Local Dispersive And Strichartz Estimates For The Schrödinger Operator On The Heisenberg Group

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Local Dispersive And Strichartz Estimates For The Schrödinger Operator On The Heisenberg Group

Hajer Bahouri, Isabelle Gallagher

To cite this version:

Hajer Bahouri, Isabelle Gallagher. Local Dispersive And Strichartz Estimates For The Schrödinger Operator On The Heisenberg Group. 2020. �hal-03058429�

SCHR ¨ODINGER OPERATOR ON THE HEISENBERG GROUP

HAJER BAHOURI AND ISABELLE GALLAGHER

Abstract. It was proved by H. Bahouri, P. G´erard and C.-J. Xu in [9] that the Schr¨odinger equation on the Heisenberg group H^d, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates onH^d for the linear Schr¨odinger equation, by a refined study of the Schr¨odinger kernel St

on H^d. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel onH^dderived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernelStconcentrates on quantized horizontal hyperplanes ofH^d.

Contents

1. Introduction 1

2. Fourier analysis on H^d 7

3. On the kernel of the heat operator onH^d 10

4. On the kernel of the Schr¨odinger operator on H^d 13 5. Proof of the local dispersive and Strichartz estimates 16 6. Concentration properties of the Schr¨odinger kernel onH^d 17

References 22

Keywords: Heisenberg group, Schr¨odinger equation, dispersive estimates, Strichartz esti- mates.

AMS Subject Classification (2000): 43A30, 43A80.

1. Introduction

1.1. Setting of the problem. It is well-known that the solution to the free Schr¨odinger equation onRⁿ

(S)

i∂tu−∆u= 0 u_|t=0=u₀ can be explicitly written with a convolution kernel fort6= 0

(1.1) u(t,·) =u0? e^−i|·|

2 4t

(−4πit)ⁿ² ·

The proof of this explicit representation stems by a combination of Fourier and complex analysis arguments, from the expression of the heat kernel onRⁿ. More precisely, taking the partial Fourier transform of (S) with respect to the variablex and integrating in time the resulting ODE, we get

u(t, ξ) = eb ^it|ξ|2u_b0(ξ),

Date: December 11, 2020.

1

where for any functiong∈L¹(Rⁿ) we have defined bg(ξ)^def= F(g)(ξ)^def=

Z

Rⁿ

e^−ihx,ξig(x)dx .

The heart of the matter to prove (1.1) then consists in computing in the sense of distri- butions the inverse Fourier transform of the complex Gaussian

(1.2) (F⁻¹e^it|·|2)(x) = e^−i|x|

2 4t

(−4πit)ⁿ² ·

The proof of Formula (1.2) is based on two observations: first, that for anyx in Rⁿ, the two maps

z∈C7−→H₁(z)^def= 1 (2π)ⁿ

Z

Rⁿ

e^ihx,ξie^−z|ξ|2dξ and z∈C7−→H₂(z)^def= 1 4πz

n 2

e⁻

|x|2 4z

are holomorphic on the domainDof complex numbers with positive real part. Accordingly with the expression of the heat kernel, these two functions coincide on the intersection of the real line withD, and thus they coincide on the whole domainD. Second, if (z_p)_p∈Nde- notes a sequence of elements ofDwhich converges to−itfort6= 0, the use of the Lebesgue dominated convergence theorem ensures thatH₁(z_p) andH₂(z_p) converge inS⁰(Rⁿ), asp tends to infinity, which achieves the proof of (1.2).

Formula (1.1) implies by Young’s inequality the following dispersive estimate:

(1.3) ∀t6= 0, ku(t,·)k_L^∞_(Rⁿ₎≤ 1

(4π|t|)ⁿ²ku₀k_L1(Rⁿ)·

Such estimate plays a key role in the study of semilinear and quasilinear equations which appear in numerous physical applications. Combined with an abstract functional analysis argument known as the T T^∗-argument, it yields a range of inequalities involving space- time Lebesgue norms, known as Strichartz estimates¹. Whenu₀ is for instance in L²(Rⁿ), the above dispersive estimate (1.3) gives rise to the following Strichartz estimate for the solution to the free Schr¨odinger equation

(1.4) kuk_Lq(R;L^p(Rⁿ))≤C(p, q)ku₀k_L2(Rⁿ), where (p, q) satisfies the scaling admissibility condition

(1.5) 2

q +n p = n

2 with q ≥2 and (n, q, p)6= (2,2,∞).

The interest for this issue has soared in the last decades. We refer for instance to the monographs [3,31] and the references therein for an overview on this topic in the euclidean framework.

In the present work, we aim at investigating this phenomenon for the Schr¨odinger equation on the Heisenberg group H^d involving the sublaplacian. Recall that in [9], the first author along with P. G´erard and C.-J. Xu proved that no dispersion occurs for this equation, and in particular exhibited an example for which the Schr¨odinger operator onH^dbehaves as a transport equation with respect to one direction, known as the vertical direction. More precisely they established the following result which shows that a global dispersive estimate of the type (1.3) cannot be expected on H^d. We refer to the coming paragraph for the notation.

Proposition 1.1 ([9]). There exists a functionu₀ in the Schwartz classS(H^d)such that the solution to the free Schr¨odinger equation onH^d satisfies

(1.6) ∀t∈R, ∀(Y, s)∈H^d, u(t, Y, s) =u₀(Y, s+ 4td).

1For further details, one can consult the papers of Ginibre-Velo [23], Keel-Tao [25] and Strichartz [30].

This result rules out an estimate of the type (1.3) in the setting of the Heisenberg group but does not exclude a degraded estimate: for instance in the case whenu0 is compactly supported, then the solution remains compactly supported, in a set transported along the vertical line, so a local L^∞ norm decays to zero with time. Inspired by the euclidean strategy displayed above, we shall indeed be able to establish local decay in the spirit of (1.3). The precise result is stated in the next paragraph. As in the euclidean case, such a local dispersive estimate stems from the explicit expression of the Schr¨odinger kernelSt

onH^d, which turns out to be of type (1.2) in a horizontal strip ofH^d (see Theorem 2).

Note also that a lack of dispersion was highlighted for the Schr¨odinger propagator (as- sociated with the sublaplacian) in the framework of H-type groups ([12]) or more generally in the case of 2-step stratified Lie groups ([8]). More precisely, if pdenotes the dimension of the center of the H-type group, M. Del Hierro proved in [12] sharp dispersive inequalities for the Schr¨odinger equation solution (with a|t|^−(p−1)/2 decay). Concerning the more gen- eral case of 2-step stratified Lie groups, the authors along with C. Fermanian-Kammerer [8]

emphasized the key role played by the canonical skew-symmetric form in determining the rate of decay of the solutions of the Schr¨odinger equation: they established that if p de- notes the dimension of the center of a 2-step stratified Lie groupG and k the dimension of the radical of its canonical skew-symmetric form, then the solutions of the Schr¨odinger equation onGsatisfy dispersive estimates with a rate of decay at most of order|t|^−k+p−12 . 1.2. Basic facts about the Heisenberg group. Recall that thed-dimensional Heisen- berg groupH^d can be defined asT^?R^d×RwhereT^?R^d is the cotangent bundle, endowed with the noncommutative product law²

(1.7) (Y, s)·(Y⁰, s⁰)^def= Y +Y⁰, s+s⁰+ 2hη, y⁰i −2hη⁰, yi,

where w = (Y, s) = (y, η, s) and w⁰ = (Y⁰, s⁰) = (y⁰, η⁰, s⁰) are elements of H^d. The variable Y is called the horizontal variable, while the variable s is known as the vertical variable.

The space H^d is provided with a smooth left invariant measure, the Haar measure, which in the coordinate system (Y, s) is simply the Lebesgue measure. In particular, one can define the following (noncommutative) convolution product for any two integrable functionsf and g:

(1.8) f ? g(w)^def= Z

H^d

f(w·v⁻¹)g(v)dv= Z

H^d

f(v)g(v⁻¹·w)dv , and the usual Young inequalities are valid:

(1.9) kf ? gk_Lr(H^d)≤ kfk_Lp(H^d)kgk_Lq(H^d), whenever 1≤p, q, r≤ ∞ and 1 r = 1

p+1 q −1. The dilation onH^dis defined for all a >0 by

(1.10) δa(Y, s)^def= (aY, a²s). Since, for alla >0 and anyf ∈L¹(H^d),

Z

H^d

f δ_a(w)dw=a^−(2d+2) Z

H^d

f(w)dw , the homogeneous dimension ofH^dis Q^def= 2d+ 2.

The natural distance onH^dcompatible with the product law (1.7) is called the Kor´anyi distance and is defined by

(1.11) d_H(w, w⁰)^def= ρ_H(w⁻¹·w⁰),

2We refer to the monographs [7,18,19,29,32,33] and the references therein for further details.

for allw,w⁰ inH^d, where ρ_H stands for the distance to the origin (1.12) ρ_H(w) =ρ_H(Y, s)^def= |Y|⁴+s²

1 4 .

In the followingB_H(w₀, R) denotes the Heisenberg ball centered atw₀ and of radiusR for the distanced_H defined by (1.11), namely

B_H(w0, R)^def= ⁿw∈H^d/ d_H(w, w0)< R^o.

Observing that the distanced_H is invariant by left translation, that is to say

∀(w, w⁰, w0)∈(H^d)³, d_H τw0(w), τw0(w⁰)=d_H(w, w⁰) whereτw0 denotes the left translation defined by

(1.13) τ_w0(w)^def= w₀·w ,

one can readily check thatτw0 B_H(0, R)=B_H(w0, R).

Most classical analysis tools ofRⁿ can be adapted toH^d, resorting to the following left invariant vector fields

X_j ^def= ∂_yj+ 2η_j∂_s and Ξ_j ^def= ∂_ηj−2y_j∂_s with j∈ {1, . . . , d},

known as the horizontal left invariant vector fields. In particular, the sublaplacian is given by

∆_H ^def=

d

X

j=1

(X_j²+ Ξ²_j).

For instance the Schwartz space S(H^d), which is nothing else than S(R^2d+1), can be characterized by means of ∆_H and ρ_H.

1.3. Main results. The main goal of this article is to establish local dispersive estimates for the free linear Schr¨odinger equation on H^dassociated with the sublaplacian

(S_H)

i∂tu−∆_Hu= 0 u_|t=0=u₀.

As in the euclidean case, one can readily establish that the Cauchy problem (S_H) admits a unique, global in time solution ifu₀∈L²(H^d), by resorting to Fourier-Heisenberg analysis tools or to functional calculus of the self-adjoint operator−∆_H (see Section 4 for further details). Denoting by (U(t))t∈Rthe solution operator, namelyU(t)u₀is the solution of (S_H) at time t associated with the datau0, then similarly to the euclidean case (U(t))t∈R is a one-parameter group of unitary operators onL²(H^d).

The first result we establish states as follows.

Theorem 1. Given w0 ∈H^d, let u0 be a function in D(B_H(w0, R0)). Then the solution to the Cauchy problem (S_H) associated to u₀ disperses locally for large |t|, in the sense that, for any positive constantκ <√

4d, the following estimate holds for all2≤p≤ ∞:

(1.14) ku(t,·)k

L^p(B_H(w0,κ|t|¹²))≤ Mκ

|t|^Q2

!1−²_p

ku₀k_Lp0

(H^d), for all|t| ≥Tκ,R0, where

(1.15) Tκ,R0

def= √ R0

4d−κ 2

and Mκ def

= 1

(4π)^Q2 Z

R

2τ sinh 2τ

d

expκ²τ 2

dτ ,

andp⁰ is the conjugate exponent top.

Remark 1.2. The counterexample(1.6)due to Bahouri, G´erard and Xu in[9]is given by

(1.16) u(t, Y, s) =

Z

R

e^i(s+4dt)λe^−λ|Y|2g(λ)λ^ddλ

withginD(]0,∞[). Althoughu0 does not belong to D(H^d), one can easily check that for anyδ >0, as soon as |s+ 4dt|> δ|t|then for any integerN there is a positive constantC depending onN and u₀ such that

(1.17) |u(t, w)| ≤ C

|s+ 4dt|^N ≤ C (δ|t|)^N ·

On the other hand Estimate(1.17)fails, for any integerN ≥1, in the case whens=−4td.

This shows the sharpness of the bound on the constantκ appearing in Theorem1. More generally it was established in[6]that for any integer `, denoting byL<sup>(d−1)</sup><sub>`</sub> the Laguerre polynomial of order`and typed−1 (see for instance[13,24,28]), then

u^(`)(t, Y, s) = Z

R

ei(s+4t(2`+d))λe<sup>−|λ||Y|2</sup>L<sup>(d−1)</sup><sub>`</sub> (2|λ||Y|²)g(λ)λ^ddλ

is a solution to(S_H), and this solution satisfies (1.17) when|s+ 4(2`+d)t|> δ|t|,and|t|

is large enough.

As in the euclidean case outlined above, the (local) dispersive estimate (1.14) stems from Young inequalities (1.9) using an explicit formula of the type (1.1) for the Schr¨odinger kernel on H^d. However, the study of the kernel S_t of the Schr¨odinger operator on H^d is more involved than in the euclidean case, because on the one hand the Fourier transform on H^d is an intricate tool and on the other hand St does not enjoy a formulation of type (1.2) globally on H^d. In fact, as will be seen in Section4 (see Proposition 4.1), one can compute St in the sense of distributions, using the Fourier-Heisenberg analysis tools developed in [5], and also it turns out (see Theorem 4) thatS_t concentrates on quantized horizontal hyperplanes of H^d. It follows that the explicit formula of the type (1.1) that we obtain here is only local. More precisely, our result states as follows. Its sharpness is discussed in Paragraph1.4.

Theorem 2. The kernel associated with the free Schr¨odinger equation (S_H) reads for allt6= 0

(1.18) S_t(Y, s) = 1 (−4iπt)^Q2

Z

R

2τ sinh 2τ

^d exp

−τ s

2t −i |Y|²τ 2ttanh 2τ

dτ , provided that|s|<4d|t|.

Remark 1.3. Theorem2highlights the separate roles of the horizontal and vertical vari- ables ofH^d. Note that in[27], D. M¨uller already emphasized the distinguished role of the horizontal variable in the study of the Fourier restriction theorem onH^d.

Even though the dispersive estimate (1.14) we establish for the Schr¨odinger operator on H^d is only local, we are able to prove that the solutions of the Schr¨odinger equa- tion (S_H) enjoy locally Strichartz estimates in the spirit of (1.4). More precisely, we have the following result.

Theorem 3. Under the notations of Theorem 1, given κ <√

4d and (p, q) belonging to the admissible set

(1.19) A^def= ⁿ(p, q)/2 q + Q

p = Q

2 with 2≤p≤ ∞^o,

there exists a positive constant C(q, κ) such that, for all u₀ ∈ L²(H^d) supported in the ballB_H(w0, R0), for some w0 ∈H^d, the solution to the Cauchy problem(S_H)satisfies the

following local Strichartz estimate

(1.20) kuk

L^q(]−∞,−CκR²₀]∪[CκR²₀,+∞[;L^p(B_H(0,κ√

|t|)))≤C(q, κ)ku₀k_L2(B_H(w0,R0)), whereC_κ= (

√

4d−κ)⁻².

Note that the Strichartz estimate (1.20) is invariant by scaling (through the scal- ing u(t, w) 7→ u(λ²t, δλw)). Let us underline that there is a duality between the size of the support of u₀ and the time for which the Strichartz estimates holds. Indeed, let- tingR0 go to zero, we find that for an initial data concentrated around somew0 ∈H^d, the Strichartz estimate is almost global in time. Conversely, lettingR₀ go to infinity, the time from which (1.20) occurs is close to infinity. Let us also emphasize that the counterex- amples introduced in Remark 1.2 show somehow the optimality of our result, since for these counterexamples a global integrability both with respect to t and s independently is excluded.

1.4. Refined study of the Schr¨odinger kernel on H^d. Theorem 2 asserts that St, for t 6= 0, is a decaying smooth function on the strip |s| < 4d|t| (with a decay rate of order|t|^−Q/2). One may wonder ifSt (t6= 0) which, according to Proposition 4.1, belongs toS⁰(H^d) can be identified with a function on the horizontal hyperplaness=±4d|t|. The answer to this question is negative as asserted by the following result.

Theorem 4. With the previous notations, for all ±w_{`</sub> <sup>def</sup>= (0,±4(2`+d)|t|), where t6= 0 and`∈N, there exists an initial datau<sup>±,`</sup>₀ ∈ S(H^d)such thatu^{±,`</sup>(t,·)<sup>def</sup>= U(t)u<sup>±,`}₀ satisfies (1.21) u<sup>±,`</sup>(t,±w<sub>`}) =u^{±,`</sup><sub>0</sub> (0) =hδ<sub>0</sub>, u<sup>±,`}₀ i_S0(H^d)×S(H^d).

Remark 1.4. Actually, the above theorem can be easily generalized to any element of the horizontal hyperplaness=±4(2`+d)|t|,t6= 0and `∈N, namely(Y₀,±4(2`+d)|t|), where Y<sub>0</sub> is some fixed element of T<sup>?</sup>R<sup>d</sup>. The Cauchy data generating a solution which concentrates on hyperplaness=±4(2`+d)|t|are linked to the counterexamples introduced in Remark1.2.

The above result shows the optimality of the bound 4d|t| in Theorem 2. However, we are able to improve this bound when we restrict (S_H) to some subspaces of Cauchy data as in the next statement.

Theorem 5. There exists an orthogonal decomposition ofL²(H^d)

(1.22) L²(H^d) =⊕_m∈

N^dL²_m(H^d)

such that the restrictionS_t<sup>(`)</sup>ofS<sub>t</sub>to the subspaceV`(H^d)^def= ⊕<sub>|m|≥`</sub>L<sup>2</sup><sub>m</sub>(H<sup>d</sup>)is well defined as soon as|s|<4(2`+d)|t|, and satisfies for any positive constantκ <^p4(d+ 2`),

sup

|s|≤κ²|t|

sup

Y∈T^∗R^d

1

|t|^Q2

|S_t<sup>(`)</sup>(Y, s)| ≤C(`, κ).

Remark 1.5. Decomposition(1.22) is strongly tied to the spectral representation of the sublaplacian −∆_H. In order to give a flavor of the above result, let us point out that, as we shall see in Section 2, the Fourier-Heisenberg transform exchanges −∆_H with the harmonic oscillator. Then in some sense, (1.22) consists in a decomposition of L²(H^d) along Hermite-type functions, via the Fourier-Heisenberg transform.

1.5. Main steps of the proof of the main results and layout of the paper. Since the Schr¨odinger equation onH^dis invariant under left translations, one can assume without loss of generality in the proof of Theorem 1 that w₀ = 0. By Young inequalities (1.9), Theorem 1 readily follows from Theorem 2 (reducing the assumption |s| < 4d|t| to the fact that ρ_H(w) < ^p4d|t|). To prove Theorem 1, we are thus reduced to establishing Theorem 2. Roughly speaking the proof of Theorem 2 is achieved in three steps. In the first step, using the Fourier-Heisenberg analysis on tempered distributions developed in [5] (see also Section2.2in this paper), we establish that the kernelStof the Schr¨odinger operator onH^dbelongs toS⁰(H^d) (Proposition4.1). It is well-known since the paper of B.

Gaveau [20], that the solution to the heat equation onH^dassociated with the sublaplacian writes for allt >0

u(t,·) = 1 t^Q2

u₀?(h◦δ^√_t),

whereδ^√_t is the dilation operator defined in (1.10) and h is the function in the Schwartz classS(H^d) given by

(1.23) h(Y, s)^def= 1 (4π)^Q2

Z

R

2τ sinh 2τ

d

exp

iτ s

2 − |Y|²τ 2 tanh 2τ

dτ .

Then the second step is devoted to the proof of the fact that the fundamental solution of the heat equation onH^d coming from Fourier analysis on H^d coincides with the explicit formula (1.23) established by B. Gaveau [20] (see Proposition3.1). This step uses Melher’s formula, along with the Fourier approach developed in [4,5]. The last step concludes the proof following the general method of the euclidean case via complex analysis, described above (see Section4.2). It is in this final step that the restriction|s|<4d|t|appears.

As usual, the proof of the local Strichartz estimates stated in Theorem 3 is straight- forward from the local dispersive estimate (1.14) thanks to standard functional analysis arguments.

Finally, the refined study of the Schr¨odinger kernel on H^d(through Theorems 4and 5) is derived by a combination of Fourier-Heisenberg tools and the spectral analysis of the harmonic oscillator.

Let us describe the organization of the paper. Section2is dedicated to a brief description of the Fourier transformF_HonH^dand the space of frequenciesHb

d, as well as the extension of F_H to tempered distributions – which is at the heart of the matter in this paper. In Section 3, we recover the explicit formula of the heat kernel on H^d established by B.

Gaveau in [20], using Fourier analysis onH^d. In Section4, we investigate the kernel of the Schr¨odinger operator onH^d and prove Theorem2, while Section5is devoted to the proof of Theorem 1 thanks to Theorem 2. Then, we establish the local Strichartz estimates (Theorem3). In Section6, we undertake a refined study ofSt and establish Theorems4 and5 making use of the Fourier-Heisenberg approach developed in [4,5].

To avoid heaviness, all along this article C will denote a positive constant which may vary from line to line. We also useA.B to denote an estimate of the formA≤CB.

Acknowledgements. The authors thank Nicolas Lerner and Jacques Faraut very warmly for their help and input concerning the proof of Theorem5.

2. Fourier analysis on H^d

2.1. The Fourier transform on H^d. The Fourier transform on H^d is defined using irreducible unitary representations ofH^d. It is thus not a complex-valued function on some

“frequency space” as in the euclidean case, but a family of bounded operators onL²(R^d) (see for instance [2,11,14,16,29,32,33] for further details). Recently, in [4] and [5] the

authors introduced an equivalent, intrinsic definition of the Fourier transform on H^d in terms of functions acting on a frequency set denotedHe

ddef

= N^2d×R\ {0}. More precisely, denoting the elements of this set byw_b^def= (n, m, λ), the Fourier transform of an integrable function onH^d is defined in the following way:

(2.1) ∀w_b ∈He^d, F_Hf(w)_b ^def= Z

H^d

e^isλW(w, Y_b )f(Y, s)dY ds , withW the Wigner transform of the (renormalized) Hermite functions (2.2) W(w, Y_b )^def=

Z

R^d

e^2iλhη,ziH_n,λ(y+z)H_m,λ(−y+z)dz , H_m,λ(x)^def= |λ|^d4Hm(|λ|¹²x), with (H_m)_m∈

N^d the Hermite orthonormal basis of L²(R^d) given by the eigenfunctions of the harmonic oscillator:

−(∆− |x|²)H_m= (2|m|+d)H_m. We recall that

(2.3) H_m(x)^def= 1 2^|m|m!

¹₂ Y^d

j=1

−∂_jH₀(x) +x_jH₀(x)^mj, withH₀(x)^def= π^−d4e^−|x|

2

2 ,m!^def= m₁!· · ·m_d! and |m|^def= m₁+· · ·+m_d. In [4], the authors show that the completion of the setHe

dfor the distance d(bw,_b w_b⁰)^def= λ(n+m)−λ⁰(n⁰+m⁰)

`<sup>1</sup>(N<sup>d</sup>)+(n−m)−(n<sup>0</sup>−m<sup>0</sup>)|<sub>`</sub>1(N^d)+d|λ−λ⁰| is the set

Hb

ddef

= He

d∪Hb

d

0 with Hb

d 0

def= R^d∓×Z^d and R^d∓

def= (R−)^d∪(R+)^d.

In this setting, the classical statements of Fourier analysis hold in a similar way to the euclidean case. In particular, the inversion and Fourier-Plancherel formulae read:

(2.4) f(w) = 2^d−1

π^d+1 Z

eH

de^isλW(w, Y_b )F_Hf(w)_b dw_b and

(2.5) (F_Hf|F_Hg)

L²(eH

d)= π^d+1

2^d−1(f|g)_L2(H^d),

where the measuredw_b is defined in the following way³: for any function θ onHe

d, Z

He

dθ(w)_b dw_b ^def= Z

R

X

(n,m)∈N^2d

θ(n, m, λ)|λ|^ddλ .

Straightforward computations give

(2.6) F_H(−∆_Hf)(w) = 4|λ|(2|m|_b +d)F_H(f)(w)_b . According to (2.4)-(2.5), one can easily check that

(2.7) L²(H^d) =⊕_m∈

N^dL²_m(H^d), in the following way: any functionf ∈L²(H^d) can be split as (2.8) f = ^X

m∈N^d

fm with fm(Y, s) = 2^d−1 π^d+1

X

n∈N^d

Z

R

e^isλW((n, m, λ), Y)F_Hf(n, m, λ)|λ|^ddλ,

3As shown in [5], the measured^wbcan be extended by 0 onHb

d 0.

and

kfk²_L2(

H^d) = ^X

m∈N^d

kf_mk²_L2(

H^d).

Let us also note that if f and g are two functions of L¹(H^d) then for any w_b = (n, m, λ) inHe

d,there holds

(2.9) F_H(f ? g)(w) = (F_{b H}f· F_Hg)(w)_b ^def= ^X

p∈N^d

F_Hf(n, p, λ)F_Hg(p, m, λ).

As we shall see, the heat and Schr¨odinger kernels on H^d are radial, in the sense that they are invariant under the action of the unitary group of T^?R^d. In addition, being functions of −∆_H they are even. In fact, the Fourier transform of radial functions turns out to be simpler than in the general case: if f is a radial function in L¹(H^d), then for any (n, m, λ)∈He

d,

(2.10) F_H(f)(n, m, λ) =F_H(f)(n, m, λ)δ_n,m =F_H(f)(|n|,|n|, λ)δ_n,m, with, for all`∈N,

(2.11) F_H(f)(`, `, λ) =

`+d−1

`

−1Z

H^d

e^−isλe^−|λ||Y|2L^(d−1)_{`</sub> (2|λ||Y|<sup>2</sup>)f(Y, s)dY ds , whereL<sup>(d−1)</sup><sub>`} stands for the Laguerre polynomial⁴ of order`and type d−1.

Obviously the inversion formula writes in that case (2.12) f(w) = 2^d−1

π^d+1 X

`∈N

Z

R

e^isλWf(`, λ, Y)F<sub>H</sub>(f)(`, `, λ)|λ|^ddλ , where

(2.13) Wf(`, λ, Y)^def= ^X

n∈N^d

|n|=`

W(n, n, λ, Y) = e^−|λ||Y|2L^(d−1)_` (2|λ||Y|²).

2.2. The Fourier transform on S⁰(H^d). The new approach of the Fourier-Heisenberg transform developed in [4] enabled the authors in [5] to extend F_H toS⁰(H^d), the set of tempered distributions: note that since the Schwartz classS(H^d) coincides withS(R^2d+1) then similarlyS⁰(H^d) is nothing else than S⁰(R^2d+1). Roughly speaking, the first step to achieve this extension consists in characterizingS(Hb^d), the range ofS(H^d), byF_H. It will be useful to recall in the following that according to [5], the spaceS(Hb^d) can be equipped with semi-norms k · k

N,S(bH

d) and that in particular, for all θ ∈ S(Hb

d) and N ∈N, there existsC_N such that for all ˆw= (n, m, λ)∈He^d

(2.14) |θ( ˆw)| ≤CN(1 + 4|λ|(2|m|+d))^−Nkθk

N,S(Hb

d).

We refer to [5] for the definition ofS(H^bd) and further details. Then the result follows by duality, as in the euclidean case, once shown that the Fourier transformF_H is a bicontin- uous isomorphism between the spacesS(H^d) andS(Hb

d)⁵.

4The interested reader can consult for instance [1,10,13] and the references therein.

5Note that a first attempt in the description of the range ofS(H^d) by the Fourier-Heisenberg transform goes back to the pioneering works by D. Geller in [21,22], where asymptotic series are used. One can also consult the paper of F. Astengo, B. Di Blasio and F. Ricci [2].

The mapF_Hcan thus be continuously extended fromS⁰(H^d) intoS⁰(Hb^d) in the following way:

(2.15) F_H:







S⁰(H^d) −→ S⁰(Hb^d)

T 7−→ ^hθ7→ hF_HT, θi_S0(H^d)×S(H^d)=hT,^tF_Hθi_S0(H^d)×S(H^d)

i,

where, according to (2.4),

(2.16) ^tF_Hθ(y, η, s)^def= π^d+1 2^d−1(F⁻¹

H θ)(y,−η,−s). In particular one can compute the Fourier transform of the Dirac mass:

F_H(δ0) =1_{{(n,m,λ)/ n=m}}, that is to say, for anyθin S(Hb^d)

(2.17) hF_H(δ0), θi

S⁰(bH

d)×S(Hb

d)= ^X

n∈N^d

Z

R

θ(n, n, λ)|λ|^ddλ .

It will be useful later on to notice that S⁰(Hb

d) contains (after suitable identification) all functions with moderate growth, that are defined as the locally integrable functionsθ onHb

d such that for some large enough integerN,the map

(2.18) (n, m, λ)7−→ 1 +|λ|(n+m|+d) +|n−m|^−Nθ(n, m, λ)

belongs to L^∞(He^d). It is proved in [5] that any such function can be identified with a tempered distribution onHb

d.

Let us end this introduction on Fourier analysis on the Heisenberg group by recalling that ifT is a tempered distribution on H^d, then for allf inS(H^d) and allw inH^d,

(T ? f)(w) =hT,fˇ◦τ_w⁻¹i_S0(H^d)×S(H^d), and

(2.19) (f ? T)(w) =hT,fˇ◦τ_w^ri_S0(H^d)×S(H^d), where

(2.20) fˇ(w)^def= f(w⁻¹),

and τw denotes the left translation operator byw defined in (1.13) while τ_w^r is the right translation operator byw defined by

(2.21) τ_w^r(w⁰)^def= w⁰·w .

3. On the kernel of the heat operator on H^d

A striking consequence of Fourier analysis on H^d developed in [5] is that it provides another proof of the fact that the fundamental solution of the heat equation onH^d coin- cides with the explicit formula established by B. Gaveau in [20]. First note the following representation coming from Fourier-Heisenberg analysis.

Proposition 3.1. Ifu denotes the solution to the free heat equation on H^d (H_H)

∂tu−∆_Hu = 0 u_|t=0 = u₀,

whereu0 is a given integrable function onH^d,then for allt >0 there holds u(t,·) =u0? ht,

whereh_t is defined by h_t(Y, s)^def= 2^d−1

π^d+1 X

n∈N^d

Z

R

e^isλW (n, n, λ), Ye−4t|λ|(2|n|+d)|λ|^ddλ .

Proof. Applying F_H to the heat equation and taking advantage of (2.6), we get for all (n, m, λ) in H^e

d,





 du_bH

dt (t, n, m, λ) = −4|λ|(2|m|+d)u_bH(t, n, m, λ) ub_H|t=0 = F_Hu0.

By time integration, this implies that for all (n, m, λ) in H^e

d, (3.1) u_bH(t, n, m, λ) = e−4t|λ|(2|m|+d)F_Hu₀(n, m, λ). According to (2.9), we deduce that

ub_H(t, n, m, λ) = (F_Hu0·θt)(n, m, λ) with θt(n, m, λ)^def= e−4t|λ|(2|n|+d)

δn,m, whereδ_n,m denotes the Kronecker symbol, which implies that

u(t,·) =u0? ht with (F_Hht)(n, m, λ)^def= e−4t|λ|(2|n|+d)

δn,m.

This concludes the proof of the proposition thanks to the inversion formula (2.4).

Remark 3.2. Performing the change of variable tλ 7→ λ in the heat kernel given by Proposition3.1readily implies that for all t >0

h_t(Y, s) = 1 t^Q2

h Y

√ t,s

t

, with

(3.2) h(Y, s)^def= 2^d−1 π^d+1

X

m∈N^d

Z

R

e^isλW (n, n, λ), Ye−4|λ|(2|m|+d)δ_n,m|λ|^ddλ .

The following remarkable result due to B. Gaveau ([20]) asserts that the heat operator onH^d has a convolution kernel in S(H^d).

Theorem 6 ([20]). There exists a function h in S(H^d) such that for any u₀ ∈ L¹(H^d), the solution to(H_H)writes for all t >0

u(t,·) =u₀? h_t, whereht is defined by

(3.3) h_t(Y, s) = 1

t^Q2 h

Y

√t,s t

and the functionh is given by the formula (3.4) h(Y, s)^def= 1

(4π)^Q2 Z

R

2τ sinh 2τ

^d exp

iτ s

2 − |Y|²τ 2 tanh 2τ

dτ .

Proof. Our purpose here is to establish that the formula coming from Fourier analysis given by (3.2), namely

h(y, η, s) = 2^d−1 π^d+1

X

m∈N^d

Z

R×R^d

eisλ+2iλhη,zi

e−4|λ|(2|m|+d)

Hm,λ(y+z)Hm,λ(−y+z)dz|λ|^ddλ

coincides with the explicit expression of the heat kernel (3.4) given by Theorem 6. The proof relies on Melher’s formula (see [17])

(3.5) ^X

m∈N

Pm(x)Pm(x)_e r^m= 1

√

1−r² exp2xxr_e −(x²+x_e²)r² 1−r²

,

that holds true for allx,x_einRandrin ]−1,1[,whereP_mdenotes the Hermite polynomial of orderm defined by

P_m(x)^def= π¹⁴H_m(x)e^|x|

2 2 , withH_m the Hermite function introduced in (2.3).

To this end, we shall use the following lemma.

Lemma 3.3. Under the above notations, there holds for all (y, z) ∈ R² and all positive real numbersλand t,

X

m∈N

e^−2mtλHm,λ(z−y)Hm,λ(z+y) = 1 π¹²

λ 1−e^−4tλ

¹₂

exp−λz²tanh(tλ)− λy² tanh(tλ)

.

Proof. Applying the Mehler formula (3.5) to the rescaled Hermite functions (2.3) yields X

m∈N

e^−2tmλH_m,λ(z−y)H_m,λ(z+y) = 1 π¹²

e^−λ(z2+y2) λ 1−e^−4tλ

¹₂

×exp 1 1−e^−4tλ

2λ(z²−y²)e^−2tλ−2λ(z²+y²)e^−4tλ. The result follows from the following identities:

−λ(z²+y²) + 1 1−e^−4tλ

2λ(z²−y²)e^−2tλ−2λ(z²+y²)e^−4tλ

=− λ

1−e^−4tλ z²(1−e^−2tλ)²+y²(1 + e^−2tλ)², (1−e^−2tλ)²

1−e^−4tλ = (e^tλ−e^−tλ)²

e^2tλ−e^−2tλ = e^tλ−e^−tλ

e^tλ+ e^−tλ = tanh(tλ), and

(1 + e^−2tλ)²

1−e^−4tλ = (e^tλ+e^−tλ)²

e^2tλ−e^−2tλ = e^tλ+ e^−tλ

e^tλ−e^−tλ = 1 tanh(tλ)·

The lemma is proved.

Let us return to the proof of Theorem6. In order to establish that the two formulae (3.2) and (3.4) coincide, it suffices to consider the case whend= 1: (3.2) becomes

(3.6) h(y, η, s) = 1 π²

X

m∈N

Z

R²

e^isλ+2iληze−4|λ|(2m+1)H_m,λ(y+z)H_m,λ(−y+z)dz|λ|dλ . Then applying Lemma3.3witht= 4|λ|, we find that

h(y, η, s) = 1 π⁵²

Z

R²

eisλ+2iληz−4|λ| |λ|

1−e^−16|λ|

¹₂

×exp

−|λ|z²tanh(4|λ|)− |λ|y² tanh(4|λ|)

|λ|dλdz .

Performing the change of variables|λ|¹²z7→z, this gives rise to h(y, η, s) = 1

π⁵² Z

R

e^{isλ−4|λ|−}

|λ|y2

tanh(4|λ|)F e^{−tanh(4|λ|)|·|2} 2|λ|¹²η |λ|

1−e^−16|λ|

¹₂

|λ|¹²dλ .

Since

F e^{−tanh(4|λ|)|·|2} 2|λ|¹²η=

s π

tanh(4|λ|)e⁻

|λ|η2 tanh(4|λ|) , we discover that

h(y, η, s) = 1 π²

Z

R

e^{isλ−4|λ|−}

|λ|(y2+η2) tanh(4|λ|)

tanh(4|λ|)(1−e^−16|λ|)

−¹

2 |λ|dλ . But

tanh(4|λ|)(1−e^−16|λ|) = 4 e^−8|λ|sinh²(4|λ|),

which ends the proof of Theorem6.

4. On the kernel of the Schr¨odinger operator on H^d

4.1. Representation of the free Schr¨odinger equation. Contrary to the heat equa- tion (and as in the euclidean case recalled in the introduction), the kernel of the Schr¨odinger operator does not belong to the Schwartz classS(H^d). Nevertheless, one can solve explic- itly the Schr¨odinger equation (S_H) by means of the Fourier-Heisenberg transform intro- duced in Section2, in the following way.

Proposition 4.1. The solution to the free Schr¨odinger equation (S_H)

i∂tu−∆_Hu= 0 u_|t=0=u₀, reads for allt6= 0 and allu₀ ∈ S(H^d)

(4.1) u(t,·) =u₀? S_t,

whereSt denotes the tempered distribution onH^d defined for allϕinS(H^d) by hS_t, ϕi_S0(H^d)×S(H^d) =he4it|λ|(2|n|+d)δ_n,m, θi

S⁰(Hb

d)×S(Hb

d), withϕ=^tF_Hθ, according to Notation(2.16).

Proof. Arguing as for the proof of Proposition3.1, we start by applyingF_H to (S_H), which thanks to (2.6) implies that





 idu_bH

dt (t, n, m, λ) = −4|λ|(2|m|+d)u_bH(t, n, m, λ) ub_H|t=0 = F_Hu₀,

and leads by integration to

(4.2) u_bH(t, n, m, λ) = e4it|λ|(2|m|+d)F_Hu0(n, m, λ), for all (n, m, λ) in H^ed. Then taking advantage of (2.9), we find that

ub_H(t, n, m, λ) = (F_Hu₀·Θ_t)(n, m, λ) with Θ_t(n, m, λ)^def= e4it|λ|(2|n|+d)δ_n,m. One can easily check that Θ_t is a function with moderate growth in the sense of (2.18), and thus as it was proved in [5], it is a tempered distribution on H^bd. This ensures that the Schr¨odinger kernel Stbelongs to S⁰(H^d).

Finally combining (2.16) together with (2.19), we readily gather that for allu₀inS(H^d) and allw inH^d,

(4.3)

(u₀? S_t)(w) =hS_t,uˇ₀◦τ_w^ri_S0(H^d)×S(H^d)=he4it|λ|(2|n|+d)δ_n,m, θi

S⁰(Hb

d)×S(Hb

d)

= ^X

n∈N^d

Z

R

e4it|λ|(2|n|+d)θ(n, n, λ)|λ|^ddλ , where ˇu0◦τ_w^r =^tF_Hθ, which completes the proof of the proposition.