We prove that if $E\subseteq Ĝ$ does not contain parallelepipeds of arbitrarily large dimension then for any open, non-empty $S\subseteq G$ there exists a constant c > 0 such that $\parallel f{1}_S{\parallel }_2\ge c\parallel f{\parallel }_2$ for all $f\in L^2\left(G\right)$ whose Fourier transform is supported on E. In particular, such functions cannot vanish on any open, non-empty subset of G. Examples of sets which do not contain parallelepipeds of arbitrarily large dimension include all Λ(p) sets.

A reproducing system is a countable collection of functions ${\varphi }_j:j\in$ such that a general function f can be decomposed as $f={\sum }_{j\in }{c}_j\left(f\right){\varphi }_j$, with some control on the analyzing coefficients $c_j\left(f\right)$. Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint in the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on L²(G)....

Nous étudions le comportement à l’infini des intégrales de Poisson liées aux groupes de déplacements de Cartan.

We show that the transference method of Coifman and Weiss can be extended to Hardy and Sobolev spaces. As an application we obtain the de Leeuw restriction theorems for multipliers.

By combining some results of C. S. Herz on the Fourier algebra with the notion of contractions of Lie groups, we prove theorems which allow transference of $L^p$ multipliers either from the Lie algebra or from the Cartan motion group associated to a compact Lie group to the group itself.

It is shown that if $G$ is a connected metrizable compact Abelian group and $1\<p\<\infty$, any (possibly discontinuous) translation invariant linear form on $L^p\left(G\right)$ is a scalar multiple of the Haar measure. This result extends the theorem of G.H. Meisters and W.M. Schmidt (J. Funct. Anal. 13 (1972), 407-424) on $L^2\left(G\right)$. Our method permits in fact to consider any superreflexive translation invariant Banach lattice on $G$, which is the adopted point of view. We study the representation of an element $f$ of this invariant lattice...

By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into $L^{2l}$ of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.

Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that $T\otimes I{d}_X$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^p\left(G\right)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions...

L’espace $P{F}_p\left(G\right)$ des $p$-pseudofonctions sur un groupe localement compact $G$ est le complété de $L_1\left(G\right)$ pour la norme de convoluteur de $L_p\left(G\right)$. Dans le cas où le groupe $G$ est moyennable alors le banach dual à $P{F}_p\left(G\right)$ s’identifie avec une certaine algèbre $B_p\left(G\right)$ de fonctions continues sur $G$. L’algèbre $B_p\left(G\right)$ est déjà connue mais ici on montre que $B_p$ est un foncteur de groupes localement compacts. Pour $p=2$ alors $P{F}_2\left(G\right)$ est l’algèbre $C^{*}$ de $G$ dont le dual est $FS\left(G\right)$, l’algèbre de transformées de Fourier-Stieltjes. Donc, pour un groupe moyennable, élément...

We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x ↦ x̂ are: (i) Is $K_{\nu }=sup{\left|\right|(e-x)}^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,max\left|x̂\right|\le \nu$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_{\delta }=sup\left|\right|x^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,min\left|x̂\right|\ge \delta$ bounded? Both questions are related to a “uniform spectral radius” of the algebra, $r_{\infty }\left(A\right)$, introduced by Björk. Question (i) has an affirmative answer if and only if $r_{\infty }\left(A\right)<1$, and this result is extended to more general nonlinear extremal problems...