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where S ∈ OC(H), F ∈ H′ and the indeterminate U ∈ H′. Such equations or S are said to be hypoelli... more where S ∈ OC(H), F ∈ H′ and the indeterminate U ∈ H′. Such equations or S are said to be hypoelliptic, if any solution U belongs to OC(H) whenever F belongs to OC(H). In the case H′ = D′(Rd), we note that OC(H) = E′(Rd) the space of compact support distributions on R, and OC(H) = E(R) the space of C∞-functions on R. L. Ehrempreis [7] and next L. Hörmander [9] have characterized hypoelliptic distributions by giving necessary and sufficient conditions on their usual Fourier transforms. Analogous result is obtained by Trimèche [18] for the Dunkl convolution on R. Similar characterizations are established for other convolutions, see for example [1], [11] and [19]. In the case H′ = S′(Rd), and according to Schwartz [16], OC(H) is the space of rapidly decreasing distributions on R which we denote by OC(R), and OC(H) is the space of very slowly increasing C∞-functions and is denoted by OC(R). Let us also denote by OM (R) the space of multiplication operators on S′(Rd). For this situation, ...

Journal De Mathematiques Pures Et Appliquees, 2005

Soit T un courant positif fermé de dimension p dans C n. Soit H une sous-variété analytique de di... more Soit T un courant positif fermé de dimension p dans C n. Soit H une sous-variété analytique de dimension n − q (1 q p) dans C n. On note T , H la tranche de T en H si elle existe. Pour r > 0, on pose n T (r) = r −2p |z| r T ∧ β p et n T (H, r) = r −2(p−q) |z| r T , H ∧ β p−q. Dans cet article, on établit une majoration de n T (H, r) par une fonction de n T (r) dans le cas H = f −1 ({a}) et f : C n → C q holomorphe et dans le cas H ∈ G n−q,n la grassmannienne des sous-espaces vectoriels de dimension n − q dans C n. Pour la minoration, on montre que n T (H, r) croît au moins aussi rapidement que n T (r) pour H ∈ G n−1,n sauf pour un ensemble pluripolaire.

Integral Transform Spec Funct, 2005

A theorem of Morgan states that, for α > 2 and β = α/(α − ... more A theorem of Morgan states that, for α > 2 and β = α/(α − 1), if f is a measurable function on ℝ such that and , where a > 0, b > 0 and (αα) (b β) > [sin (π/2) (β − 1)], then f = 0. We prove a L –L -version of this result for any non-compact real semi-simple Lie group.

Comptes Rendus Mathematique, 2003

We consider a real semi-simple Lie group G with finite center and a maximal compact subgroup K of... more We consider a real semi-simple Lie group G with finite center and a maximal compact subgroup K of G. Let G = K exp(a +)K be a Cartan decomposition of G. For x ∈ G denote x the norm of the a +-component of x in the Cartan decomposition of G. Let a > 0, b > 0 and 1 p, q ∞. In this Note we give necessary and sufficient conditions on a, b such that for all K-bi-invariant measurable function f on G, if e a x 2 f ∈ L p (G) and e b λ 2 F(f) ∈ L q (a * +) then f = 0 almost everywhere. To cite this article: S.

Bulletin De La Societe Mathematique De France, 1990

Methods and Applications of Analysis, 2007

The aim of this paper is to characterize hypoelliptic convolution-equations in S (R) for the Dunk... more The aim of this paper is to characterize hypoelliptic convolution-equations in S (R) for the Dunkl theory on the real line. For this we determine the spaces of convolution and multiplication operators in S (R) for the Dunkl convolution and we show that the Fourier-Dunkl transform is a topological isomorphism between them.

The Ramanujan Journal, 2009

We prove real Paley-Wiener type theorems for the Dunkl transform F D on the space S (R d) of temp... more We prove real Paley-Wiener type theorems for the Dunkl transform F D on the space S (R d) of tempered distributions. Let T ∈ S (R d) and κ the Dunkl Laplacian operator. First, we establish that the support of F D (T) is included in the Euclidean ballB(0, M) = {x ∈ R d , x ≤ M}, M > 0, if and only if for all R > M we have lim n→+∞ R −2n n κ T = 0 in S (R d). Second, we prove that the support of F D (T) is included in R d \ B(0, M), M > 0, if and only if for all R < M, we have lim n→+∞ R 2n F −1 D (y −2n F D (T)) = 0 in S (R d). Finally, we study real Paley-Wiener theorems associated with C ∞-slowly increasing function.

International Journal of Mathematics and Mathematical Sciences, 2004

We consider a real semisimple Lie groupGwith finite center andKa maximal compact subgroup ofG. We... more We consider a real semisimple Lie groupGwith finite center andKa maximal compact subgroup ofG. We prove anLp−Lqversion of Hardy's theorem for the spherical Fourier transform onG. More precisely, leta,bbe positive real numbers,1≤p,q≤∞, andfaK-bi-invariant measurable function onGsuch thatha−1f∈Lp(G)andeb‖λ‖2ℱ(f)∈Lq(𝔞+*)(hais the heat kernel onG). We establish that ifab≥1/4andporqis finite, thenf=0almost everywhere. Ifab<1/4, we prove that for allp,q, there are infinitely many nonzero functionsfand ifab=1/4withp=q=∞, we havef=const ha.

Comptes Rendus Mathematique, 2003

We consider a real semi-simple Lie group G with finite center and a maximal compact subgroup K of... more We consider a real semi-simple Lie group G with finite center and a maximal compact subgroup K of G. Let G = K exp(a +)K be a Cartan decomposition of G. For x ∈ G denote x the norm of the a +-component of x in the Cartan decomposition of G. Let a > 0, b > 0 and 1 p, q ∞. In this Note we give necessary and sufficient conditions on a, b such that for all K-bi-invariant measurable function f on G, if e a x 2 f ∈ L p (G) and e b λ 2 F(f) ∈ L q (a * +) then f = 0 almost everywhere. To cite this article: S.

where S ∈ OC(H), F ∈ H′ and the indeterminate U ∈ H′. Such equations or S are said to be hypoelli... more where S ∈ OC(H), F ∈ H′ and the indeterminate U ∈ H′. Such equations or S are said to be hypoelliptic, if any solution U belongs to OC(H) whenever F belongs to OC(H). In the case H′ = D′(Rd), we note that OC(H) = E′(Rd) the space of compact support distributions on R, and OC(H) = E(R) the space of C∞-functions on R. L. Ehrempreis [7] and next L. Hörmander [9] have characterized hypoelliptic distributions by giving necessary and sufficient conditions on their usual Fourier transforms. Analogous result is obtained by Trimèche [18] for the Dunkl convolution on R. Similar characterizations are established for other convolutions, see for example [1], [11] and [19]. In the case H′ = S′(Rd), and according to Schwartz [16], OC(H) is the space of rapidly decreasing distributions on R which we denote by OC(R), and OC(H) is the space of very slowly increasing C∞-functions and is denoted by OC(R). Let us also denote by OM (R) the space of multiplication operators on S′(Rd). For this situation, ...

Journal De Mathematiques Pures Et Appliquees, 2005

Soit T un courant positif fermé de dimension p dans C n. Soit H une sous-variété analytique de di... more Soit T un courant positif fermé de dimension p dans C n. Soit H une sous-variété analytique de dimension n − q (1 q p) dans C n. On note T , H la tranche de T en H si elle existe. Pour r > 0, on pose n T (r) = r −2p |z| r T ∧ β p et n T (H, r) = r −2(p−q) |z| r T , H ∧ β p−q. Dans cet article, on établit une majoration de n T (H, r) par une fonction de n T (r) dans le cas H = f −1 ({a}) et f : C n → C q holomorphe et dans le cas H ∈ G n−q,n la grassmannienne des sous-espaces vectoriels de dimension n − q dans C n. Pour la minoration, on montre que n T (H, r) croît au moins aussi rapidement que n T (r) pour H ∈ G n−1,n sauf pour un ensemble pluripolaire.

Integral Transform Spec Funct, 2005

A theorem of Morgan states that, for α &amp;amp;amp;amp;amp;amp;amp;amp;gt; 2 and β = α/(α − ... more A theorem of Morgan states that, for α &amp;amp;amp;amp;amp;amp;amp;amp;gt; 2 and β = α/(α − 1), if f is a measurable function on ℝ such that and , where a &amp;amp;amp;amp;amp;amp;amp;amp;gt; 0, b &amp;amp;amp;amp;amp;amp;amp;amp;gt; 0 and (αα) (b β) &amp;amp;amp;amp;amp;amp;amp;amp;gt; [sin (π/2) (β − 1)], then f = 0. We prove a L –L -version of this result for any non-compact real semi-simple Lie group.

Comptes Rendus Mathematique, 2003

We consider a real semi-simple Lie group G with finite center and a maximal compact subgroup K of... more We consider a real semi-simple Lie group G with finite center and a maximal compact subgroup K of G. Let G = K exp(a +)K be a Cartan decomposition of G. For x ∈ G denote x the norm of the a +-component of x in the Cartan decomposition of G. Let a > 0, b > 0 and 1 p, q ∞. In this Note we give necessary and sufficient conditions on a, b such that for all K-bi-invariant measurable function f on G, if e a x 2 f ∈ L p (G) and e b λ 2 F(f) ∈ L q (a * +) then f = 0 almost everywhere. To cite this article: S.

Bulletin De La Societe Mathematique De France, 1990

Methods and Applications of Analysis, 2007

The aim of this paper is to characterize hypoelliptic convolution-equations in S (R) for the Dunk... more The aim of this paper is to characterize hypoelliptic convolution-equations in S (R) for the Dunkl theory on the real line. For this we determine the spaces of convolution and multiplication operators in S (R) for the Dunkl convolution and we show that the Fourier-Dunkl transform is a topological isomorphism between them.

The Ramanujan Journal, 2009

We prove real Paley-Wiener type theorems for the Dunkl transform F D on the space S (R d) of temp... more We prove real Paley-Wiener type theorems for the Dunkl transform F D on the space S (R d) of tempered distributions. Let T ∈ S (R d) and κ the Dunkl Laplacian operator. First, we establish that the support of F D (T) is included in the Euclidean ballB(0, M) = {x ∈ R d , x ≤ M}, M > 0, if and only if for all R > M we have lim n→+∞ R −2n n κ T = 0 in S (R d). Second, we prove that the support of F D (T) is included in R d \ B(0, M), M > 0, if and only if for all R < M, we have lim n→+∞ R 2n F −1 D (y −2n F D (T)) = 0 in S (R d). Finally, we study real Paley-Wiener theorems associated with C ∞-slowly increasing function.

International Journal of Mathematics and Mathematical Sciences, 2004

We consider a real semisimple Lie groupGwith finite center andKa maximal compact subgroup ofG. We... more We consider a real semisimple Lie groupGwith finite center andKa maximal compact subgroup ofG. We prove anLp−Lqversion of Hardy's theorem for the spherical Fourier transform onG. More precisely, leta,bbe positive real numbers,1≤p,q≤∞, andfaK-bi-invariant measurable function onGsuch thatha−1f∈Lp(G)andeb‖λ‖2ℱ(f)∈Lq(𝔞+*)(hais the heat kernel onG). We establish that ifab≥1/4andporqis finite, thenf=0almost everywhere. Ifab<1/4, we prove that for allp,q, there are infinitely many nonzero functionsfand ifab=1/4withp=q=∞, we havef=const ha.

Comptes Rendus Mathematique, 2003

We consider a real semi-simple Lie group G with finite center and a maximal compact subgroup K of... more We consider a real semi-simple Lie group G with finite center and a maximal compact subgroup K of G. Let G = K exp(a +)K be a Cartan decomposition of G. For x ∈ G denote x the norm of the a +-component of x in the Cartan decomposition of G. Let a > 0, b > 0 and 1 p, q ∞. In this Note we give necessary and sufficient conditions on a, b such that for all K-bi-invariant measurable function f on G, if e a x 2 f ∈ L p (G) and e b λ 2 F(f) ∈ L q (a * +) then f = 0 almost everywhere. To cite this article: S.