Tempered distributions and Fourier transform on the Heisenberg group

The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on $\H^d$is no longer a function on $\H^d$ : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Fourier transform of an integrable functionto be a mapping on the set~$\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}$endowed with a suitable distance $\wh d$.This viewpoint turns out to provide a user friendly description of the range of the Schwartz space on $\H^d$ by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward. As a first application, we give an explicit formula for the Fourier transform of smooth functions on $\H^d$ that are independent of the vertical variable. We also provide other examples.


H A J E R B A H O U R I J E A N -Y V E S C H E MIN R A P H A Ë L D A N C HIN T E M P E R E D DIST R IB U TIO 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 DIST R IB U TIO N S T E M P É R É E S E T T R A N S F O R M ATIO 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 on H 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 set H d = N d × N d × R \ {0}, that may be completed to a larger set H d for some suitable distance. This viewpoint provides a user friendly description of the range of the Schwartz space on H d by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward.

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 space R n the group of characters may be identified with the dual space (R n ) of R n through the map ξ → e i ξ,· , where ξ, · designate the value of the one-form ξ when applied to elements of R n , and the Fourier transform of an integrable function f may be seen as the function on (R n ) , defined by A fundamental fact of the distribution theory on R n is that the Fourier transform is a bi-continuous isomorphism on the Schwartz space S(R n ) -the set of smooth functions whose derivatives decay at infinity faster than any power. Hence, one can define the transposed Fourier transform t F on the so-called set of tempered distributions S (R n ), that is the topological dual of S(R n ) (see e.g. [Rud62,Sch98] for a self-contained presentation). As the whole distribution theory on R n is based on identifying locally integrable functions with linear forms by means of the Lebesgue integral, one can look for a more direct relationship between t F and F, by considering the following bilinear form on S(R n ) × S(R n ): where the cotangent bundle T R n of R n is identified with R n × (R n ) . This allows to identify t F |S((R n ) ) with F |S(R n ) , and it is thus natural to define the extension of F on S (R n ) to be t F, that is, 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 on H d by 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 form a(−∆ H ) with a ∈ S(R), belong to S(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(S (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 group H d and of the Fourier transform on H d , and a few basic properties. We shall define H d to be the set T R d × R equipped with the product law w · w def = (Y + Y , s + s + 2σ(Y, Y )) = (y + y , η + η , s + s + 2 η, y − 2 η , y ) where w = (Y, s) = (y, η, s) and w = (Y , s ) = (y , η , s ) are generic elements of H d . In the above definition, the notation · , · 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, a 2 s).
Note that dilations commute with the product law on H d , that is δ a (w · w ) = δ a (w) · δ a (w ). 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 on R 2d+1 . The corresponding Lebesgue spaces L p (H d ) are thus the sets of measurable functions f : with the standard modification if p = ∞. TOME 1 (2018) The convolution product of any two integrable functions f and g is given by 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: The standard definition of the Fourier transform then reads as follows.
The function F H (f ) which takes values in the space of bounded operators on L 2 (R d ) is, by definition, the Fourier transform of f .
An obvious drawback of Definition 1.1 is that F H f 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 by F 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) set H d endowed with some distance d.
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 translation τ w (w ) def = w · w , is spanned by the vector fields The sublaplacian associated to the vector fields (X j ) 1 j d and (Ξ j ) 1 j d is defined by and may be rewritten in terms of the usual derivatives as follows: 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 functions f in the Schwartz space (see e.g. [Hue76,Olv74]): In order to take advantage of the spectral structure of the harmonic oscillator, we introduce the corresponding eigenvectors, that is the family of Hermite functions (H n ) n∈N d defined by 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 j u(x) def = x j u(x). As usual, for any multi-index n in N d , n! def = n 1 ! ... n d ! and |n| def = n 1 + ... + n d stands for the length of n (not to be confused with the Euclidean norm that we shall sometimes denote in the same way for elements of R d ).
Recall that (H n ) n∈N d is an orthonormal basis of L 2 (R d ), and that we have Our alternative definition of the Fourier transform on H d then reads as follows: To underline the similarity between that definition and the classical one in R n , one may further compute (F H (f )(λ)H m,λ |H n,λ ) L 2 . 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 functions which are closely related to Wigner transforms of Hermite functions, namely With this new point of view, Formula (1.7) recasts as follows: Furthermore, if we endow the set H d with the measure d w defined by the relation then the classical inversion formula and Fourier-Plancherel theorem become: and the Fourier transform F H can be extended to a bicontinuous isomorphism between the spaces L 2 (H d ) and L 2 ( H d ), that satisfies Finally, for any couple (f, g) of integrable functions, the following convolution identity holds true: For the reader's convenience, we shall present a proof of Theorem 1.3 in the appendix.

Main results
As already mentioned, our final goal is to extend the Fourier transform to tempered distributions on H 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 range S( H d ) of S(H d ) by F H that allows to find out some bilinear form B H for identifying t F H with F H .
Here, we first present results pertaining to the characterization of the set S( H d ), then exhibit a large class of functions of S( H d ) that, in particular, contains the Fourier transform of the heat kernel. Third, we go to examples of tempered distributions on the frequency set H d and, finally, to computations of Fourier transform of tempered distributions (with a special attention to functions independent of the vertical variable).
To characterize F 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 on H d to decay of the Fourier transform. This is achieved in the following lemma that has been proved in [BCD]. (1 + |λ|(|n| where · N,S denotes the classical family of semi-norms of S(R 2d+1 ), namely 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 expect F H to transform multiplication by a (suitable) weight function into a derivative operator on H d . So far however, we lack a notion of differentiation on H d that could fit the scope. This is the aim of the following definition (see also Proposition A.3 in Appendix): and, if in addition θ is differentiable with respect to λ, where w ± j def = (n ± δ j , m ± δ j , λ) and δ j denotes the element of N d with 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.
Then for all f in S(H d ), the following two relations hold true on H d : The third aspect of regularity for functions in F 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 M 0 F −1 H which is done in the proof of the inversion theorem in the Schwartz space S(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 Then, P maps continuously S(H d ) to S(H d ) and, for any f in S(H d ) and w in H d , Technically, the above weird relation stems from the fact that the function W fulfills: Note that, since the left-hand side of (2.6) belongs to the space F H (S(H d )) that is the natural candidate for being S( H d ), we need to prescribe in addition to the decay properties pointed out in Lemmas 2.1 and 2.3 some condition involving Σ 0 . This motivates the following definition. We equip S( H d ) with the family of semi-norms Clearly, the above definition of semi-norms guarantees that there exist an integer K and a positive real number C so that the following inequality holds true: More importantly, we have the following isomorphism theorem.
Theorem 2.6. -The Fourier transform F H is a bicontinuous isomorphism between S(H d ) and S( H d ), and the inverse map is given by We shall discover throughout the paper that the definition of S( H d ) encodes a number of nontrivial hidden informations that are consequences of the sub-ellipticity of ∆ H . For instance, the stability of S( H d ) by the multiplication law defined in (1.13) is a consequence of the stability of S(H d ) by convolution and of Theorem 2.6. Another hidden information is the behavior of functions of S( H d ) when λ tends to 0.
Keeping the decay inequality (2.1) in mind, we endow the set H d with the distance: where | · | 1 denotes the Manhattan norm 1 on R d . With this choice, the Fourier transform of any function of S(H d ) (and even of any integrable function on H d ) is uniformly continuous on H d , and can thus be extended by continuity on the completion H d of H d , as explained in greater details in the following statement that has been proved in [BCD]: The Fourier transform f H of any integrable function on H d may be extended continuously to the whole set H d . Still denoting by f H (or F H f ) that extension, the linear map Note that for any function θ in S( H d ), having w tend to (ẋ, k) in (2.6) yields As regards convolution, we obtain, after passing to the limit in (1.13), the following noteworthy formula, valid for any two functions f and g in L 1 (H d ): Let us underline that the above product law (2.10) is commutative even though convolution of functions on the Heisenberg group is not (see (1.13)).
The next question is how to extend the measure d w to H d . In fact, we have for any positive real numbers R and ε, Therefore, it is natural to extend d w by 0 on H d 0 . With this convention, keeping the same notation for the extended measure, we have for all continuous compactly supported function θ on H d , At this stage of the paper, pointing out non-trivial examples of functions in S( H d ) is highly informative. To this end, we introduce the set As may be easily checked by the reader, the space S + d is stable under derivation and multiplication by polynomial functions of (x, k).
for some positive real number r 0 , and satisfies 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 in H 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]).
In particular, the function h such that, for all u 0 in L 1 (H d ) and t > 0, the map 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 u must fulfill At the same time, we have u(t) = u 0 h t . Hence, combining the convolution formula (1.13) and identity (2.12), we gather that Convergence of tempered distributions is defined in the standard way: we say that a sequence ( A first class of examples of tempered distributions on H d is given by "functions with moderate growth": Definition 2.11. -We denote by L 1 M ( H d ) the set of functions on H d with moderate growth, that is the set of locally integrable functions f on H d such that there exists an integer p satisfying As in the Euclidean setting, functions in L 1 M ( H d ) may be identified with tempered distributions: Theorem 2.12. -Let us consider ι be the map defined by ι : Moreover, if p is an integer such that the map The following proposition that will be proved in Section 5 provides examples of Remark 2.14. -The above proposition is no longer true for γ = d + 1. If we look at the quantity |λ|(2|n| + d) in H d as an equivalent of |ξ| 2 for R d , then it means that the homogeneous dimension of H d is 2d + 2, as for H d .
Obviously, any Dirac mass on H d is a tempered distribution. Let us also note that because We now want to exhibit tempered distributions on H d which are not functions or measures. The following proposition states that the analogue on H d of finite part distributions on R n , are indeed in S ( H d ).
is integrable. Furthermore, the linear form defined by , and its restriction to H d is the function f γ that has been defined above, that is, for any θ in S( H d ) such that θ(n, n, λ) = 0 for small enough |λ|(2|n| + d), we have Another interesting example of tempered distribution on H d is the measure µ [BCD]. In our setting, that result recasts as follows: Then µ , and for any function ψ in S(R) with integral 1 we have

ANNALES HENRI LEBESGUE
Let us finally explain how Formulae (1.2) and (1.3) may be adapted so as to extend the Fourier transform on H d to tempered distributions. We introduce: Since, for any θ in S( H d ) and w = (y, η, s) in H d , we have This prompts us to extend F H on S (H d ) by A direct consequence of this definition is the following statement: where I is defined by (2.14) and 0 is the element of H d 0 corresponding toẋ = 0 and k = 0.
Let us emphasize, that without completing the frequency set H d , 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 on T R d , The rest of the paper unfolds as follows. In Section 3, we prove Lemmas 2.3 and 2.4, and then Theorem 2.6. In Section 4, we establish Theorem 2.8. In Section 5, we study in full details the examples of tempered distributions on H d given in Propositions 2.13-2.15, and Theorem 2.12. In Section 6, we prove Proposition 2.18 and Theorem 2.19. Further remarks as well as proofs within our setting of known results are postponed in the appendix.

The range of the Schwartz class by the Fourier transform
The present sectionaims at giving a handy characterization of the range of S(H d ) by the Fourier transform. Our Ariadne thread throughout will be that we expect that, for the action of F H , regularity implies decay and decay implies regularity. The answer to the first issue has been given in Lemma 2.1 (proved in [BCD]). Here we shall concentrate on the second issue, in connection with the definition of differentiation for functions on H d , given in (2.2) and (2.3). To complete our analysis of the space F H (S(H d )), we will have to get some information on the behavior of elements of F H (S(H d )) for λ going to 0 (that is in the neighborhood of the set H d 0 ). This is Lemma 2.4 that points out an extra and fundamental relationship between positive and negative λ's for functions of F H (S(H d )).
A great deal of our program will be achieved by describing the action of the weight function M 2 and of the differentiation operator ∂ λ on W. This is the goal of the next paragraph.

Some properties for Wigner transform of Hermite functions
The following lemma describes the action of the weight function M 2 on W.
where Operator ∆ has been defined in (2.2).
The purpose of the following lemma is to investigate the action of ∂ λ on W.
Lemma 3.2. -We have, for all w in H d and Y ∈ T R d , the following formula: where, for θ : H d → C, we denoted Proof. -Let us write that As we have an integration by parts gives Now, let us compute This gives Using Relations (A.4) completes the proof of the Lemma.

Decay provides regularity
Granted with Lemmas 3.1 and 3.2, it is now easy to establish Lemma 2.3. Indeed, according to (1.9), we have Therefore, Lemma 3.1 implies that By the definition of the Fourier transform and of ∆, this gives To establish the second part of the lemma, we start from (1.9) and get Rewriting the last term according to Formula (3.1), we discover that By the definition of the Fourier transform, this concludes the proof of Lemma 2.3. On the one hand, Lemmas 2.1 and 2.3 guarantee that decay in the physical space provides regularity in the Fourier space, and that regularity gives decay. On the other hand, the relations we established so far do not give much insight on the behavior of the Fourier transform near H d 0 even though we know from Theorem 2.7 that in the case of an integrable function, it has to be uniformly continuous up to λ = 0. Getting more information on the behavior of the Fourier transform of functions in S(H d ) in a neighborhood of H d 0 is what we want to do now with the proof of Lemma 2.4. Proof of Lemma 2.4. -Since the partial Fourier transform F s with respect to the vertical variable is a bounded isomorphism on S(R 2d+1 ), it suffices to establish that F s P is a bounded operator on S(R 2d+1 ). Now, fix some function f in S(H d ), and observe that Taking the Fourier transform with respect to the variable s gives Let us consider a function χ in S(R) with value 1 near 0 and let us write The two terms on the right-hand side define bounded operators on S(R 2d+1 ). Hence, the operator F s P is bounded on S(R 2d+1 ) and thus, so does P.
Note that in the case of a function g in S(H d ), Formula (1.9) may be alternately written:

Relations (2.7) and (3.2) guarantee that
which completes the proof of Lemma 2.4.

Proof of the inversion theorem in the Schwartz space
The aim of this section is to prove Theorem 2.6. To this end, let us first note that from Inequality (2.1) and Lemmas 2.3 and 2.4, we gather that F H maps S(H d ) to S( H d ) continuously. In addition, (2.8) guarantees that all elements of S( H d ) are in Hence Theorem 1.3 ensures that F H : S(H d ) → S( H d ) is one-to-one, and that the inverse map has to be the functional F H defined in (2.9). Therefore, there only remains to prove that F H maps S( H d ) to S(H d ). To this end, it is convenient to introduce the semi-norms defined by: which are equivalent to the classical ones defined in Lemma 2.1 (see Proposition A.2).

ANNALES HENRI LEBESGUE
Hence, thanks to (2.7) (n,m)∈N 2d Swapping indices n and m in the last sum gives Remembering that |W| 1, we thus get Since the first term of the right-hand side of (3.6) tends to 0 when ε goes to 0. Employing the same argument with Σ 0 θ guarantees that we do have from which one may conclude that Together with (3.4), this implies that Hence, for any integer K, there exist an integer N K and a constant C K so that Finally, to study the action of the sublaplacian on F H (S( H d )), we write that by definition of X j and of W, we have As 2iλη j e 2iλ η,z = ∂ z j (e 2iλ η,z ), integrating by parts yields The action of Ξ j is simply described by Together with (3.8) and the definition of ∆ H in (1.6), this gives This implies that for all integers K, we have whence there exist an integer N K and a constant C K so that Putting (3.7) and (3.9) together and remembering the definition of the semi-norms on S(H d ) given in (3.3), we conclude that for all integer K, there exist an integer N K and a constant C K so that This completes the proof of Theorem 2.6.

Examples of functions in the range of the Schwartz class
The purpose of this sectionis to prove Theorem 2.8. Let us recall the notation For  We have the following lemma.
Proof. -Performing a Taylor expansion at order M + 1, we get m 1 + 1, . . . , n j + m j + 1 ± 2t, . . . , n d + m d + 1). The fact that f belongs to S + d implies that for any positive integer N , we have This gives the lemma.

TOME 1 (2018)
One can now tackle the proof of Theorem 2.8. Let us first investigate the (easier) case when the support of f is included in [0, ∞[ d × {0} × R. The first step consists in computing an equivalent (in the sense of Definition 4.1) of ∆Θ f at an order which will be chosen later on. For notational simplicity, we here set R(n) def = R(n, n) and omit the second variable of f . Now, by definition of the operator ∆, we have Lemma 4.2, and Assertions (4.3) and (4.4) imply that Let us define Clearly, all functions f are supported in [0, ∞[ d × {0} × R and belong to S d + , and the above equality rewrites (4.7) ∆Θ f (n, n, λ) Arguing by induction, it is easy to establish that for any function f in S + d supported in [0, ∞[ × {0} × R and any integers N and p, the quantity ∆ p Θ f N,0,S( H d ) is finite. Indeed, this is obvious for p = 0. Now, if the property holds true for some non negative integer p then, thanks to (4.5) and (4.7), From (4.1), (4.7) and the induction hypothesis, it is clear that if we choose M greater than p then we get that ∆ p Θ f N,0,S( H d ) is finite for all integer N .
Let us next study the action of Operator D λ . From its definition in Lemma 2.3 and the chain rule, we gather that Lemma 4.2, and Assertions (4.3) and (4.4) imply that Observe that the term corresponding to = 0 of the first and second sums above cancel out with the second and third terms of the right-hand side of (4.8), respectively. Hence, defining for 1 the functions we get, using (4.8) and (4.9), From that relation, mimicking the induction proof for ∆, we easily conclude that for any function f in S + d supported in [0, ∞[ d × {0} × R, and any integer p, the quantity D p λ Θ f N,0,S( H d ) is finite for all integer N . This completes the proof Theorem 2.8 in that particular case.
Next, let us investigate the case when the function f of S + d is supported in the set [r 0 , ∞[ d × Z d × R for some positive r 0 and satisfies (2.11). Then, by definition of the operator ∆, we have for all w = (n, m, λ) in H d , denoting k def = m − n, Compared to (4.6), the computations get wilder, owing to the square roots in the above formula. Let M be an integer (to be suitably chosen later on). Lemma 4.2, and Assertions (4.3) and (4.4) imply that for nonnegative integers p and q, we get 1 2|λ| , m), k, λ) .
In order to compute an expansion of α ± j (n, m) with respect to n j +m j +1 and n j −m j , we shall take advantage of the following two identities that are valid for any (p, q) in N 2 : (p + 1)(q + 1) = 1 2 (p + q + 1) 1 + 2 Let us introduce the notation f (p, q) = O M (p, q) to mean that for some constant C, there holds |f (p, q)| C 1 (p + q + 1) M + |p − q| 2M +2 (p + q + 1) 2M +1 . Using the following Taylor expansion with K = 2M : we gather that, up to a O 2M (p, q) term, we have (p + 1)(q + 1) = 1 2 (p + q + 1) Now we can compute the expansion of α ± (p, q). Newton's formula gives (4.10) α + (p, q) = p+q+1+ In the above expansion, some O 2M (p, q) terms are kept for notational simplicity. Now, one may check that for all functions θ and θ supported in [r 0 , ∞[ d × Z d × R and any integers M 1 and M 2 , we have for all j ∈ {1, . . . , d}, Then Assertion (4.10) implies that for any function g in S + d supported in the set [r 0 , ∞[ d × Z d × R, and any j in {1, . . . , d}, we have (4.12) α + (n j , m j )Θ g ( w) and (4.13) α − (n j , m j )θ g ( w) Using (4.11), this gives and, if 1 M , Similarly, From the definition of Operator ∆, we thus infer that there exist functions f of S + d supported in [r 0 , ∞[ d × Z d × R and satisfying (2.11), such that for all M 0, At this stage, one may prove by induction as in the previous case that is finite for all integers N and p. Let us finally study the action of D λ . From its definition, setting k = m − n, we get − (n j + 1)(m j + 1)f (|λ|(R(n, m) + 2δ j ), k, λ) .
Hence, using (4.12) and (4.13) and noticing that the coefficient a 1 involved in the expansion of α ± (n j , m j ) is equal to 1/2, we conclude that there exist some functions and At this stage, one can complete the proof as in the previous cases. Let us finally record two interesting asymptotic properties of the operators ∆ and D λ when λ tends to 0.

Examples of tempered distributions
A first class of examples is given by the functions belonging to the space L 1 M ( H d ) of Definition 2.11. This is exactly what states Theorem 2.12 that we are going to prove now.
Proof of Theorem 2.12. -Inequality (2.13) just follows from the definition of the semi-norms on H d . So let us focus on the proof of the first part of the statement that actually holds true in the largest class of locally integrable functions.
Let f be a function of L 1 loc ( H d ) such that ι(f ) = 0. In order to prove that f = 0 a.e., it suffices to check that for all K > 0 and b > a > 0, we have To this end, we introduce the bounded function : and smooth it out with respect to λ by setting and χ stands for some smooth even function on R, supported in the interval [−1, 1] and with integral 1.
Note that by definition, g is supported in the set C a,b,K . Therefore, if ε < a then g ε is supported in C a−ε,b+ε,K(1+ε/a) . This readily ensures that g ε N,0,S( H d ) is finite for all integer N (as regards the action of operator Σ 0 , note that g ε (n, m, λ) = 0 whenever |λ| < a − ε).
In order to prove that g ε belongs to S( H d ), it suffices to use the following lemma the proof of which is left to the reader: Because g ε is in S( H d ) for all 0 < ε < a, our assumption on f ensures that we have Let K = 3K/2. Whenever 0 < ε a/2, we have for all (n, m, λ) ∈ H d and λ = 0, which guarantees that On the one hand, we have, changing variable λ = λ(2|n|+d) and using that γ > d+1, On the other hand, changing again variable λ = λ|(2|n| + d), we see that At this stage, we need a suitable bound of the integrand just above. This will be achieved thanks to the following lemma.
Lemma 5.2. -There exists an integer k such that for any function θ in S( H d ), we have This implies that As γ < d + 3/2, combining with (5.1) completes the proof of the proposition.

Examples of computations of Fourier transforms
The present sectionaims at pointing out a few examples of computations of Fourier transform that may be easily achieved within our approach.
Let us start with Proposition 2.18. The first identity is easy to prove. Indeed, according to (1.10), we have As (H n,λ ) n∈N is an orthonormal basis of L 2 (R d ), we get which is exactly the first identity.
In order to prove the second identity, we start again from the definition of the Fourier transform on S (H d ), and get Let us underline that because t F H θ belongs to S(H d ), the above integral makes sense. Besides, (2.15) implies that By Theorem 2.7 and Lemma 5.2 we have, for any integrable function f on H d , Thus we get . This concludes the proof of the proposition.
In order to prove Theorem 2.19, we need the following obvious property of the Fourier transform.  [BCD] (and its proof) in terms of tempered distributions. We recall here the statement for the reader convenience.
Theorem 6.2. -Let χ be a function of S(R) with value 1 at 0, and compactly supported Fourier transform. Then for any function g in L 1 (T R d ) and any sequence (ε n ) n∈N of positive real numbers tending to 0, we have Proof of Theorem 2.19. -As g ⊗ χ(ε n ·) tends to g ⊗ 1 in S (H d ), Proposition 6.1 ensures that Moreover, according to Theorem 6.2, we have, for any θ in S( H d ), As g is integrable on T R d , Proposition 2.1 of [BCD] implies that the (numerical) product Gθ is a continuous function that satisfies This matches the hypothesis of Lemma 3.1 in [BCD], and thus Together with (6.3), this proves the theorem.

Appendix A. Useful tools and more results
For the reader's convenience, we here recall (and sometimes prove) some results that have been used repeatedly in the paper. We also provide one more result concerning the action of the Fourier transform on derivatives.

A.1. Hermite functions
In addition to the creation operator C j def = −∂ j + M j already defined in the introduction, we used the following annihilation operator: It is very classical (see e.g. [Olv74]) that As, obviously, we discover that M j H n = 1 2 2n j H n−δ j + 2n j + 2 H n+δ j and ∂ j H n = 1 2 2n j H n−δ j − 2n j + 2 H n+δ j . theorem in R d+1 , the Fourier transform F H may be seen as the composition of three invertible and bounded operators on L 2 , and we have This gives (1.11) and (1.12). For the proof of (1.13), we refer for instance to [BCD]. This concludes the proof of Theorem 1.3.

A.3. Properties related to the sub-ellipticity of the sublaplacian
Under Notation (1.5), denote by X = (X 1 , . . . , X 2d ) the family of horizontal left invariant vector fields (where we agree that X j+d = Ξ j for j in {1, . . . , d}), and define, for any multi-index α in {1, . . . , 2d} k : Then setting, for a nonnegative integer k we have the following well-known result (see the proof in e.g. [CCX93,Hör85]): Theorem A.1. -For any positive integer , we have for some constant C > 0, This will enable us to establish the following proposition which states that the usual semi-norms on the Schwartz class and the semi-norms using the structure of H d are equivalent. Then the two families of semi-norms defined on S(H d ) by are equivalent to the classical family of semi-norms on S(R 2d+1 ).  2m j + 2 θ(n, m + δ j , λ) + 2m j θ(n, m − δ j , λ) .

ANNALES HENRI LEBESGUE
For multiplication by y j , we proceed along the same lines. By definition of W, we have − 2m j W(n, m − δ j , λ, Y ) − 2m j + 2 W(n, m + δ j , λ, Y )).
Then reverting to the definition of F H gives the last part of the proposition. BIBLIOGRAPHY