Tempered distributions and Fourier transform on the Heisenberg group

H A J E R B A H O U R I J E A N - Y V E S C H E M I N R A P H A Ë L D A N C H I N

T E M P E R E D DIS T R I B U T IO N S A N D F O U R IE R T R A N S F O R M O N T H E H E IS E N B E R G G R O U P

DIS T R I B U T IO N S T E M P É R É E S E T

T R A N S F O R M AT IO N D E F O U R IE R S U R L E G R O U P E D E H E IS E N B E R G

Abstract. — We aim at extending the Fourier transform on the Heisenberg group H^d, to tempered distributions. Our motivation is to provide the reader with a hands-on approach that allows for further investigating Fourier analysis and PDEs onH^d.

As in the Euclidean setting, the strategy is to show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. To achieve it, the Fourier transform of an integrable function is viewed as a uniformly continuous mapping on the setHe^d=N^d×N^d×R\ {0}, that may be completed to a larger setHe^dfor some suitable distance. This viewpoint provides a user friendly description of the range of the Schwartz space onH^dby the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward.

Keywords:Fourier transform, Heisenberg group, frequency space, tempered distributions, Schwartz space.

2010Mathematics Subject Classification:43A30, 43A80.

DOI:https://doi.org/10.5802/ahl.1

(*) The authors are deeply indebted to the anonymous referees for their very complete reading of the manuscript. Their long list of remarks and constructive suggestions greatly contributed to improve the final version of this work.

To highlight the strength of our approach, we give examples of computations of Fourier transforms of tempered distributions that do not correspond to integrable or square integrable functions. The most striking one is a formula for the Fourier transform of functions onH^d that are independent of the vertical variable, an open question, to the best of our knowledge.

Résumé. — Dans cet article, on s’intéresse à l’extension de la transformation de Fourier sur le groupe de Heisenberg H^d, aux distributions tempérées. Notre but est de donner au lecteur un cadre adapté à l’étude de l’analyse de Fourier et des EDP surH^d.

Comme dansRⁿ, il s’agit en premier lieu de montrer que la transformation de Fourier est un isomorphisme sur l’espace de Schwartz, puis d’étendre sa définition par dualité aux distributions tempérées. Pour ce faire, on définit la transformée de Fourier d’une fonction intégrable comme étant une fonction uniformément continue sur l’ensembleHe^d=N^d×N^d×R\ {0}, qui peut être complété en un ensemble He^d pour une distance adéquate. Ce point de vue donne une caractérisation simple de l’image de l’espace de Schwartz sur H^d par la transformation de Fourier, permettant ainsi d’étendre sa définition à l’ensemble des distributions tempérées.

Pour illustrer la puissance de notre approche, on donne quelques exemples de calculs explicites de transformées de Fourier de fonctions ou distributions tempérées qui ne correspondent pas à des fonctions intégrables ou de carrés intégrables. Le plus spectaculaire est l’obtention d’une formule explicite pour la transformée de Fourier de fonctions régulières indépendantes de la variable verticale. Pour répondre à cette question de façon simple, avoir pris le soin de le compléter au préalable l’espace des fréquencesHe^d s’avère fondamental.

1. Introduction

The present work aims at extending Fourier analysis on the Heisenberg group to tempered distributions. It is by now very classical that in the case of a commutative group, the Fourier transform is a function on the group of characters. In the Euclidean spaceRⁿ the group of characters may be identified with the dual space (Rⁿ)^? of Rⁿ through the mapξ7→e^{ihξ,· i}, wherehξ,· i designate the value of the one-formξ when applied to elements of Rⁿ, and the Fourier transform of an integrable function f may be seen as the function on (Rⁿ)^?, defined by

(1.1) F(f)(ξ) = f(ξ)^{b def}=

Z

Rⁿ

e^−ihξ,xif(x) dx.

A fundamental fact of the distribution theory onRⁿ is that the Fourier transform is a bi-continuous isomorphism on the Schwartz space S(Rⁿ) – the set of smooth functions whose derivatives decay at infinity faster than any power. Hence, one can define the transposed Fourier transform ^tF on the so-called set of tempered distributionsS⁰(Rⁿ), that is the topological dual of S(Rⁿ) (see e.g. [Rud62, Sch98]

for a self-contained presentation). As the whole distribution theory onRⁿis based on identifying locally integrable functions with linear forms by means of the Lebesgue integral, one can look for a more direct relationship between^tF andF, by considering the following bilinear form onS(Rⁿ)× S(Rⁿ):

(1.2) B_R(f, φ)^def=

Z

T^?Rⁿ

f(x)e^−ihξ,xiφ(ξ) dxdξ,

where the cotangent bundleT^?Rⁿ ofRⁿ is identified with Rⁿ×(Rⁿ)^?. This allows to identify^tF|S((Rⁿ)^?) with F|S(Rⁿ), and it is thus natural to define the extension of

F onS⁰(Rⁿ) to be ^tF, that is,

(1.3) ∀(T, φ)∈ S⁰(Rⁿ)× S(Rⁿ), hFT, φiS⁰(Rⁿ)×S(Rⁿ)

def= hT,FφiS⁰(Rⁿ)×S(Rⁿ). We aim at implementing that procedure on the Heisenberg group H^d. As in the Euclidean case, to achieve our goal, it is fundamental to have a handy characterization of the range of the Schwartz space onH^dby the Fourier transform. The first attempt in that direction goes back to the pioneering works by D. Geller in [Gel77, Gel80] (see also [ADBR13, Far10, LT14] and the references therein). However, the description of F(S(H^d)) that is given therein relies on complicated asymptotic series, so that whether it allows to extend the Fourier transform to tempered distribution is unclear.

We here aim at presenting a rather elementary description of F(S(H^d)) that is based on the approach that we developed recently in[BCD]. Our motivation is threefold. First, our approach enables us to recover easily different classical but highly nontrivial results like the fact that the fundamental solution of the heat flow or kernels of multipliers of the forma(−∆_H) witha∈ S(R), belong toS(H^d). Second, we are able to compute explicitly the Fourier transform of a number of functions which are not integrable like, for instance, that of functions that are independent of the vertical variable. Third, having a complete description of both F(S(H^d)) and F(S⁰(H^d)) gives us a solid basis to tackle more complicated Fourier analysis issues, with applications to Partial Differential Equations.

At this stage of the paper, we have to recall the definitions of the Heisenberg groupH^d and of the Fourier transform onH^d, and a few basic properties. We shall defineH^d to be the set T^?R^d×R equipped with the product law

w·w^{0 def}= (Y +Y⁰, s+s⁰+ 2σ(Y, Y⁰)) = (y+y⁰, η+η⁰, s+s⁰+ 2hη, y⁰i −2hη⁰, yi) where w = (Y, s) = (y, η, s) and w⁰ = (Y⁰, s⁰) = (y⁰, η⁰, s⁰) are generic elements of H^d. In the above definition, the notation h ·,· i designates the duality bracket between (R^d)^? and R^d and σ is the canonical symplectic form on R^2d, seen as T^?R^d. This gives H^d the structure of a non commutative group for which w⁻¹ =−w. The reader will find more details in the books [BFKG12, FH84, FR14, Fol89, FS82, Ste93, Tay86, Tha98] and in the references therein.

In accordance with the above product formula, one can define the set of the dilations on the Heisenberg group to be the family of operators (δ_a)_a>0 given by (1.4) δ_a(w) = δ_a(Y, s)^def= (aY, a²s).

Note that dilations commute with the product law on H^d, that is δ_a(w ·w⁰) = δ_a(w)·δ_a(w⁰). Furthermore, as the determinant of δ_a (seen as an automorphism of R^2d+1) is a^2d+2, the homogeneous dimension of H^d is N ^def= 2d+ 2.

The Heisenberg group is endowed with a smooth left invariant Haar measure, which, in the coordinate system (y, η, s) is just the Lebesgue measure onR^2d+1. The corresponding Lebesgue spaces L^p(H^d) are thus the sets of measurable functions f :H^d→C such that

kfk_L^p_(H^d₎ ^def=

Z

H^d

|f(w)|^pdw

¹_p

<∞, if 16p <∞, with the standard modification ifp=∞.

The convolution product of any two integrable functionsf and g is given by f ? g(w)^def=

Z

H^d

f(w·v⁻¹)g(v) dv =

Z

H^d

f(v)g(v⁻¹·w) dv.

As in the Euclidean case, the convolution product is an associative binary operation on the set of integrable functions. Even though it is no longer commutative, the following Young inequalities hold true:

kf ? gk_L^r 6kfk_L^pkgk_L^q, whenever 16p, q, r6∞ and 1 r = 1

p+ 1 q −1.

TheSchwartz spaceS(H^d) corresponds to the Schwartz spaceS(R^2d+1) (an equiva- lent definition involving the Heisenberg structure will be provided in Appendix A.3).

As the Heisenberg group is noncommutative, defining the Fourier transform of integrable functions onH^d, by a formula similar to (1.1) is no longer relevant. One has to use a more elaborate family of irreducible representations of H^d, all of them being unitary equivalent (see e.g. [Tay86, Chapter 2]) to theSchrödinger representa- tion (U^λ)_λ∈R\{0} which is the family of group homomorphismsw7→U_w^λ between H^d and the unitary groupU(L²(R^d)) of L²(R^d) defined for all w= (y, η, s) in H^d andu inL²(R^d) by

U_w^λu(x)^def= e−iλ(s+2hη,x−yi)u(x−2y).

The standard definition of the Fourier transform then reads as follows.

Definition 1.1. — Forf inL¹(H^d) and λ in R\ {0}, we define F_H(f)(λ)^def=

Z

H^d

f(w)U_w^λdw.

The function F_H(f) which takes values in the space of bounded operators on L²(R^d) is, by definition, the Fourier transform of f.

An obvious drawback of Definition 1.1 is thatF_Hf is not a complex valued function on some “frequency space”, but a much more complicated object. Consequently, one can hardly expect to have a simple characterization of the range of the Schwartz space byF_H, allowing for our extending the Fourier transform to tempered distributions.

To overcome that difficulty, we proposed in our recent paper[BCD] an alternative (equivalent) definition that makes the Fourier transform of any integrable function on H^d, a continuous function on another (explicit) setH^bdendowed with some distanced.^b

Before giving our definition, we need to introduce some notation. Let us first recall that the Lie algebra of left invariant vector fields on H^d, that is vector fields commuting with any left translationτ_w(w⁰)^def= w·w⁰, is spanned by the vector fields (1.5) S ^def= ∂_s, X_j ^def= ∂_yj + 2η_j∂_s and Ξ_j ^def= ∂_ηj −2y_j∂_s, 16j 6d.

The sublaplacian associated to the vector fields (X_j)_16j6d and (Ξ_j)_16j6d is defined by

(1.6) ∆_H ^def=

d

X

j=1

(X_j²+ Ξ²_j),

and may be rewritten in terms of the usual derivatives as follows:

∆_Hf(Y, s) = ∆_Yf(Y, s) + 4

d

X

j=1

(η_j∂yj−yj∂ηj)∂_sf(Y, s) + 4|Y|²∂_s²f(Y, s).

The sublaplacian plays a fundamental role in the Fourier theory of the Heisenberg group. The starting point is the following relation that holds true for all functionsf in the Schwartz space (see e.g.[Hue76, Olv74]):

(1.7) F_H(∆_Hf)(λ) = 4F_H(f)(λ)◦∆^λ_osc with ∆^λ_oscu(x)^def=

d

X

j=1

∂_j²u(x)−λ²|x|²u(x).

In order to take advantage of the spectral structure of the harmonic oscillator, we introduce the corresponding eigenvectors, that is the family of Hermite func- tions (H_n)_n∈N^d defined by

H_n^def=

1 2^|n|n!

¹₂

CⁿH₀ with C^ndef=

d

Y

j=1

C_j^nj and H₀(x)^def= π^−d2e^−|x|

2 2 ,

where C_j ^def= −∂_j +M_j stands for the creation operator with respect to the j-th variable and M_j is the multiplication operator defined by M_ju(x)^def= x_ju(x).

As usual, for any multi-index n in N^d, n! ^def= n₁!... n_d! and |n| ^def= n₁ +...+n_d stands for the length ofn (not to be confused with the Euclidean norm that we shall sometimes denote in the same way for elements ofR^d).

Recall that (H_n)_n∈N^d is an orthonormal basis of L²(R^d), and that we have (−∂_j²+M_j²)Hn = (2nj + 1)Hn and thus −∆¹_oscHn = (2|n|+d)Hn. For λ in R\ {0}, we finally introduce the rescaled Hermite function H_n,λ(x) ^def=

|λ|^d4H_n(|λ|¹²x). It is obvious that (H_n,λ)_n∈Nd is still an orthonormal basis of L²(R^d) and that

(1.8) (−∂_j²+λ²M_j²)H_n,λ = (2n_j+ 1)|λ|H_n,λ and −∆^λ_oscH_n,λ= (2|n|+d)|λ|H_n,λ. Our alternative definition of the Fourier transform onH^d then reads as follows:

Definition 1.2. — Let H^{ed def}= N^2d×R\ {0}, and denote by w_b = (n, m, λ) a generic point of H^ed. For f in L¹(H^d), we define the map F_Hf (also denoted by f^b_H) to be

F_Hf :







He^d −→ C

wb 7−→ (F_H(f)(λ)H_m,λ|H_n,λ)_L2.

To underline the similarity between that definition and the classical one inRⁿ, one may further compute (F_H(f)(λ)H_m,λ|H_n,λ)_L2. One can observe that, after a change of variable, the Fourier transform recasts in terms of the mean value of f modulated by some oscillatory functionswhich are closely related to Wigner transforms of Hermite

functions, namely

F_Hf(w) =_b

Z

H^d

e^isλW(w, Y_b )f(Y, s) dY ds (1.9)

with W(w, Yb )^def=

Z

R^d

e^2iλhη,ziH_n,λ(y+z)H_m,λ(−y+z) dz.

(1.10)

With this new point of view, Formula (1.7) recasts as follows:

F_H(∆_Hf)(w) =_b −4|λ|(2|m|+d)f^b_H(w)._b

Furthermore, if we endow the set H^ed with the measuredw_b defined by the relation

Z

He^d

θ(w)_b dw_b ^def= ^X

(n,m)∈N^2d

Z

R

θ(n, m, λ)|λ|^ddλ,

then the classical inversion formula and Fourier–Plancherel theorem become:

Theorem 1.3. — Let f be a function in S(H^d). Then we have the inversion formula

(1.11) f(w) = 2^d−1 π^d+1

Z eH^d

e^isλW(w, Yb )f^b_H(w) db wb for any w inH^d,

and the Fourier transform F_H can be extended to a bicontinuous isomorphism be- tween the spaces L²(H^d) and L²(H^ed), that satisfies

(1.12) kf^b_Hk²_L2(

eH^d)= π^d+1

2^d−1kfk²_L2(H^d).

Finally, for any couple(f, g)of integrable functions, the following convolution identity holds true:

(1.13) F_H(f ? g)(n, m, λ) = (f^b_H·gb_H)(n, m, λ)

with (f^b_H ·gb_H)(n, m, λ)^def= ^X

`∈N^d

fb_H(n, `, λ)gb<sub>H</sub>(`, m, λ).

For the reader’s convenience, we shall present a proof of Theorem 1.3 in the appendix.

2. Main results

As already mentioned, our final goal is to extend the Fourier transform to tempered distributions onH^d. If we follow the standard approach of the Euclidean setting that is described by (1.2) and (1.3), then the main difficulty is to get a description of the rangeS(H^ed) of S(H^d) by F_H that allows to find out some bilinear form B_H for identifying ^tF_H with F_H.

Here, we first present results pertaining to the characterization of the set S(H^ed), then exhibit a large class of functions of S(H^ed) that, in particular, contains the Fourier transform of the heat kernel. Third, we go to examples of tempered distri- butions on the frequency setH^ed and, finally, to computations of Fourier transform of tempered distributions (with a special attention to functions independent of the vertical variable).

To characterizeF_H(S(H^d)), we first have to keep in mind that, as in the Euclidean setting, we expect the Fourier transform to change the regularity of functions onH^d to decay of the Fourier transform. This is achieved in the following lemma that has been proved in [BCD].

Lemma 2.1. — For any integerp, there exist an integerN_p and a positive constant C_p such that for all w_b inH^ed and all f in S(H^d), we have

(2.1) (1 +|λ|(|n|+|m|+d) +|n−m|)^p|f^b_H(n, m, λ)|6C_pkfk_Np,S, where k · k_N,S denotes the classical family of semi-norms of S(R^2d+1), namely

kfkN,S def

= sup

|α|6N

(1 +|Y|²+s²)^N/2∂_Y,s^α f

L^∞.

The second property we expect to have is that the Fourier transform changes decay into regularity. As in the Euclidean case, this is closely related to how it acts on weight functions: we expectF_H to transform multiplication by a (suitable) weight function into a derivative operator on H^ed. So far however, we lack a notion of differentiation on H^ed that could fit the scope. This is the aim of the following definition (see also Proposition A.3 in Appendix):

Definition 2.2. — For any functionθ :H^ed→C we define (2.2) ∆θ(^b w)b ^def= − 1

2|λ|(|n+m|+d)θ(w)b

+ 1 2|λ|

d

X

j=1

q

(nj + 1)(mj + 1)θ(wb⁺_j ) +√

njmjθ(wb_j⁻)

and, if in addition θ is differentiable with respect toλ, (2.3) D^b_λθ(w)b ^def= dθ

dλ(w) +b d 2λθ(w)b

+ 1 2λ

d

X

j=1

√

n_jm_jθ(w_b⁻_j )−^q(n_j+1)(m_j+1)θ(w_b⁺_j )

,

where wb_j^±def= (n±δ_j, m±δ_j, λ)andδ_j denotes the element ofN^dwith all components equal to 0 except the j-th which has value 1.

The notation in the above definition is justified by the following lemma that will be proved in Subsection 3.2.

Lemma 2.3. — LetM² andM₀ be the multiplication operators defined onS(H^d) by

(2.4) (M²f)(Y, s)^def= |Y|²f(Y, s) and M₀f(Y, s)^def= −isf(Y, s).

Then for all f in S(H^d), the following two relations hold true on H^ed: F_HM²f =−∆F^b _Hf and F_H(M₀f) =D^b_λF_Hf.

The third aspect of regularity for functions inF_H(S(H^d)) we have to underline is the link between their values for positive λ and negative λ. This link is crucial in the computation of M0F_H⁻¹ which is done in the proof of the inversion theorem in the Schwartz spaceS(H^d). That property has no equivalent in the Euclidean setting, and is described in the following lemma:

Lemma 2.4. — Let us consider on S(H^d) the operator P defined by (2.5) P(f)(Y, s)^def= 1

2

Z s

−∞

(f(Y, s⁰)−f(Y,−s⁰)) ds⁰.

Then, P maps continuously S(H^d) to S(H^d) and, for any f in S(H^d) andw_b in H^ed, we have 2iF_H(Pf) =Σ^b₀(F_Hf) with

(2.6) Σ^b₀(θ)(w)_b ^def= θ(n, m, λ)−(−1)^|n+m|θ(m, n,−λ)

λ .

Technically, the above weird relation stems from the fact that the function W fulfills:

(2.7) ∀(n, m, λ, Y)∈H^ed×T^?R^d, W(n, m, λ, Y) = (−1)^|n+m|W(m, n,−λ, Y).

Note that, since the left-hand side of (2.6) belongs to the spaceF_H(S(H^d)) that is the natural candidate for beingS(H^ed), we need to prescribe in addition to the decay properties pointed out in Lemmas 2.1 and 2.3 some condition involving Σ^b₀. This motivates the following definition.

Definition 2.5. — We defineS(H^ed)to be the set of functionsθon H^edsuch that:

• for any (n, m)in N^2d, the map λ 7−→θ(n, m, λ) is smooth on R\ {0},

• for any non negative integer N, the functions ∆^bNθ, D^b_λ^Nθ and Σ^b₀D^bN_λθ decay faster than any power of d^b₀(w)_b ^def= |λ|(|n+m|+d) +|m−n|.

We equip S(H^ed) with the family of semi-norms kθk_N,N0,S(eH^d)

def= sup

w∈b bH^d

1 +d^b₀(w)_b ^N|∆^bN0θ(w)|_b +|D^b_λ^N0θ(w)|_b +|Σ^b₀D^b_λ^N0θ(w)|_b . Clearly, the above definition of semi-norms guarantees that there exist an integerK and a positive real numberC so that the following inequality holds true:

(2.8) kθk_L1(

eH^d) 6Ckθk_K,0,S(

He^d).

More importantly, we have the following isomorphism theorem.

Theorem 2.6. — The Fourier transform F_H is a bicontinuous isomorphism be- tween S(H^d) and S(H^ed), and the inverse map is given by

(2.9) F^e_Hθ(w)^def= 2^d−1 π^d+1

Z eH^d

e^isλW(w, Y_b )θ(w) d_b w._b

We shall discover throughout the paper that the definition of S(H^ed) encodes a number of nontrivial hidden informations that are consequences of the sub-ellipticity of ∆_H. For instance, the stability ofS(H^ed) by the multiplication law defined in (1.13)

is a consequence of the stability ofS(H^d) by convolution and of Theorem 2.6. Another hidden information is the behavior of functions of S(H^ed) whenλ tends to 0.

Keeping the decay inequality (2.1) in mind, we endow the setH^edwith the distance:

d(bw,b wb⁰)^def= |λ(n+m)−λ⁰(n⁰+m⁰)|₁+|(n−m)−(n⁰−m⁰)|+d|λ−λ⁰|, where | · |₁ denotes the Manhattan norm`¹ onR^d.

With this choice, the Fourier transform of any function ofS(H^d) (and even of any integrable function onH^d) isuniformly continuous on H^ed, and can thus be extended by continuity on the completion H^bd of H^ed, as explained in greater details in the following statement that has been proved in [BCD]:

Theorem 2.7. — The completion of (H^ed,d)^b is the metric space(H^bd,d)^b where Hb^ddef= H^ed∪H^bd0 with H^bd0

def= R^d∓×Z^d and R^d∓

def= (R⁻)^d∪(R⁺)^d

and the extended distance (still denoted byd) is given for all^b wb = (n, m, λ)and wb⁰ = (n⁰, m⁰, λ⁰)in H^ed, and for all ( ˙x, k)and ( ˙x⁰, k⁰) inH^bd0 by

d(bw,_b w_b⁰) = |λ(n+m)−λ⁰(n⁰+m⁰)|₁+|(m−n)−(m⁰ −n⁰)|+d|λ−λ⁰|, db(w,b ( ˙x, k)) =d^b(( ˙x, k),w)b ^def= |λ(n+m)−x|˙ ₁+|m−n−k|+d|λ|,

db(( ˙x, k),( ˙x⁰, k⁰)) =|x˙ −x˙⁰|₁+|k−k⁰|.

The Fourier transform f^b_H of any integrable function on H^d may be extended continuously to the whole set H^bd. Still denoting by f^b_H (or F_Hf) that extension, the linear map F_H :f 7→f^b_H is continuous from the space L¹(H^d)to the space C₀(H^bd) of continuous functions on H^bd tending to 0 at infinity.

The above theorem prompts us to consider the space S(H^bd) of functions on H^bd which are continuous extensions of elements of S(H^ed), endowed with the semi-norms k · k_N,N0,S(eH^d). Those semi-norms will be denoted by k · k_N,N0,S(bH^d) in all that follows.

Note that for any function θ in S(H^bd), having w_b tend to ( ˙x, k) in (2.6) yields θ( ˙x, k) = (−1)^|k|θ(−x,˙ −k).

As regards convolution, we obtain, after passing to the limit in (1.13), the following noteworthy formula, valid for any two functionsf and g inL¹(H^d):

(2.10) F_H(f ? g)

Hb^d₀ = (F_Hf)

bH^d₀ Hb^d₀

· (F_Hg)

bH^d₀

with (θ₁^bH

d

·0 θ₂)( ˙x, k)^def= ^X

k⁰∈Z^d

θ₁( ˙x, k⁰)θ₂( ˙x, k−k⁰).

Let us underline that the above product law (2.10) is commutative even though convolution of functions on the Heisenberg groupis not (see (1.13)).

The next question is how to extend the measure dwb to H^bd. In fact, we have for any positive real numbers R and ε,

Z

Hb^d

1{|λ| |n+m|+|m−n|6R}1^|λ|6εdwb =

Z ε

−ε

X

n,m

1{|λ| |n+m|+|m−n|6R}

!

|λ|^ddλ 6CR^2dε.

Therefore, it is natural to extend dw_b by 0 on H^bd0. With this convention, keeping the same notation for the extended measure, we have for all continuous compactly supported function θ on H^bd,

Z bH^d

θ(w) d_b w_b ^def=

Z

He^d

θ(w) d_b w._b

At this stage of the paper, pointing out non-trivial examples of functions inS(H^bd) is highly informative. To this end, we introduce the set S_d⁺ of smooth functions f on [0,∞[^d×Z^d×R such that for any integerp, we have

sup

(x1,...,xd,k,λ)∈[0,∞[^d×Z^d×R

|α|6p

(1 +x₁+...+xd+|k|)^p∂_x,λ^α f(x₁, . . . , xd, k, λ)<∞.

As may be easily checked by the reader, the spaceS_d⁺ is stable under derivation and multiplication by polynomial functions of (x, k).

Theorem 2.8. — Let f be a function in S_d⁺. Let us define for w_b = (n, m, λ) inH^ed,

Θf(w)b ^def= f(|λ|R(n, m), m−n, λ) with R(n, m)^def= (nj +mj+ 1)_16j6d. Iff is supported in[0,∞[^d× {0} ×R, or iff is supported in[r₀,∞[^d×Z^d×R for some positive real number r₀, and satisfies

(2.11) f(x,−k,−λ) = (−1)^|k|f(x, k, λ) for all k ∈Z^d, λ >0 and x >0, then Θ_f belongs toS(H^bd).

A particularly striking consequence of Theorem 2.8 is that it provides a trivial proof of Hulanicki’s theorem in [Hul84] and of the fact that the fundamental solution of the heat equation inH^d for the sublaplacian belongs to S(H^d) (a nontrivial result that is usually deduced from the explicit formula established by B. Gaveau in [Gav77]).

Corollary 2.9. — For any function a in S(R), there exists a function h_a inS(H^d) such that

∀f ∈ S(H^d), a(−∆_H)f =f ? h_a.

In particular, the functionh such that, for all u₀ in L¹(H^d)and t >0, the map u:t7−→u₀? h_t with h_t(y, η, s)^def= 1

t^d+1h y

√t, η

√t,s t

!

fulfills

∂_tu−∆_Hu= 0 in ]0,+∞[×H^d and lim

t→0⁺u(t) = u₀ in L¹(H^d), belongs toS(H^d).

Proof. — Let us focus on the second result, proving the first one being similar.

Then applying the Fourier transform with respect to the Heisenberg variable gives that umust fulfill

(2.12) u_bH(t, n, m, λ) = e−4t|λ|(2|m|+d)

ub₀(n, m, λ).

At the same time, we have u(t) = u₀ ? h_t. Hence, combining the convolution for- mula (1.13) and identity (2.12), we gather that

bh_H(w) =b e−4|λ|(2|n|+d)

1^{n=m}.

Then applying Theorem 2.8 to the function e^{−4(x1+...+xd)}1^{k=0} ensures that h^b_H belongs to S(H^bd), and the inversion theorem 2.6 eventually implies that h is in

S(H^d).

Before explaining how to extend the Fourier transform to tempered distributions, let us specify what a tempered distribution on H^bd is.

Definition 2.10. — Tempered distributions onH^bdare elements of the setS⁰(H^bd) of continuous linear forms on the Fréchet space S(H^bd).

Convergence of tempered distributions is defined in the standard way: we say that a sequence (T_n)n∈N of S⁰(H^bd) converges to a tempered distributionT if

∀θ ∈ S(H^bd), lim

n→∞hT_n, θi_S0(bH^d)×S(bH^d)=hT, θi_S0(Hb^d)×S(Hb^d).

A first class of examples of tempered distributions on H^bd is given by “functions with moderate growth”:

Definition 2.11. — We denote by L¹_M(H^bd) the set of functions on H^bd with moderate growth, that is the set of locally integrable functions f on H^bd such that there exists an integer psatisfying

Z

Hb^d

(1 +|λ|(n+m|+d) +|n−m|)^−p|f(w)|db w <b ∞.

As in the Euclidean setting, functions inL¹_M(H^bd) may be identified with tempered distributions:

Theorem 2.12. — Let us considerι be the map defined by ι:







L¹_M(H^bd) −→ S⁰(H^bd) ψ 7−→ ι(ψ) :^hθ 7→^R

bH^dψ(w)θ(_b w) d_b w_bⁱ. Then ι is a one-to-one linear map.

Moreover, if pis an integer such that the map

(n, m, λ)7−→(1 +|λ|(n+m|+d) +|n−m|)^−pf(n, m, λ) belongs toL¹(H^bd), then we have

(2.13) |hι(f), θi|6(1 +|λ|(n+m|+d) +|n−m|)^−pf

L¹(Hb^d)kθk_p,0,S(

bH^d). The following proposition that will be proved in Section 5 provides examples of functions in L¹_M(H^bd).

Proposition 2.13. — Let γ < d+ 1. Then the set L¹_M(H^bd) contains the func- tion fγ defined by

f_γ(n, m, λ)^def= (|λ|(2|m|+d))^−γ δ_n,m, (n, m, λ)∈H^ed.

Remark 2.14. — The above proposition is no longer true forγ =d+ 1. If we look at the quantity|λ|(2|n|+d) inH^bd as an equivalent of|ξ|² forR^d, then it means that the homogeneous dimension of H^bd is 2d+ 2, as forH^d.

Obviously, any Dirac mass onH^bd is a tempered distribution. Let us also note that because

|θ(n, n, λ)|6(1 +|λ|(2|n|+d))^−d−2kθk_d+2,0,S(

Hb^d), the linear form

(2.14) I :

( S(H^bd) −→ C θ 7−→ ^P_n∈Nd

R

Rθ(n, n, λ)|λ|^ddλ is a tempered distribution onH^bd.

We now want to exhibit tempered distributions onH^bd which are not functions or measures. The following proposition states that the analogue on H^bd of finite part distributions on Rⁿ, are indeed in S⁰(H^bd).

Proposition 2.15. — Let γ be in the interval ]d+ 1, d+ 3/2[ and denote by ^b0 the element (0,0) of H^bd0. Then for any function θ inS(H^bd), the function defined a.e.

on H^bd by

(n, m, λ)7−→δn,m

θ(n, n, λ) +θ(n, n,−λ)−2θ(^b0)

|λ|^γ(2|n|+d)^γ

!

, is integrable. Furthermore, the linear form defined by

*

Pf 1

|λ|^γ(2|n|+d)^γ

!

, θ

+

def= 1 2

Z

Hb^d

θ(n, n, λ) +θ(n, n,−λ)−2θ(^b0)

|λ|^γ(2|n|+d)^γ

!

δ_n,mdw_b is inS⁰(H^bd), and its restriction to H^ed is the functionfγ that has been defined above, that is, for any θ in S(H^bd)such thatθ(n, n, λ) = 0for small enough |λ|(2|n|+d), we

have *

Pf 1

|λ|^γ(2|n|+d)^γ

!

, θ

+

=

Z

Hb^d

f_γ(w)_b θ(w) d_b w._b

Another interesting example of tempered distribution on H^bd is the measure µ

bH^d0

defined in Lemma 3.1 of [BCD]. In our setting, that result recasts as follows:

Proposition 2.16. — Let the measureµ

bH^d₀ be defined for all functionθ inS(H^bd), by

hµ

bH^d0

, θi=

Z bH^d₀

θ( ˙x, k) dµ

bH^d0

( ˙x, k)^def= 2^{−d X}

k∈Z^d

Z

(R−)^d

θ( ˙x, k) d ˙x+

Z

(R+)^d

θ( ˙x, k) d ˙x

!

. Thenµ

Hb^d₀ belongs toS⁰(H^bd), and for any functionψ in S(R)with integral 1we have limε→0

1 εψ λ

ε

!

=µ

bH^d₀ inS⁰(H^bd).

Let us finally explain how Formulae (1.2) and (1.3) may be adapted so as to extend the Fourier transform onH^d to tempered distributions. We introduce:

B_H :

( S(H^d)× S(H^bd) −→ C (f, θ) 7−→ ^R

H^d×Hb^df(Y, s)e^isλW(w, Yb )θ(w) dwb dwb

and ^tF_H :

( S(H^bd) −→ S(H^d)

θ 7−→ ^R

Hb^de^isλW(w, Yb )θ(w) db w.b

Since, for anyθ inS(H^bd) andw= (y, η, s) in H^d, we have (2.15) (^tF_Hθ)(y, η, s) = π^d+1

2^d−1(F_H⁻¹θ)(y,−η,−s).

Theorem 2.6 ensures that^tF_H is a continuous isomorphism betweenS(H^bd) andS(H^d).

Now, we observe that for any f in S(H^d) and θ in S(H^bd), we have B_H(f, θ) =

Z

H^d

f(w)(^tF_Hθ)(w) dw=

Z

Hb^d

(F_Hf)(w)θ(b w) db w.b

This prompts us to extendF_H on S⁰(H^d) by hF_HT, θi_S0(bH^d)×S(bH^d)

def= hT,^tF_Hθi_S⁰_(H^d_)×S(H^d₎ for all θ ∈ S(H^bd).

A direct consequence of this definition is the following statement:

Proposition 2.17. — The map F_H defined just above is continuous and one-to- one from S⁰(H^d)onto S⁰(H^bd). Furthermore, its restriction to L¹(H^d)coincides with Definition 1.2.

Just to compare with the Euclidean case, let us give some examples of simple computations of Fourier transform of tempered distributions on H^d.

Proposition 2.18. — We have

F_H(δ₀) = I and F_H(1) = π^d+1 2^d−1δ

b⁰

,

where I is defined by (2.14) and ^b0is the element of H^bd0 corresponding to x˙ = 0 and k= 0.

Let us emphasize, that without completing the frequency setH^ed, it would not be even possible to state the second result of the above proposition. The same happens in the following statement when computing the Fourier transform of a function independent of the vertical variable.

Theorem 2.19. — We have for any integrable function g onT^?R^d, F_H(g⊗1) = (G_Hg)µ

bH^d0