=0$) the estimate

  $$\begin{aligned} \Vert u\Vert _{L^\infty _s L^{2}_t L^{2}_{Y}} \le C \Vert u_0\Vert _{L^2({\mathop {\mathbb H}\nolimits }^d)} \end{aligned}$$

  which has the flavor of the energy equality $\Vert u\Vert _{L^\infty _t L^2({\mathop {\mathbb H}\nolimits }^d)} = \Vert u_0\Vert _{L^2({\mathop {\mathbb H}\nolimits }^d)}$ but is of course of a different nature.

• In the case when $\displaystyle \frac{2}{q}+\frac{2d}{p} < \frac{Q}{2}$ we have a larger range of admissible Strichartz pairs, but with a loss of derivatives compared to the euclidean framework. Such phenomena have been proved to hold in Riemannian compact manifolds as well as in the case of variable-coefficient evolution equations with low regularity (see for instance [4, 5, 18, 19, 44] and the references therein).

• Note that the Strichartz estimate (1.19) is invariant by scaling (through the scaling $u (t,w)\mapsto u (\Lambda ^{-2} t, \delta _{\Lambda ^{-1}}w)$ and $f (t,w)\mapsto \Lambda ^{-2} f (\Lambda ^{-2} t, \delta _{\Lambda ^{-1}}w)$.)

• A natural open question is to obtain Strichartz estimates without loss of derivatives for a larger range of admissible Strichartz pairs. It is difficult to state a conjecture on the range of indices for which the estimates should be valid and the heart of the matter relies on the possibility to relax or not the constraint $p\ge 2$. Actually, as will be seen later, our approach is inspired by the remarkable paper of Müller [38] where a counter-example is provided for In order to investigate the sharpness of our results, one first needs to have more insight about the restriction result of Müller for $p_d\le p<2$ which is a very challenging issue (see Sect. 4.2 for further details).

In the case of the wave equation we obtain the following Strichartz estimate.

Theorem 2

With the above notation, given (p, q) belonging to the admissible set

$$\begin{aligned} {\mathcal A}^{{\text{ W }}}\buildrel \hbox { def}\over =\Big \{(p,q)\in [2,\infty ]^2\, /\, p \le q \quad \text{ and } \quad \frac{1}{q}+\frac{2d}{p} \le \frac{Q}{2}-1\Big \} \,, \end{aligned}$$

there is a constant $C_{p, q}$ such that the solution to the wave equation $(W_{\mathop {\mathbb H}\nolimits })$ associated with radial data satisfies the following Strichartz estimate:

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^\infty _s L^{q}_t L^{p}_{Y}} \le C_{p, q} \Big (&\Vert u_0\Vert _{H^{ \frac{Q}{2} - \frac{1}{q}-\frac{2d}{p}}({\mathop {\mathbb H}\nolimits }^{d})} + \Vert u_1\Vert _{H^{ \frac{Q}{2} - \frac{1}{q}-\frac{2d}{p}- 1}({\mathop {\mathbb H}\nolimits }^{d})}+ \Vert f\Vert _{L^1({\mathop {\mathbb R}\nolimits }, H^{ \frac{Q}{2} - \frac{1}{q}-\frac{2d}{p} - 1}({\mathop {\mathbb H}\nolimits }^{d}))}\Big )\,. \end{aligned} \end{aligned}$$

Remark 1.5

Note that in the limit case when $\displaystyle \frac{1}{q}+\frac{2d}{p}= \frac{Q}{2} -1,$ the set of admissible Strichartz pairs is wider than for the Schrödinger equation. In some sense, this is not surprising since as was already highlighted in Remark 1.3, compared with the Euclidean framework there is an exchange in the behavior between the wave and the Schrödinger equations on ${\mathop {\mathbb H}\nolimits }^{d}$, and as is well-known, in the Euclidean case the Schrödinger equation enjoys better dispersive estimates than the wave equation.

Our strategy of proof of the estimates is closely related to the method developed in [42] (the reader may consult [43] and the references therein for an overview on this subject in the Euclidean framework, as well as Sect. 2 below) consisting in reducing the problem to the study of the restriction operator on a manifold in Fourier space — with additional non negligible technicalities owing to the complexity of the Fourier transform on the Heisenberg group. That is actually the main achievement of this paper. At this stage, one should mention the Fourier restriction theorem on ${\mathop {\mathbb H}\nolimits }^{d}$ due to Müller ( [38]), where the author investigated the restriction of the Heisenberg Fourier transform on the unit sphere and emphasized the separate roles of the horizontal and vertical variables of ${\mathop {\mathbb H}\nolimits }^{d}$.

Our limitation in Theorems 1 and 2 to radial data is intimately linked to the complexity of the manifolds in the Fourier side outside the radial framework: see Remark 4.1 for more on this.

Other results extending the restriction theorem of Müller to more general nilpotent groups through spectral analysis have been considered in [20, 21] and [35, 36]. Finally, let us mention that applications of non commutative Fourier analysis have been also used to study the heat equation associated to sublaplacians on groups, see for instance [1]. For our purposes, we need Fourier restriction estimates in a direct product of the Heisenberg group and the real line, which will be obtained by combining the methods of Müller [38] and Tomas-Stein [45].

1.4 Layout

The proof of Theorems 1 and 2 is addressed in Sect. 5. A short illustration of the proof in the (well-known) Euclidean case is provided in Sect. 2 for the convenience of the reader. The Fourier transform on ${\mathop {\mathbb H}\nolimits }^d$ and the space of frequencies $\widehat{\mathop {\mathbb H}\nolimits }^d$ are defined and described in Sect. 3, while Sect. 4 is dedicated to the study of the restriction of the Heisenberg Fourier transform to the unit sphere of the frequency space $\widehat{\mathop {\mathbb H}\nolimits }^d$: this is not strictly necessary to the proof of our main results but will be a way of introducing our methods, by recovering the results of Müller [38] in a slightly simpler setting. Finally in the Appendix we recall some properties of $\lambda$-twisted convolutions which are needed in the proof.

To avoid heaviness, all along this article C will denote a positive constant which may vary from line to line. We also use $f\lesssim g$ to denote an estimate of the form $f\le C g$.

2 Fourier restriction theorem and its applications in the Euclidean space

In this section we recall some classical results on the Fourier restriction problem and its application to PDEs in the classical, Euclidean setting for the convenience of the reader, since we shall follow a similar approach in our framework. To keep the notation consistent with the case of the Heisenberg group that follows, we distinguish ${\mathop {\mathbb R}\nolimits }^{n}$ and its dual $\widehat{\mathop {\mathbb R}\nolimits }^{n}$, which is of course isomorphic to ${\mathop {\mathbb R}\nolimits }^{n}$ itself.

2.1 Restriction theorems

The Fourier transform ${\mathcal F}(f)$ of a function f in $L^1({\mathop {\mathbb R}\nolimits }^n)$ is continuous, thus it makes sense to restrict ${\mathcal F}(f)$ to any subset of $\widehat{\mathop {\mathbb R}\nolimits }^n$. However, the Fourier transform of a function in $L^2({\mathop {\mathbb R}\nolimits }^n)$ is, in general, only in $L^2(\widehat{\mathop {\mathbb R}\nolimits }^n)$, hence completely arbitrary on a set $\widehat{S}$ of $\widehat{\mathop {\mathbb R}\nolimits }^n$ of measure zero.

Indeed, in general, the Fourier transform of a function in $L^{p}$ for $p>1$ cannot be restricted to an hyperplane. As one can easily check, the function $f:{\mathop {\mathbb R}\nolimits }^{n}\rightarrow {\mathop {\mathbb R}\nolimits }$ defined by

$$\begin{aligned} \displaystyle f(x)= \frac{e^{- |x'|^2}}{1+|x_1|}\, ,\qquad x=(x_1,x')\in {\mathop {\mathbb R}\nolimits }^{n}, \end{aligned}$$

(2.1)

belongs to $L^p({\mathop {\mathbb R}\nolimits }^n)$, for all $p>1$, but its Fourier transform does not admit a restriction on the hyperplane $\widehat{S}$ of $\widehat{\mathop {\mathbb R}\nolimits }^{n}$ defined by $\widehat{S}=\{\xi \in \widehat{\mathop {\mathbb R}\nolimits }^n \, /\, \xi _1=0 \}$.

Tomas and Stein made the surprising discovery that one can restrict the Fourier transform of $L^p({\mathop {\mathbb R}\nolimits }^n)$ functions, for $p>1$ (and close to 1), to hypersurfaces $\widehat{S}$ that are “sufficiently curved”, as for instance the sphere. More generally, given a hypersurface $\widehat{S} \subset \widehat{\mathop {\mathbb R}\nolimits }^n$ endowed with a smooth measure $d\sigma$, the restriction problem asks for which pairs (p, q) an inequality of the form

$$\begin{aligned} \Vert {\mathcal F}( f)|_{\widehat{S}}\Vert _{L^q (\widehat{S}, d \sigma )}\le C \Vert f\Vert _{L^p ({\mathop {\mathbb R}\nolimits }^n)} \end{aligned}$$

(2.2)

holds for all f in ${\mathcal S}({\mathop {\mathbb R}\nolimits }^n)$.

Despite all the recent progresses in this field, this question is not completely settled in its general form and remains a topical issue. For a general survey on these questions we refer to the book of Stein [41] and the text of Tao [43]. In what follows, we focus on the case $q=2$.

By a duality argument, the above question for $q=2$ is equivalent to asking whether the adjoint operator $R^{*}_{S}$ defined by

$$\begin{aligned} R^{*}_{S}g\buildrel \hbox { def}\over ={\mathcal F}^{-1} (gd\sigma ) \end{aligned}$$

is continuous from $L^{2}(\widehat{S}, d\sigma )$ to $L^{p'}({\mathop {\mathbb R}\nolimits }^n)$, where $p'$ is the dual exponent of p.

A basic counterexample shows that the range of p for which the estimate holds cannot be the entire interval $1\le p\le 2$; for details we refer to [42].

Example 1

(Knapp) Let $\widehat{S}$ be the $(n-1)$-dimensional sphere in $\widehat{\mathop {\mathbb R}\nolimits }^{n}$ endowed with the standard measure $d\mu$. Let $g_{\delta }$ be the characteristic function of a spherical cap

$$\begin{aligned} \widehat{C}_{\delta }\buildrel \hbox { def}\over =\{x\in \widehat{S} : |x\cdot e_{n}|<\delta \}\,. \end{aligned}$$

With some computation one can prove that as $\delta \rightarrow 0$,

$$\begin{aligned} \Vert g_{\delta }\Vert _{L^{2}(\widehat{S},d\mu )}\sim \delta ^{(n-1)/2},\qquad \Vert {\mathcal F}^{-1}(g_{\delta }) \Vert _{L^{p'}( {\mathop {\mathbb R}\nolimits }^{n})}\ge C \delta ^{n-1}\delta ^{-(n+1)/p'}\,, \end{aligned}$$

hence the estimate can hold only if $p'\ge (2n+2)/(n-1)$, i.e., if $p\le (2n+2)/(n+3)$.

The above range is indeed the correct one in the case of a surface with non vanishing curvature. This is the statement of the so-called Tomas-Stein theorem.

Theorem 3

([45]) Let $\widehat{S}$ be a smooth compact hypersurface in $\widehat{\mathop {\mathbb R}\nolimits }^{n}$ with non vanishing Gaussian curvature at every point, and let $d\sigma$ be a smooth measure on $\widehat{S}$. Then there holds for every $f\in {\mathcal S}({\mathop {\mathbb R}\nolimits }^n)$ and every $p\le (2n+2)/(n+3)$,

$$\begin{aligned} \Vert {\mathcal F}(f)|_{\widehat{S}}\Vert _{L^2 (\widehat{S}, d \sigma )}\le C_p \Vert f\Vert _{L^p ({\mathop {\mathbb R}\nolimits }^n)}\, . \end{aligned}$$

A similar result is possible for surfaces with vanishing Gaussian curvature (that are not flat). In this case the range of p is smaller depending on the order of tangency of the surface to its tangent space. The assumption about compactness of $\widehat{S}$ can be removed by replacing $d\sigma$ with a compactly supported smooth measure.

2.2 Application of restriction theorems to some PDEs

Restriction estimates have several applications, from spectral theory to number theory. Here we recall some of these to PDEs: indeed, the restriction theorem can be efficiently applied to obtain Strichartz estimates on the solutions to some PDEs. Here we focus on the Schrödinger and wave equations, for which these estimates were first discovered by Strichartz in his seminal work [42].

Let us first consider the classical Schrödinger equation (S) in ${\mathop {\mathbb R}\nolimits }^{n}$, recalled in the introduction page 1. Given a solution u(t, x) of this equation, the Fourier transform ${\widehat{u}}(t,\xi )$ with respect to the spatial variable x satisfies

$$\begin{aligned} i\partial _{t}{\widehat{u}}(t,\xi )=-|\xi |^{2} {\widehat{u}}(t,\xi ), \qquad {\widehat{u}}(0,\xi )=\widehat{u}_{0}(\xi ). \end{aligned}$$

(2.3)

Solving the corresponding ODE and taking the inverse Fourier transform one has

$$\begin{aligned} u(t,x)=\int _{\widehat{\mathop {\mathbb R}\nolimits }^{n}}e^{i(x\cdot \xi + t|\xi |^{2})}\widehat{u}_{0}(\xi )d\xi \, . \end{aligned}$$

(2.4)

Formula (2.4) can be interpreted as the restriction of the Fourier transform on the paraboloid $\widehat{S}$ in the space of frequencies $\widehat{\mathop {\mathbb R}\nolimits }^{n+1}=\widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb R}\nolimits }^{n}$, defined as

$$\begin{aligned} \widehat{S}\buildrel \hbox { def}\over =\Big \{(\alpha ,\xi )\in \widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb R}\nolimits }^{n}\mid \alpha =|\xi |^{2} \Big \}\, . \end{aligned}$$

Let us endow $\widehat{S}$ with the measure $d\sigma =d\xi$ induced by the projection $\pi : \widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb R}\nolimits }^{n}\rightarrow \widehat{\mathop {\mathbb R}\nolimits }^n$ onto the second factor. More formally one should write⁴$d\sigma =(\pi |_{\widehat{S}}^{-1})_{\sharp }d\xi$. Notice that $\pi |_{\widehat{S}}$ is invertible and $d\sigma$ is not the intrinsic surface measure of $\widehat{S}$, which is written in coordinates as $d\mu =\sqrt{1+2|\xi |}d\xi$.

Given $\widehat{u}_{0}:\widehat{\mathop {\mathbb R}\nolimits }^{n}\rightarrow \mathop {\mathbb C}\nolimits$ define $g:\widehat{S} \rightarrow \mathop {\mathbb C}\nolimits$ as $g=\widehat{u}_{0}\circ \pi |_{\widehat{S}}$. In other words $g(|\xi |^{2},\xi )=\widehat{u}_{0}(\xi )$. By construction, for $\widehat{u}_{0}\in L^{2}(\widehat{\mathop {\mathbb R}\nolimits }^{n})$ one has $g\in L^{2}(\widehat{S},d\mu )$ and $\Vert u_{0}\Vert _{L^{2}(\widehat{\mathop {\mathbb R}\nolimits }^{n})}=\Vert g\Vert _{L^{2}(\widehat{S},d\mu )}$. Then

$$\begin{aligned} u(t,x)=\int _{{\mathop {\mathbb R}\nolimits }^{n}}e^{i(x\cdot \xi + t|\xi |^{2})}\widehat{u}_{0}(\xi )d\xi = \int _{\widehat{S}}e^{i y\cdot z} g(z)d\sigma (z) \end{aligned}$$

where $y=(t,x)$ and $z=(\alpha ,\xi )$. Theorem 3, in dual form, tells us that

$$\begin{aligned} \Vert {\mathcal F}^{-1}(gd\sigma ) \Vert _{L^{p'}(\widehat{\mathop {\mathbb R}\nolimits }^{n+1})}\le C_p \, \Vert g\Vert _{L^{2}(\widehat{S},d\mu )}\, , \end{aligned}$$

(2.5)

for all $g\in L^{2}(\widehat{S},d\mu )$ and all $p' \ge 2(n+2)/n$ (we stress that we apply the result in dimension $n+1$, i.e., in ${\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb R}\nolimits }^{n}={\mathop {\mathbb R}\nolimits }^{n+1}$).

Hence applying the statement to g related to a initial data $u_{0}$ such that $\widehat{u}_{0}$ is supported on a unit ball (which can be translated in a compact support for $d\sigma$) one has by the Plancherel formula

$$\begin{aligned} \Vert u\Vert _{L^{p'}({\mathop {\mathbb R}\nolimits }^{n+1})}\le C\Vert u_{0}\Vert _{L^{2}({\mathop {\mathbb R}\nolimits }^{n})}\, , \end{aligned}$$

(2.6)

for all $p' \ge 2(n+2)/n$.

A scaling argument and the density of spectrally localized functions in $L^2({\mathop {\mathbb R}\nolimits }^{n})$, give for $p'= 2+\frac{4}{n}$ and all $u_0 \in L^{2}({\mathop {\mathbb R}\nolimits }^{n})$

$$\begin{aligned} \Vert u\Vert _{L^{\frac{2n+4}{n}}({\mathop {\mathbb R}\nolimits }, L^{\frac{2n+4}{n}} ({\mathop {\mathbb R}\nolimits }^{n}))}\le C \Vert u_{0}\Vert _{L^{2}({\mathop {\mathbb R}\nolimits }^{n})}\, . \end{aligned}$$

(2.7)

One can similarly prove a Strichartz estimate for the wave equation (W) in the Euclidean space recalled on page 2, by using the representation formula (1.4). The solution can be seen as a sum of two parts, each of which is the restriction of the Fourier transform on one of the two halves of the cone

$$\begin{aligned} \widehat{S}_{\pm }\buildrel \hbox { def}\over =\Big \{(\alpha ,\xi )\in \widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb R}\nolimits }^{n}\mid \alpha ^{2}=|\xi |^{2}, \pm \alpha >0 \Big \}\, , \end{aligned}$$

each of which endowed with the measure defined by the projection $\pi : \widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb R}\nolimits }^{n}\rightarrow \widehat{\mathop {\mathbb R}\nolimits }^n$ onto the second factor (cf. the discussion above).

Now let us first assume that $\gamma _{\pm }$ is frequency localized in a unit ring $\mathcal C_1$ centered at zero. Then for any $p' \ge 2(n+2)/n$ we have

$$\begin{aligned} \Vert u\Vert _{L^{p'}({\mathop {\mathbb R}\nolimits }^{n+1})} \le C \Vert {\mathcal F}_{\mathop {\mathbb H}\nolimits }^{-1}\gamma _{\pm } \Vert _{L^{2}({\mathop {\mathbb R}\nolimits }^n)} \,. \end{aligned}$$

As above, for $p'=(2n+2)/(n-2)$, we conclude by scaling arguments and the density in $L^2({\mathop {\mathbb R}\nolimits }^n)$ of functions whose Fourier transform is compactly supported in rings centered at zero.

Remark 2.1

Notice that to apply the Fourier restriction to evolution PDEs and obtain Strichartz estimates, one applies the result to a surface in the space ${\mathop {\mathbb R}\nolimits }^{n+1}={\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb R}\nolimits }^{n}$, namely the paraboloid and the cone for the Schrödinger and wave equation, respectively.

When dealing with equations defined on the Heisenberg group ${\mathop {\mathbb H}\nolimits }^{d}$, one is naturally lead to consider surfaces in the space ${\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb H}\nolimits }^{d}$, which is not equal to ${\mathop {\mathbb H}\nolimits }^{d'}$ for some $d'$. Hence it is not enough to know restriction theorems in $\widehat{\mathop {\mathbb H}\nolimits }^{d}$ (cf. Sect. 4) but one needs to adapt these results to surfaces in $\widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb H}\nolimits }^{d}$ (cf. Sect. 5).

3 Fourier analysis on ${\mathop {\mathbb H}\nolimits }^d$

3.1 The Fourier transform on ${\mathop {\mathbb H}\nolimits }^d$

As the Heisenberg group is noncommutative, defining the Fourier transform of integrable functions on ${\mathop {\mathbb H}\nolimits }^d$ by means of characters is not relevant. The standard way consists in using irreducible representations of ${\mathop {\mathbb H}\nolimits }^d$, and in that case the Heisenberg Fourier transform ${\mathfrak {F}}_{\mathop {\mathbb H}\nolimits }f (\lambda )$ is not a complex valued function on some “frequency space” as in the Euclidean case, but a family of bounded operators on $L^2({\mathop {\mathbb R}\nolimits }^d)$ (see Corwin and Greenleaf [24] for instance for more details). Starting from the so-called Schrödinger representation, in [8] and [7] the authors introduce an alternative definition of the Fourier transform on ${\mathop {\mathbb H}\nolimits }^d$ in terms of functions acting on some frequency set $\widetilde{\mathop {\mathbb H}\nolimits }^d$. This point of view (which turns out to be equivalent to the classical definition) consists in defining the Fourier transform of an integrable function f on ${\mathop {\mathbb H}\nolimits }^d$ by projecting ${\mathfrak {F}}_{\mathop {\mathbb H}\nolimits }(\lambda )$ onto the orthonormal basis of $L^2({\mathop {\mathbb R}\nolimits }^d)$ given by Hermite functions. This enables to see the Fourier transform of a function f in $L^1({\mathop {\mathbb H}\nolimits }^d)$ as the mean value of f modulated by some oscillatory functions in the following way:

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }f(\widehat{w}) \buildrel \hbox { def}\over =\int _{{\mathop {\mathbb H}\nolimits }^d} \overline{e^{is\lambda } {\mathcal W}(\widehat{w},Y)}\, f(Y,s) \,dY\,ds\,, \end{aligned}$$

(3.1)

for any $\widehat{w}\buildrel \hbox { def}\over =(n,m,\lambda )$ in $\widetilde{\mathop {\mathbb H}\nolimits }^d\buildrel \hbox { def}\over =\mathop {\mathbb N}\nolimits ^{2d}\times {\mathop {\mathbb R}\nolimits }\setminus \{0\}$, with ${\mathcal W}$ the Wigner transform of the (renormalized) Hermite functions

$$\begin{aligned} {\mathcal W}(\widehat{w},Y) \buildrel \hbox { def}\over =\int _{{\mathop {\mathbb R}\nolimits }^d} e^{2i\lambda \langle \eta ,z\rangle } H_{n,\lambda }(y+z) H_{m,\lambda } (-y+z)\,dz\,. \end{aligned}$$

(3.2)

Here $H_{m,\lambda }$ stands for the renormalized Hermite function on ${\mathop {\mathbb R}\nolimits }^d$, namely $H_{m,\lambda } (x)\buildrel \hbox { def}\over =|\lambda |^{\frac{d}{4}} H_m(|\lambda |^{\frac{1}{2}} x)$, with $(H_{m})_{m\in {\mathop {\mathbb N}\nolimits ^d}}$ the Hermite orthonormal basis of $L^2({\mathop {\mathbb R}\nolimits }^d)$ given by the eigenfunctions of the harmonic oscillator:

$$\begin{aligned} -(\Delta -|x|^2) H_m= (2|m|+d) H_m\, , \end{aligned}$$

specifically

$$\begin{aligned} H_m \buildrel \hbox { def}\over =\Bigl (\frac{1}{2^{|m|} m!}\Bigr ) ^{\frac{1}{2}} \prod _{j=1}^d \big (-\partial _j H_0+ x_jH_0\big )^{m_j} \, , \end{aligned}$$

(3.3)

with $H_0(x)\buildrel \hbox { def}\over =\pi ^{-\frac{d}{4}} e^{-\frac{|x|^2}{2}}$, $m!\buildrel \hbox { def}\over =m_1!\cdots m_d!\,$ and $\,|m|\buildrel \hbox { def}\over =m_1+\cdots +m_d.$

In this setting, the classical statements of Fourier analysis hold in a similar way to the Euclidean case, namely the inversion and Fourier-Plancherel formulae read

$$\begin{aligned} f(w) = \frac{2^{d-1}}{\pi ^{d+1} } \int _{\widetilde{\mathop {\mathbb H}\nolimits }^d} e^{is\lambda } {\mathcal W}(\widehat{w}, Y){\mathcal F}_{\mathop {\mathbb H}\nolimits }f(\widehat{w}) \, d\widehat{w} \end{aligned}$$

(3.4)

and

$$\begin{aligned} ({\mathcal F}_{\mathop {\mathbb H}\nolimits }f|{\mathcal F}_{\mathop {\mathbb H}\nolimits }g)_{L^2(\widetilde{\mathop {\mathbb H}\nolimits }^d)} = \frac{\pi ^{d+1}}{2^{d-1}} (f|g)_{L^2({\mathop {\mathbb H}\nolimits }^d)}\, , \end{aligned}$$

(3.5)

with the notation

$$\begin{aligned} \int _{\widetilde{\mathop {\mathbb H}\nolimits }^d} \theta (\widehat{w})\,d\widehat{w} \buildrel \hbox { def}\over =\int _{{\mathop {\mathbb R}\nolimits }} \sum _{(n,m)\in \mathop {\mathbb N}\nolimits ^{2d} }\theta (n,m,\lambda ) |\lambda |^d\,d\lambda \,. \end{aligned}$$

(3.6)

By straightforward computations we find that

$$\begin{aligned} -\Delta _{\mathop {\mathbb H}\nolimits }\bigl (e^{is\lambda } {\mathcal W}(\widehat{w},Y)\bigr ) = 4|\lambda | (2|m|+d) e^{is\lambda } {\mathcal W}(\widehat{w},Y) \, ,\end{aligned}$$

(3.7)

for any $\widehat{w}=(n,m,\lambda )$ in $\widetilde{\mathop {\mathbb H}\nolimits }^d$, which readily implies that

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }( \Delta _{\mathop {\mathbb H}\nolimits }f) (\widehat{w}) =-4| \lambda | (2|m|+d){\mathcal F}_{\mathop {\mathbb H}\nolimits }( f)(\widehat{w})\,. \end{aligned}$$

This formula allows to give a definition of a function whose Fourier transform is compactly supported, in the following way.

Definition 3.1

We say that a function f on ${\mathop {\mathbb H}\nolimits }^d$ is frequency localized in a ball ${\mathcal B}_\Lambda$ centered at 0 of radius $\Lambda$ if there exists an even function $\psi$ in ${\mathcal D}({\mathop {\mathbb R}\nolimits })$ supported in ${\mathcal B}_1$ and equal to 1 near 0 such that⁵

$$\begin{aligned} f = \psi (- \Lambda ^{-2}\Delta _{\mathop {\mathbb H}\nolimits }) \, f \, , \end{aligned}$$

which is equivalent to stating that for any $\widehat{w}=(m,n,\lambda )$ in $\widetilde{{\mathop {\mathbb H}\nolimits }}^d$,

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f)(n,m,\lambda ) = \psi (\Lambda ^{-2} 4|\lambda |(2|m|+d)) \, {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f)(n,m,\lambda )\, . \end{aligned}$$

Similarly we say that a function f on ${\mathop {\mathbb H}\nolimits }^d$ is frequency localized in a ring ${\mathcal C}_\Lambda$ centered at 0 of small radius $\Lambda /2$ and large radius $\Lambda$ if there exists an even function $\phi$ in ${\mathcal D}({\mathop {\mathbb R}\nolimits })$ supported in ${\mathcal C}_1$ and equal to 1 in a ring ${\mathcal C}'$ contained in ${\mathcal C}_1$ such that

$$\begin{aligned} f = \phi (- \Lambda ^{-2}\Delta _{\mathop {\mathbb H}\nolimits }) \, f \, , \end{aligned}$$

which is equivalent to stating that for any $\widehat{w}=(m,n,\lambda )$ in $\widetilde{{\mathop {\mathbb H}\nolimits }}^d$,

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f)(n,m,\lambda ) = \phi (\Lambda ^{-2} 4|\lambda |(2|m|+d)) \, {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f)(n,m,\lambda )\, . \end{aligned}$$

One of the interests of this definition lies in the following proposition, whose proof may be found in [11] and [12].

Lemma 3.2

With the above notation,

• if f is frequency localized in ${\mathcal B}_\Lambda$, then for all $1 \le p \le q \le \infty$, $k\in \mathop {\mathbb N}\nolimits$ and $\beta \in \mathop {\mathbb N}\nolimits ^{2d}$ with $|\beta |=k,$ there is a constant $C_k$ depending only on k such that

  $$\begin{aligned} \Vert {\mathcal X}^\beta f\Vert _{L^q({\mathop {\mathbb H}\nolimits }^d)}\le C_k \Lambda ^{k+Q (\frac{1}{p}-\frac{1}{q}) } \Vert f\Vert _{L^p({\mathop {\mathbb H}\nolimits }^d)} \, , \end{aligned}$$

  (3.8)

  where ${\mathcal X}^\beta$ denotes a product of $|\beta |$ vectors fields of type (1.12);

• if f is frequency localized in ${\mathcal C}_\Lambda$, then for all $p \ge 1$ and $s\in {\mathop {\mathbb R}\nolimits }$, there is a constant $C_s$ depending only on s such that

  $$\begin{aligned} \frac{1}{C_s} \Lambda ^{s} \Vert f\Vert _{L^p({\mathop {\mathbb H}\nolimits }^d)} \le \Vert (-\Delta _{\mathop {\mathbb H}\nolimits })^{\frac{s}{2}} f\Vert _{L^p({\mathop {\mathbb H}\nolimits }^d)} \le C_s \Lambda ^{ s} \Vert f\Vert _{L^p({\mathop {\mathbb H}\nolimits }^d)}\, \cdot \end{aligned}$$

  (3.9)

It will be useful later on to observe that for any function f in $L^1({\mathop {\mathbb H}\nolimits }^d)$ and any positive real number a, there holds

$$\begin{aligned} \forall \widehat{w}=(n,m,\lambda )\in \widetilde{\mathop {\mathbb H}\nolimits }^d\,,\ {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f\circ \delta _a) (\widehat{w}) = a^{-Q}{\mathcal F}_{\mathop {\mathbb H}\nolimits }(f)(n,m,a^{-2}\lambda )\,. \end{aligned}$$

(3.10)

Let us also emphasize that if f and g are two functions of $L^1({\mathop {\mathbb H}\nolimits }^d)$ then for any $\widehat{w}=(n,m,\lambda )$ in $\widetilde{\mathop {\mathbb H}\nolimits }^d,$

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f\star g) (\widehat{w}) = ( {\mathcal F}_{\mathop {\mathbb H}\nolimits }f \cdot {\mathcal F}_{\mathop {\mathbb H}\nolimits }g)(\widehat{w})\buildrel \hbox { def}\over =\sum _{p\in \mathop {\mathbb N}\nolimits ^{d}} {\mathcal F}_{\mathop {\mathbb H}\nolimits }f(n,p,\lambda ){\mathcal F}_{\mathop {\mathbb H}\nolimits }g(p,m,\lambda )\,.\nonumber \\ \end{aligned}$$

(3.11)

In the radial framework (recall that f is radial if it is invariant under the action of the unitary group U(d) of $T^\star {\mathop {\mathbb R}\nolimits }^d$), which is our concern in this paper, it turns out that for any function f in $L_\mathrm{{rad}}^1({\mathop {\mathbb H}\nolimits }^d)$ there holds

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f) (n, m, \lambda )= {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f) (n, m, \lambda )\delta _{n,m}= {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f) (|n|, |n|, \lambda ) \delta _{n,m} \,. \end{aligned}$$

(3.12)

The interested reader can consult for instance [10, 27, 39]. Actually the Fourier transform ${\mathcal F}_{\mathop {\mathbb H}\nolimits }$ acts in the following way on radial functions:

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f) (\ell , \ell , \lambda ) = \begin{pmatrix} \ell +d-1 \\ \ell \end{pmatrix}^{-1}\int _{{\mathop {\mathbb H}\nolimits }^d} \overline{e^{is\lambda } \widetilde{\mathcal W}(\ell ,\lambda ,Y)} f(Y,s) \,dY\,ds\, , \end{aligned}$$

with (see for example [10, 26, 39] for further details)

$$\begin{aligned} \widetilde{\mathcal W}(\ell ,\lambda ,Y) \buildrel \hbox { def}\over =\sum _{\begin{array}{c} {n\in \mathop {\mathbb N}\nolimits ^d}\\ {|n|= \ell } \end{array}} {\mathcal W}(n,n,\lambda ,Y ) =e^{- |\lambda ||Y|^2 } L_\ell ^{(d-1)} ( 2 |\lambda | |Y|^2)\, , \end{aligned}$$

(3.13)

where $L_\ell ^{(d-1)}$ stands for the Laguerre polynomial of order $\ell$ and type $d-1$ given for $x\ge 0$ by

$$\begin{aligned} L_\ell ^{(d-1)}(x)\buildrel \hbox { def}\over =\sum ^\ell _{k=0} (-1)^k \begin{pmatrix} \ell +d-1\\ \ell -k \end{pmatrix} \frac{x^k}{k!} \, \cdot \end{aligned}$$

Note that the family of functions $(\Theta _\lambda ^{(\ell )} )_{\ell \in {\mathbb N}}$ mentioned in the introduction of this paper, satisfying the transport equation (1.16), is defined by the formula

$$\begin{aligned} \Theta _\lambda ^{(\ell )} (Y,s) \buildrel \hbox { def}\over =e^{is\lambda } \widetilde{\mathcal W}(\ell ,\lambda ,Y)\, . \end{aligned}$$

Equation (1.16) then follows simply from the fact that

$$\begin{aligned} -\Delta _{\mathbb H} \Theta _\lambda ^{(\ell )} = 4|\lambda | (2\ell + d) \Theta _\lambda ^{(\ell )}\, . \end{aligned}$$

Obviously the inversion and Fourier-Plancherel formulae write in that case

$$\begin{aligned} f(w) = \frac{2^{d-1}}{\pi ^{d+1} } \sum _{\ell \in \mathop {\mathbb N}\nolimits } \int _{{\mathop {\mathbb R}\nolimits }} e^{is\lambda } \widetilde{\mathcal W}(\ell , \lambda , Y){\mathcal F}_{\mathop {\mathbb H}\nolimits }(f) (\ell , \ell , \lambda )\, |\lambda |^d\,d\lambda \end{aligned}$$

and

$$\begin{aligned} (f|g)_{L^2({\mathop {\mathbb H}\nolimits }^d)}= \frac{2^{d-1}}{\pi ^{d+1} } \sum _{\ell \in \mathop {\mathbb N}\nolimits } \begin{pmatrix} \ell +d-1 \\ \ell \end{pmatrix} \int _{{\mathop {\mathbb R}\nolimits }} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f) (\ell , \ell , \lambda ) \, \overline{{\mathcal F}_{\mathop {\mathbb H}\nolimits }(g) (\ell , \ell , \lambda )} \, |\lambda |^d\,d\lambda \, . \end{aligned}$$

Moreover since for any element R of U(d), the automorphism $\theta _R$ of ${{\mathop {\mathbb H}\nolimits }}^d$ defined by

$$\begin{aligned} \theta _R(Y,s) \buildrel \hbox { def}\over =(R(Y), s) \end{aligned}$$

preserves the Haar measure of ${{\mathop {\mathbb H}\nolimits }}^d$, we have

$$\begin{aligned} (f \star g)\circ \theta _R= (f\circ \theta _R)\star (g\circ \theta _R)\, ,\end{aligned}$$

which implies that the space $L^1_\mathrm{rad}({{\mathop {\mathbb H}\nolimits }}^d)$ equipped with its standard structure of linear space and with the convolution product is a commutative sub-algebra of $L^1({{\mathop {\mathbb H}\nolimits }}^d)$. We deduce that in this framework, (3.11) reduces to

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f\star g) (\ell , \ell , \lambda ) = {\mathcal F}_{\mathop {\mathbb H}\nolimits }f(\ell , \ell , \lambda ) {\mathcal F}_{\mathop {\mathbb H}\nolimits }g(\ell , \ell , \lambda ) \,. \end{aligned}$$

(3.14)

Finally it will be important to observe that there holds for all $w=(Y,s)$ in ${\mathop {\mathbb H}\nolimits }^d$, in the radial setting,

$$\begin{aligned} \sum _{\begin{array}{c} {n\in \mathop {\mathbb N}\nolimits ^d}\\ {|n|= \ell } \end{array}} {\mathcal F}_{{\mathop {\mathbb H}\nolimits }}(f\circ \tau _w)(n, n, \lambda )={\mathcal F}_{{\mathop {\mathbb H}\nolimits }}(f)(\ell , \ell , \lambda ) e^{-is\lambda } e^{ -|\lambda ||Y|^2 } L_\ell ^{(d-1)}(2|\lambda ||Y|^2 ) ,\qquad \end{aligned}$$

(3.15)

where $\tau _{w}$ denotes the left translate defined by $\tau _{w}(w')\buildrel \hbox { def}\over =w \cdot w'$.

3.2 Frequency space for the Heisenberg group

In [8], the authors show that the following distance $\widehat{d}$ on $\widetilde{\mathop {\mathbb H}\nolimits }^d=\mathop {\mathbb N}\nolimits ^{2d}\times {\mathop {\mathbb R}\nolimits }\setminus \{0\}$

$$\begin{aligned} \widehat{d}(\widehat{w},\widehat{w}') \buildrel \hbox { def}\over =\bigl |\lambda (n+m)-\lambda '(n'+m')\bigr |_1 +\bigl |(n-m)-(n'-m')|_1+d|\lambda -\lambda '| \, ,\nonumber \\ \end{aligned}$$

(3.16)

where $|\cdot |_1$ denotes the $\ell ^1$ norm on ${\mathop {\mathbb R}\nolimits }^d$, is appropriate and that the completion of the set $\widetilde{\mathop {\mathbb H}\nolimits }^d$ for this distance is the set

$$\begin{aligned}&\widehat{\mathop {\mathbb H}\nolimits }^d\buildrel \hbox { def}\over =\bigl (\mathop {\mathbb N}\nolimits ^{2d} \times {\mathop {\mathbb R}\nolimits }\setminus \{0\}\bigr ) \cup \widehat{\mathop {\mathbb H}\nolimits }^d_0 \quad \hbox {with}\quad \widehat{\mathop {\mathbb H}\nolimits }^d_0\\&\buildrel \hbox { def}\over =\{(\dot{x}, k) \in {{\mathop {\mathbb R}\nolimits }_{\mp }^d}\times {\mathop {\mathbb Z}\nolimits }^d \} \quad \hbox {and}\quad {{\mathop {\mathbb R}\nolimits }_{\mp }^d}\buildrel \hbox { def}\over =({\mathop {\mathbb R}\nolimits }_-)^d\cup ({\mathop {\mathbb R}\nolimits }_+)^d\,. \end{aligned}$$

It readily stems from (3.1) that the following continuous embedding holds:

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }: L^1({\mathop {\mathbb H}\nolimits }^d) \hookrightarrow L^\infty (\widehat{\mathop {\mathbb H}\nolimits }^d)\, .\end{aligned}$$

(3.17)

Combining the Fourier-Plancherel formula (3.5) together with interpolation theory, we deduce that, for all $1 \le p \le 2$, the Hausdorff-Young inequality holds

$$\begin{aligned} \Vert {\mathcal F}_{\mathop {\mathbb H}\nolimits }f\Vert _{L^{p'}(\widehat{\mathop {\mathbb H}\nolimits }^d)}\le \Vert f\Vert _{L^{p}({\mathop {\mathbb H}\nolimits }^d)}\, , \end{aligned}$$

where $p'$ is the dual exponent of p.

This new approach enabled the authors in [7] to extend ${\mathcal F}_{\mathop {\mathbb H}\nolimits }$ to ${\mathcal S}'({\mathop {\mathbb H}\nolimits }^d)$, the set of tempered distributions: note that since the Schwartz class ${\mathcal S}({\mathop {\mathbb H}\nolimits }^d)$ coincides with ${\mathcal S}({\mathop {\mathbb R}\nolimits }^{2d+1})$ then similarly ${\mathcal S}'({\mathop {\mathbb H}\nolimits }^{d})$ is noting else than ${\mathcal S}'({\mathop {\mathbb R}\nolimits }^{2d+1})$. As in the Euclidean case, this extension is done by duality and the starting point is the characterization of ${\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)$ as the range⁶ of ${\mathcal S}({\mathop {\mathbb H}\nolimits }^d)$ by ${\mathcal F}_{\mathop {\mathbb H}\nolimits }$. Actually in [7], the authors prove that the Fourier transform ${\mathcal F}_{\mathop {\mathbb H}\nolimits }$ is a bicontinuous isomorphism between the spaces ${\mathcal S}({\mathop {\mathbb H}\nolimits }^{d})$ and ${\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d),$ and that the map ${\mathcal F}_{{\mathop {\mathbb H}\nolimits }}$ can be continuously extended from ${\mathcal S}'({\mathop {\mathbb H}\nolimits }^d)$ into ${\mathcal S}'(\widehat{\mathop {\mathbb H}\nolimits }^d)$ in the following way:

$$\begin{aligned} {\mathcal F}_{{\mathop {\mathbb H}\nolimits }} : \left\{ \begin{array}{ccl} {\mathcal S}'({\mathop {\mathbb H}\nolimits }^d) &{} \longrightarrow &{} {\mathcal S}'(\widehat{\mathop {\mathbb H}\nolimits }^d) \\ T &{} \longmapsto &{} \Bigl [\theta \mapsto \langle T,{}^t\!{\mathcal F}_{{\mathop {\mathbb H}\nolimits }}\theta \rangle _{{\mathcal S}'({\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}({\mathop {\mathbb H}\nolimits }^d)}\Bigr ]\, , \end{array} \right. \end{aligned}$$

(3.18)

where

$$\begin{aligned} {}^t{\mathcal F}_{{\mathop {\mathbb H}\nolimits }}\theta (y,\eta ,s) \buildrel \hbox { def}\over =\frac{\pi ^{d+1}}{2^{d-1}} ({\mathcal F}_{{\mathop {\mathbb H}\nolimits }}^{-1}\theta )(y,-\eta ,-s) \, . \end{aligned}$$

(3.19)

Let us also emphasize that if T is a tempered distribution on ${\mathop {\mathbb H}\nolimits }^d$, then for all f in ${\mathcal S}({\mathop {\mathbb H}\nolimits }^d)$ and all w in ${\mathop {\mathbb H}\nolimits }^d,$

$$\begin{aligned} (T\star f)(w)=\langle T,\check{f}\circ \tau _{w^{-1}}\rangle _{{\mathcal S}'({\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}( {\mathop {\mathbb H}\nolimits }^d)} \, , \end{aligned}$$

(3.20)

where $\check{f}(w)\buildrel \hbox { def}\over =f(w^{-1})$.

4 A restriction theorem on the sphere in the frequency space $\widehat{\mathop {\mathbb H}\nolimits }^d$

Our purpose here is to recover a Fourier restriction result on the sphere of $\widehat{\mathop {\mathbb H}\nolimits }^d$ due to Müller [38]. Our approach is rather different to [38] as we use the Fourier transform as a key tool in obtaining a representation of the Fourier restriction operator, whereas Müller uses a spectral representation. As will be seen in Paragraph 5.1 in the proof of Strichartz estimates for the wave and Schrödinger operators on ${\mathop {\mathbb H}\nolimits }^d$, the interest of our approach is that it can easily be applied to more general frameworks.

4.1 Study of the surface measure on the sphere of the frequency space

The aim of this section is to recover the Heisenberg Fourier transform restriction result of Müller in [38]. To this end, let us start by introducing ${\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}$ the unit sphere on $\widehat{\mathop {\mathbb H}\nolimits }^d$: denoting by $\widehat{0}$ the origin of $\widehat{\mathop {\mathbb H}\nolimits }^d$ (that is the point of $\widehat{\mathop {\mathbb H}\nolimits }^d$ corresponding to $(\dot{x}, k)=(0,0)$, with the notation of Paragraph 3.2), the sphere of $\widehat{\mathop {\mathbb H}\nolimits }^d$ centered at the origin with radius 1 is defined by

$$\begin{aligned} {\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}\buildrel \hbox { def}\over =\Bigl \{(n,n,\lambda )\in \widehat{\mathop {\mathbb H}\nolimits }^d\,/\, (2|n|+d)|\lambda |=1\Bigr \}\, \bigcup \,\Bigl \{(\dot{x},0)\in \widehat{\mathop {\mathbb H}\nolimits }^d_0\,/\, |\dot{x}|_1=1\Bigr \},\nonumber \\ \end{aligned}$$

(4.1)

and the surface measure $\displaystyle d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}}$ is given for all $\theta$ in ${\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)$ by the following formula:

$$\begin{aligned}&\langle d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}},\theta \rangle _{{\mathcal S}'(\widehat{\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)}\nonumber \\&= \! \sum _{n\in \mathop {\mathbb N}\nolimits ^d} \frac{1}{(2|n|+d)^{d+1}}\Big (\theta \big (n,n,\frac{1}{2|n|+d}\big ) +\theta \Big (n,n,\frac{-1}{2|n|+d}\Big )\Big )\,. \end{aligned}$$

(4.2)

We observe that one can show that the measure of $\widehat{\mathop {\mathbb H}\nolimits }^d_0$ with respect to $d\widehat{w}$ is zero, and thus in all that follows, we shall agree that the measure $d\widehat{w}$ has been extended by 0 to the whole of $\widehat{\mathop {\mathbb H}\nolimits }^d_0,$ and we shall keep the same notation $d\widehat{w}$ for the measure on the whole of $\widehat{\mathop {\mathbb H}\nolimits }^d$.

More generally, if ${\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}({\sqrt{R}})$ denotes the sphere of $\widehat{\mathop {\mathbb H}\nolimits }^d$ centered at the origin $\widehat{0}$ of radius ${\sqrt{R}}$, let us prove that for all $\theta$ in ${\mathcal F}_{{\mathop {\mathbb H}\nolimits }}({\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb H}\nolimits }^d))$, there holds

$$\begin{aligned}&\langle d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}({\sqrt{R}})},\theta \rangle _{{\mathcal S}'(\widehat{\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)} \! \nonumber \\&\quad = \! \!\sum _{n\in \mathop {\mathbb N}\nolimits ^d}\! \frac{R^{d}}{(2|n|+d)^{d+1}}\Big (\theta \big (n,n,\frac{R}{2|n|+d}\big ) + \theta \big (n,n,\frac{-R}{2|n|+d}\big )\Big )\,. \end{aligned}$$

(4.3)

We start indeed with the general formula

$$\begin{aligned} \int _{\widehat{\mathop {\mathbb H}\nolimits }^d}\theta (\widehat{w})\,d\widehat{w}=\int _0^\infty \biggl (\int _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}({\sqrt{R}})}\theta (\widehat{w})\,d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}}({\sqrt{R}})\biggr )\,dR\, , \end{aligned}$$

for all $\theta$ in ${\mathcal F}_{{\mathop {\mathbb H}\nolimits }}({\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb H}\nolimits }^d))$. In view of (3.6) and (3.12), we have

$$\begin{aligned} \int _{\widehat{\mathop {\mathbb H}\nolimits }^d}\theta (\widehat{w})\,d\widehat{w}=\sum _{n\in \mathop {\mathbb N}\nolimits ^d}\int _{\mathop {\mathbb R}\nolimits }\theta (n,n,\lambda )\,|\lambda |^d d\lambda \, , \end{aligned}$$

(4.4)

which thanks to the Fubini theorem and the change of variable $R=(2|n|+d)|\lambda |$ yields

$$\begin{aligned}&\sum _{n\in \mathop {\mathbb N}\nolimits ^d}\int _{\mathop {\mathbb R}\nolimits }\theta (n,n,\lambda )\,|\lambda |^d d\lambda \\&\quad = \sum _{n\in \mathop {\mathbb N}\nolimits ^d} \int _0^\infty \frac{R^{d}}{(2|n|+d)^{d+1}}\,\Big (\theta \Big (n,n,\frac{R}{2|n|+d}\Big )\\&\qquad +\theta \Big (n,n,\frac{-R}{2|n|+d}\Big )\Big ) dR \, , \end{aligned}$$

which proves (4.3).

Remark 4.1

It is important here to be in the radial framework, as in that case the distance defined in (3.16) reduces to

$$\begin{aligned} \widehat{d}(\widehat{w},\widehat{w}') = 2 \bigl | \lambda n -\lambda ' n'\bigr |_1 +d|\lambda -\lambda '| \, . \end{aligned}$$

In the non radial case, for $R>1$ then the sphere ${\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}(R)$ (of center  $\widehat{0}$ and radius R) is a much more complex set than the unit sphere ${\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}$. For instance, when $1<R<2$, one can easily check that ${\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}(R)$ admits $d+1$ connected components in $\widetilde{\mathop {\mathbb H}\nolimits }^d$. That is the reason why we focus here on the radial framework.

In order to investigate boundedness properties of the restriction of ${\mathcal F}_{\mathop {\mathbb H}\nolimits }$ to ${\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}$, we shall adapt the Euclidean proof due to Tomas-Stein (see [45], and Sect. 2 of this paper). To this end, let us first compute ${\mathcal F}^{-1}_{\mathop {\mathbb H}\nolimits }(d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}})$. By definition, the tempered distribution

$$\begin{aligned} G \buildrel \hbox { def}\over ={\mathcal F}^{-1}_{\mathop {\mathbb H}\nolimits }(d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}}) \end{aligned}$$

satisfies for all $\theta$ in ${\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)$

$$\begin{aligned} \langle d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}},\theta \rangle _{{\mathcal S}'(\widehat{\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)}=\langle G,{}^t\!{\mathcal F}_{{\mathop {\mathbb H}\nolimits }}\theta \rangle _{{\mathcal S}'({\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}({\mathop {\mathbb H}\nolimits }^d)} \, . \end{aligned}$$

Let us prove the following proposition.

Proposition 4.2

With the above notation, G is the bounded function on ${\mathop {\mathbb H}\nolimits }^d$ defined by

$$\begin{aligned} G(Y,s)= \frac{2^d}{\pi ^{d+1}} \sum _{n\in \mathop {\mathbb N}\nolimits ^d} \frac{1}{(2|n|+d)^{d+1}} \cos \Big (\frac{s}{2|n|+d}\Big ) {\mathcal W}\Big (n,n,1,\frac{Y}{\sqrt{2|n|+d}}\Big )\,\cdot \nonumber \\ \end{aligned}$$

(4.5)

Proof

According to (3.19) and to the fact that the Fourier transform ${\mathcal F}_{\mathop {\mathbb H}\nolimits }$ is a bicontinuous isomorphism between the spaces ${\mathcal S}({\mathop {\mathbb H}\nolimits }^{d})$ and ${\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d),$ we have (with $\theta \buildrel \hbox { def}\over ={\mathcal F}_{\mathop {\mathbb H}\nolimits }f$)

$$\begin{aligned} \langle d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}},\theta \rangle _{{\mathcal S}'(\widehat{\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)} = \frac{\pi ^{d+1}}{2^{d-1}} \langle G, \widetilde{f}\rangle _{{\mathcal S}'({\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}( {\mathop {\mathbb H}\nolimits }^d)} \, , \end{aligned}$$

(4.6)

with $\widetilde{f} (y,\eta ,s) \buildrel \hbox { def}\over =f(y,-\eta ,-s)$.

Now observe that

$$\begin{aligned} \theta \Big (n,n,\frac{1}{2|n|+d}\Big )= \int _{{\mathop {\mathbb H}\nolimits }^d} e^{-i\frac{s}{2|n|+d}} \,{\mathcal W}\Big (n,n,1,\frac{Y}{\sqrt{2|n|+d}}\Big )\, f(Y,s) \,dY\,ds .\qquad \end{aligned}$$

(4.7)

Indeed, we have by easy computations

$$\begin{aligned} {\mathcal W}\Big (n,n,\frac{1}{2|n|+d},Y\Big )={\mathcal W}\Big (n,n,1,\frac{Y}{\sqrt{2|n|+d}}\Big )\, , \end{aligned}$$

which gives rise to (4.7) according to the definition of ${\mathcal F}_{\mathop {\mathbb H}\nolimits }$ and to the fact that for all $\lambda$ in ${\mathop {\mathbb R}\nolimits }\setminus \{0\}$, the function ${\mathcal W}(n, n, \lambda , Y )$ is real valued.

Besides by an obvious change of variable, one also has

$$\begin{aligned} {\mathcal W}\Big (n,n,-1,\frac{Y}{\sqrt{2|n|+d}}\Big )={\mathcal W}\Big (n,n,1,\frac{Y}{\sqrt{2|n|+d}}\Big ) \, ,\end{aligned}$$

which implies that

$$\begin{aligned} \theta \Big (n,n, - \frac{1}{2|n|+d}\Big )= \int _{{\mathop {\mathbb H}\nolimits }^d} e^{i\frac{s}{2|n|+d}} \,{\mathcal W}\Big (n,n,1,\frac{Y}{\sqrt{2|n|+d}}\Big )\, f(Y,s) \,dY\,ds \, .\qquad \end{aligned}$$

(4.8)

Invoking (4.2), this gives rise to

$$\begin{aligned}&\langle d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}},\theta \rangle _{{\mathcal S}'(\widehat{\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)} = \sum _{n\in \mathop {\mathbb N}\nolimits ^d} \frac{2}{(2|n|+d)^{d+1}}\\&\quad \times \int _{{\mathop {\mathbb H}\nolimits }^d} \cos \Big (\frac{s}{2|n|+d}\Big ) \,{\mathcal W}\Big (n,n,1,\frac{Y}{\sqrt{2|n|+d}}\Big )\, f(Y,s) \,dY\,ds\, , \end{aligned}$$

which by an obvious change of variable ends the proof of Formula (4.5).

Furthermore, we have the following classical combinatorial identity

$$\begin{aligned} \# \big \{n\in \mathop {\mathbb N}\nolimits ^d, |n|= \ell \big \} = \begin{pmatrix} \ell +d-1 \\ \ell \end{pmatrix} \, ,\end{aligned}$$

(4.9)

and since the modulus of the Wigner transform of the (renormalized) Hermite functions defined by (3.2) is bounded by one, Stirling’s formula implies that G belongs to $L^\infty ({\mathop {\mathbb H}\nolimits }^d)$. The result follows. $\square$

Remark 4.3

Arguing as for the unit sphere, we readily gather that for all $\theta$ belonging to ${\mathcal F}_{{\mathop {\mathbb H}\nolimits }}({\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb H}\nolimits }^d))$, there holds

$$\begin{aligned} \langle d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}({\sqrt{R}})},\theta \rangle _{{\mathcal S}'(\widehat{\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}(\widehat{\mathop {\mathbb H}\nolimits }^d)}=\langle G_R,{}^t\!{\mathcal F}_{{\mathop {\mathbb H}\nolimits }}\theta \rangle _{{\mathcal S}'({\mathop {\mathbb H}\nolimits }^d)\times {\mathcal S}({\mathop {\mathbb H}\nolimits }^d)} \, , \end{aligned}$$

where, in view of (3.10), $G_R$ is given by

$$\begin{aligned} G_R(Y,s)\buildrel \hbox { def}\over =R^{d} (G \circ \delta _{{\sqrt{R}}})(Y,s)\,. \end{aligned}$$

(4.10)

We thus recover Formula (19) derived by Müller in [38].

4.2 The restriction theorem

Now let us state the restriction theorem, due to Müller in [38], and sketch its proof for the convenience of the reader.

Theorem 4

( [38]) If $1 \le p \le 2$, then

$$\begin{aligned} \Vert {\mathcal F}_{\mathop {\mathbb H}\nolimits }(f)_{|{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}} \Vert _{L^{2}({{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}})}\le C_p \Vert f\Vert _{L^{p}_{Y}L^{1}_{s}}\, , \end{aligned}$$

(4.11)

for all radial functions f in ${\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb H}\nolimits }^d)$.

Remark 4.4

In light of (4.2), Theorem 4 writes

$$\begin{aligned} \begin{aligned}&\Big [\sum _{n\in \mathop {\mathbb N}\nolimits ^d} \frac{1}{(2|n|+d)^{d+1}} \Big (\Big |{\mathcal F}_{\mathop {\mathbb H}\nolimits }(f)(n,n, \frac{1}{2|n|+d}\Big )\Big |^2 + \Big |{\mathcal F}_{\mathop {\mathbb H}\nolimits }(f)(n,n, \frac{-1}{2|n|+d}\Big )\Big |^2\Big )\Big ]^{\frac{1}{2}} \\&\quad \le C_p \Vert f\Vert _{L^{p}_{Y}L^{1}_{s}}\, , \end{aligned} \end{aligned}$$

for f in ${\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb H}\nolimits }^d)$. Besides, since ${\mathcal S}({\mathop {\mathbb H}\nolimits }^d)$ is dense in the space $L^{p}_{Y}L^{1}_{s}({\mathop {\mathbb H}\nolimits }^d)$, when the estimate of Theorem 4 holds we can define ${\mathcal F}_{\mathop {\mathbb H}\nolimits }f$ on ${\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}$ (a.e. with respect to $d\sigma _{{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}}$), for each function f in $L^{p}_{Y}L^{1}_{s}$. Let us also emphasize that the gain we get here with respect to the horizontal variable Y is better than the one obtained in Euclidean case, since the index p ranges from 1 to 2 with no further restriction, contrary to the Euclidean case (recall Theorem 3). Note that a counterexample for large values of p (namely $p>4d/(2d-1)$, which is the dual index to $p_d$ introduced in Remark 1.4) is provided in [38].

Remark 4.5

By duality, Inequality  (4.11) is equivalent to the following estimate

$$\begin{aligned} \Vert {\mathcal F}^{-1}_{\mathop {\mathbb H}\nolimits }(\theta _{|{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}}) \Vert _{L^{p'}_{Y}L^{\infty }_{s}}\le C_p \, \Vert \theta _{|{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}}\Vert _{L^{2}({{\mathop {\mathbb S}\nolimits }_{\widehat{\mathop {\mathbb H}\nolimits }^d}})}\, , \end{aligned}$$

(4.12)

for all $2 \le p' \le \infty$ and all $\theta$ in ${\mathcal F}_{\mathop {\mathbb H}\nolimits }({\mathcal S}_\mathrm{{rad}})({\mathop {\mathbb H}\nolimits }^d)$.

Proof of Theorem 4

First note that the case when $p=1$ is a straightforward consequence of (3.17). Then by interpolation, it suffices to investigate the case when $p=2$. For that purpose, we shall proceed using purely Fourier analysis arguments, and sketch the proof due to Müller (see [38] for further details). Our goal here is to establish the following estimate

$$\begin{aligned} \sum _{\ell \in \mathop {\mathbb N}\nolimits } \Big ( \Big |\widetilde{\Theta }\Big (\ell , \ell , \frac{1}{2\ell +d}\Big )\Big |^2 + \Big |\widetilde{\Theta }\Big (\ell , \ell , \frac{-1}{2\ell +d}\Big )\Big |^2\Big ) \le C_2 \Vert f\Vert _{L^{2}_{Y}L^{1}_{s}}^2\, , \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \widetilde{\Theta }(\ell , \ell , \lambda )&\buildrel \hbox { def}\over =\Big [ \begin{pmatrix} \ell +d-1 \\ \ell \end{pmatrix}^{-1} \frac{1}{(2\ell +d)^{d+1}} \Big ]^{\frac{1}{2}} \sum _{\begin{array}{c} {n\in \mathop {\mathbb N}\nolimits ^d}\\ {|n| = \ell } \end{array}} {\mathcal F}_{\mathop {\mathbb H}\nolimits }f (n, n, \lambda ) \\&= \Big [ \begin{pmatrix} \ell +d-1 \\ \ell \end{pmatrix}^{-1} \frac{1}{(2\ell +d)^{d+1}} \Big ]^{\frac{1}{2}} \int _{{\mathop {\mathbb H}\nolimits }^d} e^{-is\lambda } \widetilde{\mathcal W}( \ell , \lambda ,Y)\, f(Y,s) \,dY\,ds\,. \end{aligned} \end{aligned}$$

(4.13)

This amounts to proving that the operator T defined by

$$\begin{aligned} Tf \buildrel \hbox { def}\over =\Big (\widetilde{\Theta }\Big (\ell , \ell , \frac{1}{2\ell +d}\Big )\Big )_{\ell \in \mathop {\mathbb N}\nolimits } \,, \end{aligned}$$

with $\widetilde{\Theta }$ obtained from f through (4.13), is bounded from $L^{2,1}({\mathop {\mathbb H}\nolimits }^{d})$ into $\ell ^2(\mathop {\mathbb N}\nolimits )$, or equivalently that its adjoint $T^*$ is bounded from $\ell ^2(\mathop {\mathbb N}\nolimits )$ into $L^{2,\infty }({\mathop {\mathbb H}\nolimits }^{d})$.

Now for any sequence ${\underline{a}} = (a_\ell )_{\ell \in \mathop {\mathbb N}\nolimits }$ in $\ell ^2(\mathop {\mathbb N}\nolimits )$, the operator $T^*$ is given by

$$\begin{aligned} T^*({\underline{a}}) (Y,s)= \sum ^\infty _{\ell =0} a_\ell \, e^{-is\frac{1}{2\ell +d}} \, K\Big ( \ell ,\frac{1}{2\ell +d} ,Y\Big ) \, , \end{aligned}$$

with

$$\begin{aligned} K\Big ( \ell ,\frac{1}{2\ell +d} ,Y\Big )\buildrel \hbox { def}\over =\Big [ \begin{pmatrix} \ell +d-1 \\ \ell \end{pmatrix}^{-1} \frac{1}{(2\ell +d)^{d+1}} \Big ]^{\frac{1}{2}} \widetilde{\mathcal W}(\ell , \lambda ,Y)\, . \end{aligned}$$

But by Lemma 4.2 in [38], we know that for all $\ell$, m in $\mathop {\mathbb N}\nolimits$

$$\begin{aligned} \int _{T^\star {\mathop {\mathbb R}\nolimits }^d} \Big |K\Big ( \ell ,\frac{1}{2\ell +d} ,Y\Big )\Big | \, \Big |K\Big ( m,\frac{1}{2m +d} ,Y\Big )\Big | dY= {\mathcal O}\Big ( \frac{1}{\max (\ell , m)}\Big )\,.\qquad \end{aligned}$$

(4.14)

We deduce that

$$\begin{aligned} \Vert T^*({\underline{a}})\Vert _{L^{2,\infty }({\mathop {\mathbb H}\nolimits }^{d})} \lesssim \sum _{\ell \le m} \frac{ |a_\ell | |a_m|}{m} = \sum _{ m} |a_m| b_m\,, \end{aligned}$$

with $\displaystyle b_m\buildrel \hbox { def}\over =\frac{1 }{m} \sum _{\ell \le m} |a_\ell | .$ This ensures the result thanks to the following Hardy inequality (see [3]):

$$\begin{aligned} \Vert {\underline{b}}\Vert _{\ell ^p(\mathop {\mathbb N}\nolimits )}\le C_p \Vert {\underline{a}}\Vert _{\ell ^p(\mathop {\mathbb N}\nolimits )}\, , \end{aligned}$$

(4.15)

available for all $1< p < \infty$, which achieves the proof of Theorem 4. $\square$

5 Proof of Strichartz estimates

5.1 Restriction theorem on $\widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb H}\nolimits }^d$

As we explained in Sect. 2 (cf. in particular Remark 2.1), to apply efficiently restriction estimates to PDEs to get Strichartz estimates, one has to investigate the Tomas-Stein restriction theorem on submanifolds of the product $\widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb H}\nolimits }^d$, where $\widehat{\mathop {\mathbb R}\nolimits }$ stands for the dual group of ${\mathop {\mathbb R}\nolimits }$. In the following we set

$$\begin{aligned} {\mathbb D}\buildrel \hbox { def}\over ={\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb H}\nolimits }^d \quad \text{ and }\quad \widehat{\mathbb D}\buildrel \hbox { def}\over =\widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb H}\nolimits }^d\,. \end{aligned}$$

5.1.1 The Fourier transform on ${\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb H}\nolimits }^d$

A combination of Fourier analysis on the real line and on the Heisenberg group leads directly to an efficient Fourier theory on ${\mathbb D}$ whose Haar measure is obviously nothing else than the Lebesgue measure. We define the Fourier transform of f in $L^1( {\mathbb D})$ as follows:

$$\begin{aligned} {\mathcal F}_{\mathbb D} f(\alpha , \widehat{w}) \buildrel \hbox { def}\over =\int _{{\mathbb D}} \overline{e^{it\alpha } e^{is\lambda } {\mathcal W}(\widehat{w},Y)}\, f(t,Y,s) \, dt \,dY\,ds\,, \end{aligned}$$

(5.1)

for any $(\alpha , \widehat{w}) \in \widehat{\mathbb D}$. The Fourier transform ${\mathcal F}_{\mathbb D}$ inherits all the properties of ${\mathcal F}_{{\mathop {\mathbb H}\nolimits }}$ and of the Fourier transform on the real line ${\mathcal F}$. In particular, the inversion and Fourier-Plancherel formulae take the following forms:

$$\begin{aligned} f(t, w) = \frac{2^{d-2}}{\pi ^{d+2} } \int _{\widehat{\mathbb D}} e^{it\alpha } e^{is\lambda } {\mathcal W}(\widehat{w}, Y){\mathcal F}_{{\mathbb D}} f(\alpha , \widehat{w}) \, d\alpha \, d\widehat{w} \end{aligned}$$

(5.2)

and

$$\begin{aligned} ({\mathcal F}_{{\mathbb D}} f|{\mathcal F}_{{\mathbb D}} g)_{L^2(\widehat{\mathbb D})} = \frac{\pi ^{d+2}}{2^{d-2}} (f|g)_{L^2({\mathbb D})}\, . \end{aligned}$$

(5.3)

In the sequel, we shall say that a function f on ${\mathbb D}$ is radial if it is invariant under the action of U(d), in the sense that for any R of U(d) and any (t, Y, s) of ${\mathbb D}$, we have

$$\begin{aligned} f(t, R(Y),s)= f(t, Y,s) \, . \end{aligned}$$

It readily stems from Relation (3.12) that if f belongs to $L_\mathrm{{rad}}^1( {\mathbb D})$, then for all $(\alpha , \widehat{w}) \in \widehat{\mathbb D}$,

$$\begin{aligned} {\mathcal F}_{{\mathbb D}} (f) (\alpha , n, m, \lambda )= {\mathcal F}_{{\mathbb D}} (f) (\alpha , |n|, |n|, \lambda ) \delta _{n,m} \,. \end{aligned}$$

(5.4)

To avoid any confusion, we shall denote in what follows by $\star _{\mathbb D}$ the noncommutative convolution product on ${\mathbb D}$, namely

$$\begin{aligned} f \star _{\mathbb D} g ( t, w ) \buildrel \hbox { def}\over =\int _{{\mathbb D}} f (t-t', w \cdot v^{-1} ) g(t', v)\,dt' \, dv \, , \end{aligned}$$

(5.5)

which of course enjoys Young’s inequalities and satisfies

$$\begin{aligned} {\mathcal F}_{{\mathbb D}} (f\star _{\mathbb D} g) (\alpha , \widehat{w}) =\sum _{p\in \mathop {\mathbb N}\nolimits ^{d}} {\mathcal F}_{{\mathbb D}} f(\alpha , n,p,\lambda ){\mathcal F}_{{\mathbb D}} g(\alpha , p,m,\lambda )\,. \end{aligned}$$

(5.6)

Patching Fourier analysis on the real line and on the Heisenberg group, one can easily check that $L_\mathrm{{rad}}^1( {\mathbb D})$ is a commutative sub-algebra of $L^1( {\mathbb D})$ where (5.6) reduces to

$$\begin{aligned} {\mathcal F}_{{\mathbb D}} (f\star _{\mathbb D} g) (\alpha , |n|, |n|, \lambda ) ={\mathcal F}_{{\mathbb D}} f(\alpha , |n|, |n|, \lambda ){\mathcal F}_{{\mathbb D}} g(\alpha , |n|, |n|, \lambda )\,. \end{aligned}$$

(5.7)

Besides, it is worth noticing that ${\mathcal F}_{{\mathbb D}}$ is a bicontinuous isomorphism between the space ${\mathcal S}({\mathbb D})$ (which coincides with ${\mathcal S}({\mathop {\mathbb R}\nolimits }^{2d+2})$) and ${\mathcal S}(\widehat{\mathbb D})$ — which can be defined naturally from the definition of ${\mathcal S}({\widehat{\mathop {\mathbb H}\nolimits }^d})$. The map ${\mathcal F}_{{\mathbb D}}$ can then be continuously extended from ${\mathcal S}'({\mathbb D})$ into ${\mathcal S}'(\widehat{\mathbb D})$ by duality according to the following formula:

$$\begin{aligned} {\mathcal F}_{{\mathbb D}}: \left\{ \begin{array}{ccl} {\mathcal S}'({{\mathbb D}}) &{} \longrightarrow &{} {\mathcal S}'(\widehat{\mathbb D}) \\ T &{} \longmapsto &{} \Bigl [\theta \mapsto \langle T,{}^t\!{\mathcal F}_{\mathbb D}\theta \rangle _{{\mathcal S}'({\mathbb D})\times {\mathcal S}({\mathbb D})}\Bigr ]\, , \end{array} \right. \end{aligned}$$

with

$$\begin{aligned} {}^t{\mathcal F}_{\mathbb D}\theta (t, y,\eta ,s) \buildrel \hbox { def}\over =\frac{\pi ^{d+2}}{2^{d-2}} ({\mathcal F}_{\mathbb D}^{-1}\theta )(-t, y,-\eta ,-s) \, . \end{aligned}$$

(5.8)

5.1.2 A surface measure

Let us define the set

$$\begin{aligned} \Sigma \buildrel \hbox { def}\over =\Big \{(\alpha ,\widehat{w})=\big (\alpha , (n, n, \lambda ) \big ) \in \widehat{\mathbb D}\, /\, \alpha = 4 |\lambda |(2|n|+d) \Big \} \, . \end{aligned}$$

(5.9)

We endow $\Sigma$ with the measure $d\Sigma$ induced by the projection $\pi : \widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb H}\nolimits }^{d}\rightarrow \widehat{\mathop {\mathbb H}\nolimits }^{d}$ onto the second factor. Following the notation of Sect. 2, $d\Sigma =(\pi |_{\Sigma })^{-1}_{\sharp } d\widehat{w}$. More explicitly, recalling (4.4), for all $\Theta$ in ${\mathcal S}(\widehat{\mathbb D})$, we have

$$\begin{aligned} \begin{aligned} \langle d\Sigma ,\Theta \rangle _{{\mathcal S}'(\widehat{\mathbb D})\times {\mathcal S}(\widehat{\mathbb D})}&= \int _{\Sigma }\Theta (\alpha ,\widehat{w}) \, d\Sigma (\alpha ,\widehat{w}) \\&\buildrel \hbox { def}\over =\sum _{n\in \mathop {\mathbb N}\nolimits ^d} {c_n^{d+1}} \int ^\infty _0 \Big (\Theta \big (\alpha ,n,n, {\alpha }{c_n}\big ) + \Theta \big (\alpha ,n,n, {-\alpha }{c_n}\big ) \Big ) \, \alpha ^d \, d\alpha \, , \end{aligned} \end{aligned}$$

where to simplify notation we have set

$$\begin{aligned} c_n\buildrel \hbox { def}\over =\big (4(2|n|+d)\big )^{-1}\,. \end{aligned}$$

Notice that if $\Theta :\Sigma \subset \widehat{\mathbb D} \rightarrow \mathop {\mathbb C}\nolimits$ is defined as $\Theta =\theta \circ \pi |_{\Sigma }$, where $\theta :\widehat{\mathop {\mathbb H}\nolimits }^{d}\rightarrow \mathop {\mathbb C}\nolimits$, then by construction for all $1\le p\le \infty$

$$\begin{aligned} \Vert \Theta \Vert _{L^{p}(\Sigma ,d\Sigma )}=\Vert \theta \Vert _{L^{p}(\widehat{\mathop {\mathbb H}\nolimits }^{d})}\, . \end{aligned}$$

(5.10)

Our purpose here is to show that every (appropriate) function f has a Fourier transform ${\mathcal F}_{\mathbb D}f$ that restricts to $\Sigma$. Actually as in the classical case, this restriction property is best dealt with in compact subsets of $\Sigma$. Thus, we shall consider $\Sigma$ endowed with the surface measure $d\Sigma _{\mathrm{loc}}\buildrel \hbox { def}\over =\psi (\alpha )d\Sigma$ defined by

$$\begin{aligned} \begin{aligned} \int _{\Sigma } \Theta (\alpha ,\widehat{w}) \, d\Sigma _{\mathrm{loc}}(\alpha ,\widehat{w})&\buildrel \hbox { def}\over =\sum _{n\in \mathop {\mathbb N}\nolimits ^d} {c_n^{d+1}}\\&\quad \times \int ^\infty _0 \Big (\Theta \big (\alpha ,n,n, {\alpha }{c_n}\big ) + \Theta \big (\alpha ,n,n, {-\alpha }{c_n}\big ) \Big ) \, \alpha ^d \, \psi (\alpha ) \, d\alpha \, , \end{aligned}\nonumber \\ \end{aligned}$$

(5.11)

with $\psi$ any smooth, nonnegative, even function, compactly supported in ${\mathop {\mathbb R}\nolimits }$ with an $L^\infty$ norm at most 1.

Proceeding as for the restriction theorem on the sphere of $\widehat{\mathop {\mathbb H}\nolimits }^d$, let us first compute

$$\begin{aligned} G_{\Sigma _{\mathrm{loc}}} \buildrel \hbox { def}\over ={\mathcal F}_{{\mathbb D}}^{-1}(d\Sigma _{\mathrm{loc}})\,. \end{aligned}$$

Proposition 5.1

With the above notation, $G_{\Sigma _{\mathrm{loc}}}$ is the bounded function on ${\mathbb D}$ defined by

$$\begin{aligned} G_{\Sigma _{\mathrm{loc}}}(t, w)= 2\pi \, \int ^\infty _0 G_{\alpha }(w) \, e^{-i t\,\alpha } \psi (\alpha )\, d\alpha \,, \end{aligned}$$

(5.12)

where $G_R$ is given by (4.10).

Proof

Arguing as in the proof of Proposition 4.2 and using the fact that the Fourier transform ${\mathcal F}_{\mathbb D}$ is a bicontinuous isomorphism between the spaces ${\mathcal S}({\mathbb D})$ and ${\mathcal S}(\widehat{\mathbb D}),$ there holds (with $\Theta \buildrel \hbox { def}\over ={\mathcal F}_{\mathbb D} f$)

$$\begin{aligned} \langle d{\Sigma _{\mathrm{loc}}},\Theta \rangle _{{\mathcal S}'(\widehat{\mathbb D})\times {\mathcal S}(\widehat{\mathbb D})} = \Big (\frac{\pi }{2} \Big )^d \langle G_{\Sigma _{\mathrm{loc}}} , \widetilde{f}\rangle _{{\mathcal S}'({\mathbb D})\times {\mathcal S}({\mathbb D})} \, , \end{aligned}$$

(5.13)

with $\widetilde{f} (t, y,\eta ,s) \buildrel \hbox { def}\over =f(-t, y,-\eta ,-s)$. By definition, we have for any non negative real number $\alpha$

$$\begin{aligned} \, \, \Theta \big (\alpha ,n,n, {\pm \alpha }{c_n}\big )= \int _{{\mathbb D}} e^{\mp i s{\alpha }{c_n}} \,{\mathcal W}\Big (n,n,1, { {\alpha ^\frac{1}{2} {\sqrt{c_n}}\,} Y}\Big )\, e^{-i t\,\alpha }f(t,Y,s) \,dt \,dY\,ds \, , \end{aligned}$$

which implies in view of (5.11) that

$$\begin{aligned}&\langle d{\Sigma _{\mathrm{loc}}},\Theta \rangle _{{\mathcal S}'(\widehat{\mathbb D})\times {\mathcal S}(\widehat{\mathbb D})} = \sum _{n\in \mathop {\mathbb N}\nolimits ^d} 2{c_n^{d+1}}\\&\quad \times \int ^\infty _0 \int _{{\mathbb D}} \cos \Big ( s{\alpha }{c_n}\Big ) \,{\mathcal W}\Big (n,n,1, { {\alpha ^\frac{1}{2} } {\sqrt{c_n}}\,Y}\Big )\, e^{-i t\,\alpha }f(t,Y,s) \,dt \,dY\,ds \, \alpha ^d \psi (\alpha ) \, d\alpha \, . \end{aligned}$$

This achieves the proof of the result thanks to the Fubini theorem and Formula (4.10), after an obvious change of variables. $\square$

5.1.3 Restriction theorem

Our aim now is to establish the following restriction result for the “dual set” ${\Sigma _{\mathrm{loc}}}$ of $\widehat{\mathbb D}$ defined by (5.9) and endowed with the measure $d{\Sigma _{\mathrm{loc}}}$.

Theorem 5

If $1 \le q \le p \le 2$, then

$$\begin{aligned} \Vert {\mathcal F}_{{\mathbb D}}(f)_{|{\Sigma _{\mathrm{loc}}}} \Vert _{L^{2}( d{\Sigma _{\mathrm{loc}}})}\le C_{p,q} \Vert f\Vert _{L^1_s L_t^{q}L^{p}_{Y}}\, , \end{aligned}$$

(5.14)

for all radial functions f in ${\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb H}\nolimits }^d)$.

Remark 5.2

By duality, Theorem 5 may be rephrased as follows : for any $2 \le p' \le q' \le \infty$, there holds

$$\begin{aligned} \Vert {\mathcal F}_{{\mathbb D}}^{-1}( \Theta _{|\Sigma _{\mathrm{loc}}} )\Vert _{L^\infty _s L_t^{q'} L^{p'}_{Y}} \le C_{p,q} \Vert \Theta _{|{\Sigma _{\mathrm{loc}}}} \Vert _{L^{2}( d{\Sigma _{\mathrm{loc}}})} \, ,\end{aligned}$$

(5.15)

for all $\Theta \in {\mathcal F}_{{\mathbb D}}({\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb R}\nolimits }\times {\mathop {\mathbb H}\nolimits }^d))$.

Proof of Theorem 5

We handle differently the cases $1 \le p < 2$ and $p=2$. To undertake the case $1 \le p < 2$, we shall follow the Euclidean strategy outlined in Sect. 2. To this end, let us introduce $R_{\Sigma _{\mathrm{loc}}}$ the restriction operator on $\Sigma _\mathrm{loc}$ defined for any function f in ${\mathcal S}({\mathbb D})$ by

$$\begin{aligned} R_{\Sigma _{\mathrm{loc}}} f \buildrel \hbox { def}\over ={\mathcal F}_{\mathbb D}(f)_{|{\Sigma _{\mathrm{loc}}}} = \Big (\int _{{\mathbb D}} \overline{e^{is\lambda +it\alpha } {\mathcal W}(\widehat{w},Y)}\, f(t,Y,s) \,dY\,ds\,dt\Big )_{|{\Sigma _{\mathrm{loc}}}}\,, \end{aligned}$$

and by $R_{\Sigma _{\mathrm{loc}}}^*$ its adjoint. By definition

$$\begin{aligned} \begin{aligned} \langle R_{\Sigma _{\mathrm{loc}}} f,R_{\Sigma _{\mathrm{loc}}} f\rangle _{L^2 (\Sigma _\mathrm{loc})}&= \sum _{n\in \mathop {\mathbb N}\nolimits ^d} {c_n^{d+1}}\int _0^\infty \Big (\big |\Theta \big (\alpha ,n,n, {\alpha }{c_n}\big )\big |^2 \\&\quad +\big |\Theta \big (\alpha ,n,n, {-\alpha }{c_n}\big )\big |^2\Big )\psi (\alpha )\alpha ^d\,d\alpha \, , \end{aligned} \end{aligned}$$

with $\displaystyle \Theta \buildrel \hbox { def}\over ={\mathcal F}_{\mathbb D}(f )$, which readily implies that

$$\begin{aligned}&\langle R_{\Sigma _{\mathrm{loc}}} f,R_{\Sigma _{\mathrm{loc}}} f\rangle _{L^2 (\Sigma _{\mathrm{loc}})} = \sum _{n\in \mathop {\mathbb N}\nolimits ^d} {c_n^{d+1}} \nonumber \\&\quad \times \int _0^\infty \Big [\Theta \big (\alpha ,n,n, {\alpha }{c_n}\big )\int _{{\mathbb D}} e^{ {is}{\alpha c_n}+it\alpha } {\mathcal W}\Big (n,n, {\alpha }{c_n },Y\Big )\, \overline{f(t,Y,s)} \,dt\, dY\,ds\nonumber \\&\quad + \,\Theta \big (\alpha ,n,n, {-\alpha }{c_n}\big )\int _{{\mathbb D}} e^{ {-is}{\alpha c_n}+it\alpha } {\mathcal W}\big (n,n, {-\alpha }{c_n},Y\big )\, \overline{f(t,Y,s)} \,dt\, dY\,ds\Big ] \psi (\alpha )\alpha ^d\,d\alpha .\nonumber \\ \end{aligned}$$

(5.16)

In view of (5.11), this leads to

$$\begin{aligned} \langle R_{\Sigma _{\mathrm{loc}}} f,R_{\Sigma _{\mathrm{loc}}} f\rangle _{L^2 (\Sigma _{\mathrm{loc}})}=\langle R_{\Sigma _{\mathrm{loc}}}^*R_{\Sigma _{\mathrm{loc}}} f, f\rangle _{L^2 ({\mathbb D})}\,, \end{aligned}$$

(5.17)

with (for $w=(Y,s)$)

$$\begin{aligned} (R_{\Sigma _{\mathrm{loc}}}^*R_{\Sigma _{\mathrm{loc}}} f)(t,w)= \int _{\Sigma } e^{is\lambda +it\alpha } {\mathcal W}(\widehat{w},Y) {\mathcal F}_{\mathbb D}(f) (\alpha ,\widehat{w}) d{\Sigma _{\mathrm{loc}}} (\alpha ,\widehat{w})\, . \end{aligned}$$

(5.18)

Since f belongs to ${\mathcal S}_\mathrm{{rad}}({\mathbb D})$, combining (3.12) together with (3.13) and (3.15) we infer that the operator $R_{\Sigma _{\mathrm{loc}}}^*R_{\Sigma _{\mathrm{loc}}}$ writes in the radial setting⁷ :

$$\begin{aligned} (R_{\Sigma _{\mathrm{loc}}}^*R_{\Sigma _{\mathrm{loc}}} f)(t,Y,s)= \int _{\Sigma } {\mathcal F}_{{\mathbb D}}(f \circ \tau _{(-t, Y, -s)})(\alpha ,\widehat{w})\, \, d{\Sigma _{\mathrm{loc}}}(\alpha ,\widehat{w})\, . \end{aligned}$$

(5.19)

By (3.20) and (5.13), this gives rise to

$$\begin{aligned} (R_{\Sigma _{\mathrm{loc}}}^*R_{\Sigma _{\mathrm{loc}}} f)(t, Y,s) = \Big (\frac{\pi }{2} \Big )^d \, (G_{\Sigma _{\mathrm{loc}}} \star _{{\mathbb D}} \check{f})(-t,-Y,s)\, , \end{aligned}$$

(5.20)

for all f in ${\mathcal S}_\mathrm{{rad}}({\mathbb D})$.

Now applying the Hölder inequality to (5.17), we deduce that

$$\begin{aligned} \begin{aligned} \Vert R_{\Sigma _{\mathrm{loc}}} f\Vert ^2_{L^{2}({{\Sigma _{\mathrm{loc}}}})}&\le \Vert R_{\Sigma _{\mathrm{loc}}}^* R_{\Sigma _{\mathrm{loc}}} f\Vert _{L^\infty _s L_t^{q'} L^{p'}_{Y}}\Vert f\Vert _{L^1_s L_t^{q}L^{p}_{Y}}\\&\le C_d\Vert \check{f} \star _{{\mathbb D}} G_{\Sigma _{\mathrm{loc}}} \Vert _{L^\infty _s L_t^{q'} L^{p'}_{Y}}\Vert f\Vert _{L^1_s L_t^{q}L^{p}_{Y}}\,, \end{aligned} \end{aligned}$$

for some irrelevant constant $C_d$ which may change from line to line. Then as in the Euclidean case, to complete the proof of Estimate (5.14), we are reduced to proving that $R_{\Sigma _{\mathrm{loc}}}^* R_{\Sigma _{\mathrm{loc}}}$ is bounded from $L^1_s({\mathop {\mathbb R}\nolimits }, {L_t^{q}({\mathop {\mathbb R}\nolimits },L^{p}( T^*{\mathop {\mathbb R}\nolimits }^{d})))}$ into $L^\infty _s({\mathop {\mathbb R}\nolimits }, {L_t^{q'}({\mathop {\mathbb R}\nolimits },L^{p'}( T^*{\mathop {\mathbb R}\nolimits }^{d})))}$. For that purpose, let us start by observing that in light of (4.10) and (5.12), we have

$$\begin{aligned}&f\star _{{\mathbb D}} G_{\Sigma _{\mathrm{loc}}} (t,Y,s)=C_d \sum _\ell \frac{1}{(2\ell +d)^{d+1} } \\&\quad \times \sum _{\pm }\int _0^\infty f\star _{{\mathbb D}} \Big ( e^{\pm i \alpha _{\ell } - it\alpha } {\widetilde{{\mathcal W}}} \Big (\ell , \alpha _{\ell } , Y\Big ) \Big ) \alpha ^d \, \psi (\alpha ) d\alpha \,, \end{aligned}$$

where $\widetilde{\mathcal W}$ is given by (3.13) and where we have defined

An easy computation shows that for any real number $\lambda$,

$$\begin{aligned} f\star _{{\mathbb D}} \Big ( e^{\pm i\lambda s - it\alpha } {\widetilde{{\mathcal W}}} (\ell ,\lambda ,Y) \Big ) = e^{\pm i\lambda s - it\alpha } \left( f^{\alpha ,\pm \lambda } \star _{\pm \lambda } {\widetilde{{\mathcal W}}} (\ell ,\lambda ,Y) \right) \, , \end{aligned}$$

where $\star _{ \lambda }$ denotes the twisted convolution defined in (A.1), and with

$$\begin{aligned} f^{\alpha ,\beta }(Y) \buildrel \hbox { def}\over =\int e^{ i s\beta + it\alpha } f(t, Y,s) dt ds = \int e^{ i s\beta } {\mathcal F}_{\mathop {\mathbb R}\nolimits }f(-\alpha , \cdot ) ds \, . \end{aligned}$$

(5.21)

Defining $\psi _+(\alpha ) \buildrel \hbox { def}\over =\psi (\alpha ) {\mathbf{1}}_{\alpha >0}$, it follows that

$$\begin{aligned} \begin{aligned} f\star _{{\mathbb D}} G_{\Sigma _{\mathrm{loc}}} (t,s,Y)&= C_d \sum _{\ell } \frac{1}{(2\ell +d)^{d+1} } \\&\quad \times \sum _{\pm }\int _0^\infty e^{\pm is \alpha _{\ell } -it\alpha } \Big (f^{\alpha ,\pm \alpha _{\ell } }\star _{\pm \alpha _{\ell } } {\widetilde{{\mathcal W}}} \big (\ell , \alpha _{\ell } ,Y\big )\Big ) \alpha ^d \, \psi (\alpha ) \,d\alpha \\&= C_d \sum _{\ell } \sum _{\pm }\frac{1}{(2\ell +d)^{d+1}} {\mathcal F}_{\alpha \rightarrow t} \Big (e^{\pm is \alpha _{\ell } }\alpha ^d\psi _+(\alpha ) \Big (f^{\alpha ,\pm \alpha _{\ell } }\star _{\pm \alpha _{\ell } } {\widetilde{{\mathcal W}}} \big (\ell , \alpha _{\ell } ,Y \big )\Big )\Big )\, . \end{aligned} \end{aligned}$$

Now let us fix $s\in {\mathop {\mathbb R}\nolimits }$ and $Y\in T^*{\mathop {\mathbb R}\nolimits }^{d}$, and compute the $L_t^{q'}({\mathop {\mathbb R}\nolimits })$ norm of the function of t appearing on the right-hand side of the equality, for $q' \ge 2$: by the Hausdorff-Young inequality we find

$$\begin{aligned} \Vert f\star _{{\mathbb D}} G _{\Sigma _{\mathrm{loc}}} \Vert _{L_t^{q'}} \lesssim \sum _{\ell } \sum _{\pm }\frac{1}{(2\ell +d)^{d+1}} \Big \Vert e^{\pm is \alpha _{\ell } }\alpha ^d\psi (\alpha ) \Big (f^{\alpha ,\pm \alpha _{\ell } }\star _{\pm \alpha _{\ell } } {\widetilde{{\mathcal W}}} \big (\ell , \alpha _{\ell } ,Y \big )\Big )\Big \Vert _{ L_\alpha ^q} \, . \end{aligned}$$

Now noticing that as soon as $q' \ge p'\ge 2$, thanks to Minkowski’s inequality we have for any function g defined on ${\mathop {\mathbb R}\nolimits }\times T^*{\mathop {\mathbb R}\nolimits }^{d}$

$$\begin{aligned} \begin{aligned} \Vert {\mathcal F}_{\alpha \rightarrow t} g\Vert _{L^{q'}_{t} L^{p'}_{Y}}&\le \Vert {\mathcal F}_{\alpha \rightarrow t} g\Vert _{L^{p'}_{Y} L^{q'}_{t} }\\&\lesssim \Vert g\Vert _{ L^{p'}_{Y} L^{q}_{\alpha }}\\&\lesssim \Vert g\Vert _{L^{q}_{\alpha } L^{p'}_{Y} } \,, \end{aligned} \end{aligned}$$

(5.22)

we deduce that for $q' \ge p'>2$ (since we assumed here that $1 \le p < 2$)

$$\begin{aligned} \begin{aligned} \Vert f\star _{{\mathbb D}} G_{\Sigma _{\mathrm{loc}}} \Vert _{L^\infty _s L_t^{q'} L^{p'}_{Y} }&\lesssim \sum _{\ell } \sum _{\pm }\frac{1}{(2\ell +d)^{d+1}}\Big \Vert \alpha ^d\psi (\alpha ) \Big (f^{\alpha ,\pm \alpha _{\ell } }\star _{\pm \alpha _{\ell } } {\widetilde{{\mathcal W}}} \big (\ell , \alpha _{\ell } , Y \big )\Big )\Big \Vert _{ L_\alpha ^q L^{p'}_{Y} } \, . \end{aligned} \end{aligned}$$

But by (A.2), there holds

$$\begin{aligned} \Big \Vert \Big (f^{\alpha ,\pm \frac{\alpha }{4(2\ell +d)}}\star _{\pm \frac{\alpha }{4(2\ell +d)}} {\widetilde{{\mathcal W}}} \Big (\ell , \frac{\alpha }{4(2\ell +d)},Y\Big )\Big ) \Big \Vert _{L^{p'}_{Y}} \lesssim \ell ^{d-1+ \frac{2}{p'}}\,\alpha ^{- \frac{2d}{p'}} \Vert {\mathcal F}_{\mathop {\mathbb R}\nolimits }f(-\alpha , \cdot )\Vert _{L^{p}_{Y}L^{1}_{s}}\,, \end{aligned}$$

which (since $p'>2$) implies that

$$\begin{aligned} \Vert f\star _{{\mathbb D}} G_{\Sigma _{\mathrm{loc}}} \Vert _{L^\infty _s L_t^{q'} L^{p'}_{Y}} \lesssim \Big \Vert \Vert {\mathcal F}_{{\mathop {\mathbb R}\nolimits }}(f)(-\alpha , \cdot )\Vert _{L^{p}_{Y}L^{1}_{s}} \, \alpha ^{d(1- \frac{2}{p'})}\ \psi (\alpha )\Big \Vert _{L_\alpha ^{q}} \,. \end{aligned}$$

Then, applying a Hölder estimate in $\alpha$ followed by the Hausdorff-Young inequality, we get for any $a \ge 2$

$$\begin{aligned} \begin{aligned} \Vert f\star _{{\mathbb D}} G_{\Sigma _{\mathrm{loc}}} \Vert _{L^\infty _s L_t^{q'} L^{p'}_{Y}}&\lesssim \Vert {\mathcal F}_{{\mathop {\mathbb R}\nolimits }}(f)\Vert _{L^{a}_\alpha L^{p}_{Y}L^{1}_{s}} \Vert \alpha ^{d(1- \frac{2 }{p'})}\ \psi (\alpha )\Vert _{L_\alpha ^{b}} \\&\lesssim \Vert f\Vert _{L^{a'}_t L^{p}_{Y}L^{1}_{s}} \Vert \alpha ^{d(1- \frac{2 }{p'})}\ \psi (\alpha )\Vert _{L_\alpha ^{b}({\mathop {\mathbb R}\nolimits })} \, , \end{aligned} \end{aligned}$$

where of course $a'$ is the conjugate exponent of a and $\displaystyle \frac{1}{a}+ \frac{1}{b} = \frac{1}{q} \cdot$

Finally selecting $a'=q$ and thanks again to Minkowski’s inequality, we get for all real numbers $q' \ge p'>2$

$$\begin{aligned} \Vert f\star _{{\mathbb D}} G_{\Sigma _{\mathrm{loc}}} \Vert _{L^\infty _s L_t^{q'} L^{p'}_{Y}} \lesssim \Vert f\Vert _{L^1_s L_t^{q} L^{p}_{Y} } \end{aligned}$$

which completes the proof of the result in the case when $1 \le q \le p < 2$.

Finally the proof in the case when $p=2$ is in the same spirit than the one concerning the unit dual sphere of the Heisenberg group already outlined in the proof of Theorem 4. By definition, our aim here is to show that (for $q \le 2$)

$$\begin{aligned}&\sum _{\ell \in \mathop {\mathbb N}\nolimits } \int _0^\infty \Big ( \Big |\widetilde{\Theta }\Big (\alpha , \ell , \ell , \frac{\alpha }{2\ell +d}\Big )\Big |^2 + \Big |\widetilde{\Theta }\Big (\alpha , \ell , \ell , \frac{-\alpha }{2\ell +d}\Big )\Big |^2\Big )\psi (\alpha )\alpha ^d\,d\alpha \\&\quad \lesssim \Vert f\Vert ^2_{L^1_s L_t^{q} L^{2}_{Y}}\, , \end{aligned}$$

where with the notation introduced in (4.13) and on page 17

$$\begin{aligned} \begin{aligned} \widetilde{\Theta }\Big (\alpha , \ell , \ell , \lambda \Big )&\buildrel \hbox { def}\over =\Big [ \begin{pmatrix} \ell +d-1 \\ \ell \end{pmatrix}^{-1} \frac{1}{(2\ell +d)^{d+1}} \Big ]^{\frac{1}{2}} \sum _{\begin{array}{c} {n\in \mathop {\mathbb N}\nolimits ^d}\\ {|n| = \ell } \end{array}} {\mathcal F}_{\mathbb D} f (\alpha , n, n, \lambda ) \\&= \int _{{\mathbb D}} e^{-it\alpha } e^{-is\lambda } K(\ell , \lambda ,Y)\, f(t,Y,s) dt \,dY\,ds\,. \end{aligned} \end{aligned}$$

(5.23)

For that purpose, let us establish that the operator $T_{\mathbb D}$ defined on ${\mathcal S}_\mathrm{{rad}}({\mathbb D})$ by

$$\begin{aligned} T_{\mathbb D}f \buildrel \hbox { def}\over =\Big (\widetilde{\Theta }\big (\alpha , \ell , \ell , \alpha _{\ell } \big )\Big )_{\ell \in \mathop {\mathbb N}\nolimits } \,, \end{aligned}$$

where f is related to $\Theta$ through (5.23), is bounded from $L^1_s({\mathop {\mathbb R}\nolimits }, {L_t^{q}({\mathop {\mathbb R}\nolimits },L^{2}_{Y}( T^*{\mathop {\mathbb R}\nolimits }^{d})))}$ into the space $L^2(\mathop {\mathbb N}\nolimits \times {\mathop {\mathbb R}\nolimits })$ endowed with the measure $\ell ^2(\mathop {\mathbb N}\nolimits )\otimes L^2({\mathop {\mathbb R}\nolimits }^+, \psi (\alpha )\alpha ^d\,d\alpha )$, or equivalently that its adjoint $T_{\mathbb D}^*$ is bounded from $L^2(\mathop {\mathbb N}\nolimits \times {\mathop {\mathbb R}\nolimits })$ into $L^\infty _s({\mathop {\mathbb R}\nolimits }, L_t^{q'}({\mathop {\mathbb R}\nolimits },L^{2}_{Y}( T^*{\mathop {\mathbb R}\nolimits }^{d})))$.

For ${\underline{a}} (\alpha )= (a_\ell (\alpha ))_{\ell \in \mathop {\mathbb N}\nolimits }$ in  $L^2(\mathop {\mathbb N}\nolimits \times {\mathop {\mathbb R}\nolimits })$, the operator $T_{\mathbb D}^*$ is given by

$$\begin{aligned} T_{\mathbb D}^*({\underline{a}}) (t, Y,s)= \sum ^\infty _{\ell =0} \int _0^\infty a_\ell (\alpha )\, e^{-it\alpha } \, e^{-is \alpha _{\ell } } \, K\big (\ell , \alpha _{\ell } ,Y\big ) \psi (\alpha )\alpha ^d\,d\alpha \, . \end{aligned}$$

Combining (5.22) together with the Hausdorff-Young inequality, we find that for any fixed s in ${\mathop {\mathbb R}\nolimits }$, there holds

$$\begin{aligned} \Vert T_{\mathbb D}^*({\underline{a}}) (\cdot , \cdot , s)\Vert _{ L_t^{q'}L^{2}_{Y}} \lesssim \Vert g\Vert _{L^{q}_{\alpha } L^{2}_{Y} } \,, \end{aligned}$$

where

$$\begin{aligned} g(\alpha , Y)\buildrel \hbox { def}\over =\alpha ^d \psi (\alpha ) \,\sum ^\infty _{\ell =0} a_\ell (\alpha )\, K\big (\ell , \alpha _{\ell } ,Y\big )\, . \end{aligned}$$

Now taking advantage of (4.14) and performing an obvious change of variable, we get

$$\begin{aligned} \Big \Vert \sum ^\infty _{\ell =0} a_\ell (\alpha )\, K\big (\ell , \ell , \alpha _{\ell } ,Y\big )\Big \Vert ^2_{L^{2}_{Y}} \lesssim \alpha ^{d } \sum _{ m} |a_m(\alpha )| b_m (\alpha ) \,, \end{aligned}$$

where of course $\displaystyle b_m(\alpha )\buildrel \hbox { def}\over =\frac{1 }{m} \sum _{\ell \le m} |a_\ell (\alpha )| \,.$ Thanks to the Hardy inequality (4.15), this ensures that

$$\begin{aligned} \Vert g(\alpha , \cdot )\Vert _{L^{2}_{Y} } \lesssim \alpha ^{\frac{3d }{2} } \psi (\alpha ) \Vert {\underline{a}} (\alpha )\Vert _{\ell ^2(\mathop {\mathbb N}\nolimits ) }\, , \end{aligned}$$

which by Hölder’s inequality gives rise to

$$\begin{aligned} \Vert g\Vert _{L^{q}_{\alpha }L^{2}_{Y} } \lesssim \Vert \alpha ^{d } \psi ^{\frac{1 }{2} }(\alpha ) \Vert _{L^{\frac{2-q }{2 q } }( {\mathop {\mathbb R}\nolimits }) }\Vert {\underline{a}} \Vert _{L^2(\mathop {\mathbb N}\nolimits \times {\mathop {\mathbb R}\nolimits }) } \, . \end{aligned}$$

This achieves the proof of the Fourier restriction estimate (5.14). $\square$

Remark 5.3

In the case of the wave equation, we consider the sets

$$\begin{aligned} \Sigma _{\pm } \buildrel \hbox { def}\over =\Big \{(\alpha ,\widehat{w})=\big (\alpha , (n, n, \lambda ) \big ) \in \widehat{\mathbb D}\, /\, \alpha ^{2}= 4 |\lambda |(2|n|+d)\, , \, \, \pm \alpha >0 \Big \} .\qquad \end{aligned}$$

(5.24)

Each of those is endowed with the measure induced by the projection $\pi : \widehat{\mathop {\mathbb R}\nolimits }\times \widehat{\mathop {\mathbb H}\nolimits }^{d}\rightarrow \widehat{\mathop {\mathbb H}\nolimits }^{d}$ onto the second factor. Explicitly, for all $\Theta$ in ${\mathcal S}(\widehat{\mathbb D})$, we have

$$\begin{aligned} \begin{aligned} \langle d\Sigma _{\pm },\Theta \rangle _{{\mathcal S}'(\widehat{\mathbb D})\times {\mathcal S}(\widehat{\mathbb D})}&= \int _{\Sigma }\Theta (\alpha ,\widehat{w}) \, d\Sigma (\alpha ,\widehat{w}) \\&\buildrel \hbox { def}\over =\sum _{n\in \mathop {\mathbb N}\nolimits ^d} {c_n^{d+1}} \int ^\infty _0 \Big (\Theta \big (\alpha ,n,n, {\alpha ^{2}}{c_n}\big ) + \Theta \big (\alpha ,n,n, {-\alpha ^{2}}{c_n}\big ) \Big ) \, \alpha ^d \, d\alpha \, . \end{aligned} \end{aligned}$$

Following the same argument as above, one proves (5.15) for the corresponding localized measures.

5.2 Strichartz estimates for the Schrödinger operator on ${\mathop {\mathbb H}\nolimits }^{d}$

The aim of this section is to prove the Strichartz estimate stated in Theorem 1. By duality arguments, one can reduce to the free Schrödinger equation (see Appendix B for further details about the inhomogeneous framework). Let $u_0$ be a function in ${\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb H}\nolimits }^d)$ and consider the Cauchy problem

$$\begin{aligned} (S_{\mathop {\mathbb H}\nolimits })\qquad \left\{ \begin{array}{c} i\partial _t u -\Delta _{\mathop {\mathbb H}\nolimits }u = 0\\ u_{|t=0} = u_0 \, . \end{array} \right. \end{aligned}$$

As in the Euclidean case, Fourier analysis allows us to explicitly solve $(S_{\mathop {\mathbb H}\nolimits })$. More precisely, taking the partial Fourier transform with respect to the variable w in the Heisenberg group, we obtain for all $(t, \widehat{w})$ in ${\mathop {\mathbb R}\nolimits }\times \widetilde{\mathop {\mathbb H}\nolimits }^d$

$$\begin{aligned} \left\{ \begin{array}{rcl} \displaystyle i \frac{d}{dt} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u) (t, n,m, \lambda )&{}\! =\! &{}- 4|\lambda |(2|m|+d) {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u) (t, n,m, \lambda )\\ {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u)_{|t=0} &{}\!=\! &{} {\mathcal F}_{\mathop {\mathbb H}\nolimits }{u_0}\, . \end{array} \right. \end{aligned}$$

By integration, this leads to

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u) (t, n,m, \lambda )= e^{4i t |\lambda |(2|m|+d)}{\mathcal F}_{\mathop {\mathbb H}\nolimits }(u_0) (|n|, |n|, \lambda ) \delta _{n,m}\, . \end{aligned}$$

Then applying the inverse Fourier formula (3.4), we infer that the solution of the Cauchy problem $(S_{\mathop {\mathbb H}\nolimits })$ can be expressed as follows:

$$\begin{aligned} u(t, Y, s)= \frac{2^{d-1}}{\pi ^{d+1} } \int _{\widehat{\mathop {\mathbb H}\nolimits }^d} e^{is\lambda } \, {\mathcal W}(\widehat{w}, Y) \, e^{4i t |\lambda |(2|m|+d)} \, {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u_0) (|n|, |n|, \lambda ) \delta _{n,m} \, d\widehat{w}.\nonumber \\ \end{aligned}$$

(5.25)

That explicit representation of the solution to $(S_{\mathop {\mathbb H}\nolimits })$ can be re-expressed as the inverse Fourier transform in $\widehat{\mathbb D}$ of ${\mathcal F}_{\mathop {\mathbb H}\nolimits }(u_0) \, d\Sigma$, where as defined in Paragraph 5.1.2, $d\Sigma$ is the measure in $\widehat{\mathbb D}$ given for any regular function $\Phi$ on $\widehat{\mathbb D}$:

$$\begin{aligned} \int _{\widehat{\mathbb D}} \Phi (\alpha ,\widehat{w}) \, d\Sigma (\alpha ,\widehat{w})= \int _{\widehat{\mathop {\mathbb H}\nolimits }^d} \Phi (4|\lambda |(2|m|+d),\widehat{w}) \,d\widehat{w} \, ,\end{aligned}$$

(5.26)

and which is determined by Formula (5.11).

Now in order to establish Estimate (1.19), let us first discuss the case of initial data frequency localized, in the sense of Definition 3.1, in the unit ball ${\mathcal B}_1$; then we shall generalize this to any ball of radius $\Lambda$ by a scaling argument, and finally the result will follow by density.

So let us start by assuming that $u_0$ is frequency localized in the unit ball. Then by the restriction inequality (5.15), and (5.10) we have for any $2\le p \le q \le \infty$

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le C \Vert \mathcal F_{{\mathop {\mathbb H}\nolimits }} u_0 \Vert _{L^{2}(\widehat{\mathop {\mathbb H}\nolimits }^d)} = C \Vert u_0 \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \, , \end{aligned}$$

where we used Plancherel formula. Now if $u_0$ is frequency localized in the ball ${\mathcal B}_\Lambda$, then by virtue of (3.10) the function

$$\begin{aligned} u_{0,\Lambda }\buildrel \hbox { def}\over =u_0 \circ \delta _{\Lambda ^{-1}} \end{aligned}$$

is frequency localized in ${\mathcal B}_1$, and it gives rise to the solution

$$\begin{aligned} u_{\Lambda }(t,Y,s)\buildrel \hbox { def}\over =u ( \Lambda ^{-2}t,\Lambda ^{-1}Y, \Lambda ^{-2}s) \end{aligned}$$

of $(S_{\mathop {\mathbb H}\nolimits })$. Since

$$\begin{aligned} \Vert u_{\Lambda }\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} =\Lambda ^{\frac{2}{q} + \frac{2d}{p}} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \end{aligned}$$

and

$$\begin{aligned} \Vert u_{0,\Lambda }\Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} =\Lambda ^{d+1} \Vert u_0\Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)}\, , \end{aligned}$$

we infer that for $u_0$ frequency localized in the ball ${\mathcal B}_\Lambda$, there holds

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le C\Lambda ^{d+1-\frac{2}{q}-\frac{2d}{p}} \Vert u_0 \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \, . \end{aligned}$$

The result follows from the fact that for any $s \ge 0$, if $u_0$ is frequency localized in the ball ${\mathcal B}_\Lambda$, then

$$\begin{aligned} \Lambda ^s \Vert u_0 \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \lesssim \Vert u_0\Vert _{H^s({\mathop {\mathbb H}\nolimits }^d)} \, , \end{aligned}$$

and we conclude the proof of the estimate by density of spectrally localized functions in $H^s({\mathop {\mathbb H}\nolimits }^d)$. This ends the proof of Theorem 1. $\square$

5.3 Strichartz estimates for the wave operator on ${\mathop {\mathbb H}\nolimits }^{d}$

The aim of this section is to prove the Strichartz estimate stated in Theorem 2. The method is identical to the previous section: again we reduce to the free wave equation and consider for $(u_0,u_1)$ in ${\mathcal S}_\mathrm{{rad}}({\mathop {\mathbb H}\nolimits }^d)$ frequency localized in a unit ring in the sense of Definition 3.1, the Cauchy problem

$$\begin{aligned} (W_{\mathop {\mathbb H}\nolimits })\qquad \left\{ \begin{array}{c} \partial _t^2 u -\Delta _{\mathop {\mathbb H}\nolimits }u = 0\\ (u,\partial _t u)_{|t=0} = (u_0,u_1) \, . \end{array} \right. \end{aligned}$$

Taking the Fourier transform we find that for all $(t, \widehat{w})$ in ${\mathop {\mathbb R}\nolimits }\times \widetilde{\mathop {\mathbb H}\nolimits }^d$,

$$\begin{aligned} \left\{ \begin{array}{rcl} \displaystyle \frac{ d^2}{dt^2} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u) (t, n,m, \lambda )&{}\! =\! &{}- 4|\lambda |(2|m|+d) {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u) (t, n,m, \lambda )\\ ( {\mathcal F}_{\mathop {\mathbb H}\nolimits }u, {\mathcal F}_{\mathop {\mathbb H}\nolimits }\partial _t u)_{|t=0} &{}\!=\! &{} {( {\mathcal F}_{\mathop {\mathbb H}\nolimits }u_0, {\mathcal F}_{\mathop {\mathbb H}\nolimits }u_1)}\, . \end{array} \right. \end{aligned}$$

By integration, this leads to

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u) (t, n,m, \lambda )= \sum _{\pm } e^{\pm 2i t \sqrt{|\lambda |(2|m|+d})}{\mathcal F}_{\mathop {\mathbb H}\nolimits }(\gamma _{\pm }) (|n|, |n|, \lambda ) \delta _{n,m}\, , \end{aligned}$$

where similarly to (1.4) we have defined

$$\begin{aligned} {\mathcal F}_{\mathop {\mathbb H}\nolimits }(\gamma _{\pm }) (n, n, \lambda ) \buildrel \hbox { def}\over =\frac{1}{2} \Big ( {\mathcal F}_{\mathop {\mathbb H}\nolimits }(u_0) \pm \frac{1}{i\sqrt{4|\lambda | (2n+d)}}{\mathcal F}_{\mathop {\mathbb H}\nolimits }(u_1) \Big )\, . \end{aligned}$$

Then applying the inverse Fourier formula (3.4), we infer that the solution of the Cauchy problem $(W_{\mathop {\mathbb H}\nolimits })$ can be expressed as follows:

$$\begin{aligned} u(t, Y, s)= \sum _{\pm } \frac{2^{d-1}}{\pi ^{d+1} } \int _{\widehat{\mathop {\mathbb H}\nolimits }^d} e^{is\lambda } \, {\mathcal W}(\widehat{w}, Y) \, e^{\pm 2i t \sqrt{|\lambda |(2|m|+d})} \, {\mathcal F}_{\mathop {\mathbb H}\nolimits }(\gamma _{\pm }) (|n|, |n|, \lambda ) \delta _{n,m} \, d\widehat{w}. \nonumber \\ \end{aligned}$$

(5.27)

This can be written as the inverse Fourier transform in $\widehat{\mathbb D}$ of ${\mathcal F}_{\mathop {\mathbb H}\nolimits }(\gamma _{\pm }) \, d\Sigma _{\pm }$, where we recall that by Remark 5.3, for any regular function $\Phi$ on $\widehat{\mathbb D}$

$$\begin{aligned} \int _{\widehat{\mathbb D}} \Phi (\alpha ,\widehat{w}) \, d\Sigma _{\pm }(\alpha ,\widehat{w})= \int _{\widehat{\mathop {\mathbb H}\nolimits }^d} \Phi (\sqrt{4|\lambda |(2|m|+d}),\widehat{w}) \,d\widehat{w} \, .\end{aligned}$$

(5.28)

By hypothesis $\gamma _{\pm }$ is frequency localized in a unit ring. Then for any $2\le p \le q \le \infty$, we have by virtue of the restriction inequality (5.15)

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le C \Vert {\mathcal F}_{\mathop {\mathbb H}\nolimits }^{-1}\gamma _{\pm } \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \, . \end{aligned}$$

Next if $\gamma _{\pm }$ is frequency localized in the ring $\mathcal {C}_\Lambda$, then by  (3.10) the function

$$\begin{aligned} u_{0,\Lambda }\buildrel \hbox { def}\over =u_0 \circ \delta _{\Lambda ^{-1}} \end{aligned}$$

is frequency localized in the ring $\mathcal {C}_1$ and gives rise to the solution

$$\begin{aligned} u_{\Lambda }(t,Y,s)\buildrel \hbox { def}\over =u ( \Lambda ^{-1}t,\Lambda ^{-1}Y, \Lambda ^{-2}s) \end{aligned}$$

of $(W_{\mathop {\mathbb H}\nolimits })$. Since

$$\begin{aligned} \Vert u_{\Lambda }\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} =\Lambda ^{\frac{1}{q} + \frac{2d}{p}} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \end{aligned}$$

and

$$\begin{aligned} \Vert u_{0,\Lambda }\Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} =\Lambda ^{d+1} \Vert \gamma _{\pm }\Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)}\, , \end{aligned}$$

we infer that for $u_0$ frequency localized in the ring ${\mathcal C}_\Lambda$, there holds

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y} } \le C\Lambda ^{d+1-\frac{1}{q}-\frac{2d}{p}} \Vert \gamma _{\pm } \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \, . \end{aligned}$$

Then by Bernstein’s Lemma 3.2, and in particular estimate (3.9), followed by Plancherel’s inequality we infer that

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le C\Lambda ^{d-\frac{1}{q}-\frac{2d}{p}} \Big ( \Vert \nabla _{{\mathop {\mathbb H}\nolimits }^d}u_0 \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} +\Vert u_1 \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \Big )\, . \end{aligned}$$

To conclude the proof of the estimate, we argue as for the Schrödinger equation making use of the fact that for any real s, if a function f is frequency localized in the ring ${\mathcal C}_\Lambda$, then

$$\begin{aligned} \Lambda ^s \Vert f \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \le C \Vert f\Vert _{H^s({\mathop {\mathbb H}\nolimits }^d)} \, . \end{aligned}$$

This proves Theorem 2. $\square$

Appendices

Appendix A: Proof of a technical result

We have used the following lemma due to Müller (see [38]), whose proof we sketch below.

Lemma 5.4

Defining for f, g in ${\mathcal S}(T^\star {\mathop {\mathbb R}\nolimits }^d)$ and $\lambda$ in ${\mathop {\mathbb R}\nolimits }\setminus \{0\}$ the $\lambda$-twisted convolution

$$\begin{aligned} (f \star _\lambda g )(Y)\buildrel \hbox { def}\over =\int _{T^\star {\mathop {\mathbb R}\nolimits }^d} f(Y-w) g(w) e^{2i\lambda \sigma (Y,w)}dw\, , \end{aligned}$$

(A.1)

with $\sigma$ defined in (1.9), the following estimate holds: there exists a positive constant $C_{d}$ such that

$$\begin{aligned} \Vert f \star _\lambda \widetilde{\mathcal W}(\ell ,\lambda ,\cdot ) \Vert _{L^{p}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\le C_{d} \, |\lambda |^{- \frac{2d}{p'}}\, \ell ^ {(d-1)(1-\frac{2}{p'})}\, \Vert f \Vert _{L^{p}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, , \end{aligned}$$

(A.2)

for all $1\le p \le 2$ and all integers $\ell \ge 1$, where the function $\widetilde{\mathcal W}(\ell ,\lambda ,Y)$ is defined by (3.13).

Proof of Lemma 5.4

Let us start by establishing that for f, g in ${\mathcal S}(T^\star {\mathop {\mathbb R}\nolimits }^d)$ and $1\le p \le 2$, the following estimate holds:

$$\begin{aligned} \Vert f \star _\lambda g \Vert _{L^{p'}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\le C_{p,d} \, |\lambda |^{- \frac{d}{p'}} \, \Vert f \Vert _{L^{p}(T^\star {\mathop {\mathbb R}\nolimits }^d)} \Vert g \Vert ^{\frac{2}{p'}} _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)} \Vert g \Vert ^{1-\frac{2}{p'}} _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, . \end{aligned}$$

By definition,

$$\begin{aligned} (f \star _\lambda g )(Y)= \int _{T^\star {\mathop {\mathbb R}\nolimits }^d} f(Y-w) g(w) e^{2i\lambda \sigma (Y,w)}dw\, , \end{aligned}$$

which easily implies by Young’s inequalities that

$$\begin{aligned} \Vert f \star _\lambda g \Vert _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\le \Vert f \Vert _{L^{1}(T^\star {\mathop {\mathbb R}\nolimits }^d)} \Vert g \Vert _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, . \end{aligned}$$

Therefore invoking the method of real interpolation, we are reduced to showing that

$$\begin{aligned} \Vert f \star _\lambda g \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\le C_{d} \, |\lambda |^{- \frac{d}{2}} \, \Vert f \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)} \Vert g \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, , \end{aligned}$$

and this follows by an easy dilation argument from the well-known fact that $(L^2(T^\star {\mathop {\mathbb R}\nolimits }^d), \star _1)$ is a Hilbert algebra (see for instance [32]).

Let us now focus on Estimate (A.2). We first recall that

$$\begin{aligned} \widetilde{\mathcal W}(\ell ,\lambda ,Y) = \sum _{|n|= \ell } {\mathcal L}_n(|\lambda |^{\frac{1}{2}} \, Y)\, , \end{aligned}$$

where for $Z=(Z_1, \ldots , Z_d)$ in $T^\star {\mathop {\mathbb R}\nolimits }^d$, we denote ${\mathcal L}_n(Z)= e^{-|Z|^2} \Pi ^d_{j=1} L_ {n_j}( 2 |Z_j|)$ with $L_ {n_j}$ the Laguerre polynomial of order $n_j$ and type 0.

Define now for $n \in \mathop {\mathbb N}\nolimits ^d$ the operator $T_n$ on $L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)$ by

$$\begin{aligned} T_n f\buildrel \hbox { def}\over =f \star _\lambda {\mathcal L}_n(|\lambda |^{\frac{1}{2}} \cdot )\, ,\end{aligned}$$

(A.3)

so that

$$\begin{aligned} Tf= f \star _\lambda \widetilde{\mathcal W}(\ell ,\lambda ,\cdot )= \sum _{|n|= \ell } T_n f \, . \end{aligned}$$

(A.4)

Then using the fact that the Laguerre polynomials $(L_k)_{k \in \mathop {\mathbb N}\nolimits }$ are pairwise orthogonal on $[0, \infty [$ with respect to the measure $e^{-x} dx$, we infer that the family of operators $(T_n)_{|n|= \ell }$ is also pairwise orthogonal, and thus denoting by $\Vert T \Vert$ the norm of T defined by (A.4) as an operator on $L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)$, we can conclude that

$$\begin{aligned} \Vert T \Vert _{{\mathcal L}(L^2(T^\star {\mathop {\mathbb R}\nolimits }^d))} = \max _{|n|= \ell }\Vert T_n \Vert _{{\mathcal L}(L^2(T^\star {\mathop {\mathbb R}\nolimits }^d))} \, .\end{aligned}$$

(A.5)

Since $\Vert {\mathcal L}_n(|\lambda |^{\frac{1}{2}} \cdot )\Vert _{L^2(T^\star {\mathop {\mathbb R}\nolimits }^d)} = |\lambda |^{- \frac{d}{2}} \, \Vert {\mathcal L}_n\Vert _{L^2(T^\star {\mathop {\mathbb R}\nolimits }^d)}$, one obtains

$$\begin{aligned} \Vert T_n \Vert _{{\mathcal L}(L^2(T^\star {\mathop {\mathbb R}\nolimits }^d))} \lesssim |\lambda |^{- d} \, . \end{aligned}$$

In view of (A.5), this implies that

$$\begin{aligned} \Vert f \star _\lambda \widetilde{\mathcal W}(\ell ,\lambda ,\cdot ) \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\lesssim |\lambda |^{- d} \, \Vert f \Vert _{L^{2}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, . \end{aligned}$$

Finally, by definition

$$\begin{aligned} \Vert \widetilde{\mathcal W}(\ell ,\lambda ,\cdot ) \Vert _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\lesssim \ell ^{d-1} \, , \end{aligned}$$

which ensures that

$$\begin{aligned} \Vert f \star _\lambda \widetilde{\mathcal W}(\ell ,\lambda ,\cdot ) \Vert _{L^{\infty }(T^\star {\mathop {\mathbb R}\nolimits }^d)}\lesssim \ell ^{d-1} \, \Vert f \Vert _{L^{1}(T^\star {\mathop {\mathbb R}\nolimits }^d)}\, , \end{aligned}$$

and completes the proof of Estimate (A.2) by interpolation. $\square$

Appendix B: The inhomogeneous case

Denoting by $(\mathcal U(t))_{t\in {\mathop {\mathbb R}\nolimits }}$ the solution operator of the Schrödinger equation on the Heisenberg group, namely $\mathcal U(t)u_0$ is the solution of $(S_{\mathop {\mathbb H}\nolimits })$ with $f=0$ at time t associated with the data $u_0$, then similarly to the euclidean case $(\mathcal U(t))_{t\in {\mathop {\mathbb R}\nolimits }}$ is a one-parameter group of unitary operators on $L^2({\mathop {\mathbb H}\nolimits }^d)$. Moreover, the solution to the inhomogeneous equation

$$\begin{aligned} \left\{ \begin{array}{c} i\partial _t u -\Delta _{\mathop {\mathbb H}\nolimits }u = f\\ u_{|t=0} = 0\,, \end{array} \right. \end{aligned}$$

writes

$$\begin{aligned} u(t, \cdot )= -i \int ^t_0 \mathcal U(t-t') f(t', \cdot ) dt' ,\end{aligned}$$

(B.1)

Let us check that it satisfies, for all admissible pairs (p, q) in ${\mathcal A}^{\text{ S }}$,

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \lesssim \Vert f \Vert _{L_t^1 H^{\sigma }({\mathop {\mathbb H}\nolimits }^d)} \end{aligned}$$

(B.2)

with

Let us first assume that, for all t, the source term $f(t, \cdot )$ is frequency localized in in the unit ball ${\mathcal B}_1$ in the sense of Definition 3.1, and recall that according to the results of Sect. 5.2, if g is frequency localized in a unit ball, then for all $2\le p \le q \le \infty$

$$\begin{aligned} \Vert \mathcal U(t)g\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \lesssim \Vert g \Vert _{L^{2}({\mathop {\mathbb H}\nolimits }^d)} \, . \end{aligned}$$

(B.3)

Taking advantage of (B.1), we have for all $s \in {\mathop {\mathbb R}\nolimits }$,

$$\begin{aligned} \Vert u(t, \cdot , s) \Vert _{L^{p}_{Y}} \le \int _{\mathop {\mathbb R}\nolimits }\Vert \mathcal U(t) \mathcal U(-t') f(t', \cdot , s) \Vert _{L^{p}_{Y}} dt' . \end{aligned}$$

Therefore, still for all s,

$$\begin{aligned} \Vert u(\cdot ,\cdot , s) \Vert _{L_t^{q} L^{p}_{Y}} \le \int _{\mathop {\mathbb R}\nolimits }\Vert \mathcal U(\cdot ) \mathcal U(-t') f(t', \cdot , s) \Vert _{L_t^{q} L^{p}_{Y}} dt' . \end{aligned}$$

Invoking (B.3), we deduce that

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le \int _{\mathop {\mathbb R}\nolimits }\Vert \mathcal U(-t') f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' . \end{aligned}$$

Since $\mathcal U(-t')$ is unitary on $L^{2}({\mathop {\mathbb H}\nolimits }^d)$, we readily gather that

$$\begin{aligned} \Vert u \Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le \int _{\mathop {\mathbb R}\nolimits }\Vert f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' .\end{aligned}$$

(B.4)

Now if for all t, $f(t, \cdot )$ is frequency localized in a ball of size $\Lambda$, then setting

$$\begin{aligned} f_{\Lambda }(t, \cdot ) \buildrel \hbox { def}\over =\Lambda ^{-2} f (\Lambda ^{-2} t, \cdot ) \circ \delta _{\Lambda ^{-1}} \end{aligned}$$

we find that on the one hand, $f_{\Lambda }(t, \cdot )$ is frequency localized in a unit ball for all t, and on the other hand that the solution to the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{c} i\partial _t u_{\Lambda } -\Delta _{\mathop {\mathbb H}\nolimits }u_{\Lambda } = f_{\Lambda }\\ u_{|t=0} = 0\,, \end{array} \right. \end{aligned}$$

writes

$$\begin{aligned} u_{\Lambda }(t,w)= u (\Lambda ^{-2} t, \cdot ) \circ \delta _{\Lambda ^{-1}} \,. \end{aligned}$$

Now by scale invariance, we have

$$\begin{aligned} \int _{\mathop {\mathbb R}\nolimits }\Vert f_{\Lambda }(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' = \Lambda ^{\frac{Q}{2}}\int _{\mathop {\mathbb R}\nolimits }\Vert f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' \end{aligned}$$

and

$$\begin{aligned} \Vert u_{\Lambda }\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} =\Lambda ^{\frac{2}{q} + \frac{2d}{p}} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \, . \end{aligned}$$

Consequently, we get

$$\begin{aligned} \Vert u\Vert _{L^\infty _s L_t^{q} L^{p}_{Y}} \le C \int _{\mathop {\mathbb R}\nolimits }\Lambda ^{\frac{Q}{2}- \frac{2}{q} - \frac{2d}{p}} \Vert f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} dt' \, . \end{aligned}$$

Since $\displaystyle \frac{Q}{2}- \frac{2}{q} - \frac{2d}{p} \ge 0$, we have

$$\begin{aligned} \Lambda ^{\frac{Q}{2}- \frac{2}{q} - \frac{2d}{p}} \Vert f(t', \cdot ) \Vert _{{L^{2}({\mathop {\mathbb H}\nolimits }^d)}} \lesssim \Vert f(t', \cdot ) \Vert _{{H^{\frac{Q}{2}- \frac{2}{q} - \frac{2d}{p}}({\mathop {\mathbb H}\nolimits }^d)}} \,, \end{aligned}$$

which completes the proof of (B.2).

In the case of the wave equation, we have seen in Paragraph 5.3 that when the Cauchy data $u_0$ and $u_1$ are frequency localized in a ring, then u the solution to the Cauchy problem $(W_{\mathop {\mathbb H}\nolimits })$ with $f=0$ reads, with the previous notations,

$$\begin{aligned} u(t, \cdot )= \mathcal U^+ (t) \gamma ^+ + \mathcal U^- (t) \gamma ^- \, . \end{aligned}$$

This allows to investigate the inhomogeneous wave equation by similar arguments than the Schrödinger equation dealt with above. The proof is left to the reader.