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### Étude des coefficients de Fourier des fonctions de ${L}^{p}\left(G\right)$

Annales de l'institut Fourier

On étudie la décroissance à l’infini des coefficients de Fourier des fonctions $2\pi$-périodiques intégrables. Soit en particulier ${\lambda }_{n}$ une suite lacunaire d’entiers : ${\lambda }_{n+1}\ge 3{\lambda }_{n}$. On appelle suite $k$-lacunaire associée la suite ${\mu }_{N}^{k}$ des entiers qui s’écrivent sous la forme $±{\lambda }_{{n}_{1}}±{\lambda }_{{n}_{2}}±\cdots ±{\lambda }_{{n}_{k}}$, ${n}_{1}\phantom{\rule{-0.166667em}{0ex}}>{n}_{2}\phantom{\rule{-0.166667em}{0ex}}>\cdots \phantom{\rule{-0.166667em}{0ex}}>{n}_{k}$. On montre que si ${\int }_{0}^{2\pi }\phantom{\rule{-0.166667em}{0ex}}|f|\left({\mathrm{Log}}^{+}|f|{\right)}^{k/2}dx$ est fini, il en est de même de ${\sum }_{N}|\stackrel{^}{f}\left({\mu }_{N}^{k}\right){|}^{2}$. D’autre part, si ${\lambda }_{n}$ satisfait à une condition plus restrictive, quel que soit $1<p\le 2$, si ${\int }_{0}^{2\pi }|f{|}^{p}dx$ est fini il en est de même de ${\sum }_{k}\left(p-1\right){\sum }_{N}|\stackrel{^}{f}\left({\mu }_{N}^{k}\right){|}^{2}$. Ces résultats sont généralisés à d’autres groupes que $\mathbf{R}/2\pi \mathbf{Z}$, et à d’autres...

### Ensembles $\Lambda \left(p\right)$ dans le dual de ${D}^{\infty }$

Annales de l'institut Fourier

Recherche des conditions nécessaires pour qu’un sous-ensemble du dual de ${D}^{\infty }$ soit un ensemble $\Lambda \left(p\right)$, et constructions d’ensembles $\Lambda \left(p\right)$ particuliers.

### Three related problems of Bergman spaces of tube domains over symmetric cones

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

It has been known for a long time that the Szegö projection of tube domains over irreducible symmetric cones is unbounded in ${L}^{p}$ for $p\ne 2$. Indeed, this is a consequence of the fact that the characteristic function of a disc is not a Fourier multiplier, a fundamental theorem proved by C. Fefferman in the 70’s. The same problem, related to the Bergman projection, deserves a different approach. In this survey, based on joint work of the author with D. Békollé, G. Garrigós, M. Peloso and F. Ricci, we give...

### Fonction maximale et variation quadratique des martingales en présence d'un poids

Séminaire de probabilités de Strasbourg

### Projecteurs de Bergman et Szegö pour une classe de domaines faiblement pseudo-convexes et estimations ${L}^{p}$

Compositio Mathematica

### Estimates for the Bergman and Szegö projections in two symmetric domains of ${ℂ}^{n}$

Colloquium Mathematicae

### Mesures de Carleson d’ordre $\alpha$ et solutions au bord de l’équation $\overline{\partial }$

Bulletin de la Société Mathématique de France

### Hankel operators and weak factorization for Hardy-Orlicz spaces

Colloquium Mathematicae

We study the holomorphic Hardy-Orlicz spaces ${}^{\Phi }\left(\Omega \right)$, where Ω is the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in ℂⁿ. The function Φ is in particular such that $¹\left(\Omega \right){\subset }^{\Phi }\left(\Omega \right){\subset }^{p}\left(\Omega \right)$ for some p > 0. We develop maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Ω). As a consequence, we characterize those Hankel operators which are bounded from ${}^{\Phi }\left(\Omega \right)$ into ¹(Ω).

### Solutions de l’équation $\overline{\partial }$ et zéros de la classe de Nevanlinna dans certains domaines faiblement pseudo-convexes

Annales de l'institut Fourier

Il est montré que la condition de Blaschke est nécessaire et suffisante pour qu’un sous-ensemble analytique du domaine $D=\left\{z\in {\mathbf{C}}^{n};{\sum }_{1}^{n}|{z}_{i}{|}^{2{\rho }_{i}}<1\right\}$ soit l’ensemble des zéros d’une fonction de la classe de Nevanlinna.

### On truncations of Hankel and Toeplitz operators.

Publicacions Matemàtiques

We study the boundedness properties of truncation operators acting on bounded Hankel (or Toeplitz) infinite matrices. A relation with the Lacey-Thiele theorem on the bilinear Hilbert transform is established. We also study the behaviour of the truncation operators when restricted to Hankel matrices in the Schatten classes.

### A survey on uncertainty principles related to quadratic forms.

Collectanea Mathematica

### Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms.

Revista Matemática Iberoamericana

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on Rd which may be written as P(x)exp(-(Ax,x)), with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x)f(y). We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with...

### Wavelets obtained by continuous deformations of the Haar wavelet.

Revista Matemática Iberoamericana

One might obtain the impression, from the wavelet literature, that the class of orthogonal wavelets is divided into subclasses, like compactly supported ones on one side, band-limited ones on the other side. The main purpose of this work is to show that, in fact, the class of low-pass filters associated with reasonable (in the localization sense, not necessarily in the smooth sense) wavelets can be considered to be an infinite dimensional manifold that is arcwise connected. In particular, we show...

### On the Product of Functions in and ${}^{\text{1}}$

Annales de l’institut Fourier

The point-wise product of a function of bounded mean oscillation with a function of the Hardy space ${H}^{1}$ is not locally integrable in general. However, in view of the duality between ${H}^{1}$ and $BMO$, we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic...

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