A simple expression is presented that is equivalent to the norm of the Lpv → Lqu embedding of the cone of quasi-concave functions in the case 0 < q < p < ∞. The result is extended to more general cones and the case q = 1 is used to prove a reduction principle which shows that questions of boundedness of operators on these cones may be reduced to the boundedness of related operators on whole spaces. An equivalent norm for the dual of the Lorentz spaceΓp(v) = { f: ( ∫0∞ (f**)pv...

Let $S\subset {ℝ}^{n+1}$ be the graph of the function $\varphi :{[-1,1]}^n\to ℝ$ defined by $\varphi (x_1,\cdots ,x_n)={\sum }_{j=1}^n{\left|x_j\right|}^{{\beta }_j},$ with 1<${\beta }_1\le \cdots \le {\beta }_n,$ and let $\mu$ the measure on ${ℝ}^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu$ is $L^p$-$L^q$ bounded.

We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on $L^p$ radial functions. This is a consequence of endpoint bounds for a corresponding square function for Hankel multipliers. We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel multipliers and $L^p$ bounds of maximal operators generated by Hankel multipliers as corollaries. The proof is built on techniques developed by Garrigós and Seeger for characterizations of Hankel multipliers....

We establish sharp (H1,L1,q) and local (L logrL,L1,q) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map H1 to L1,∞ and L log1/2L to L1,∞, and that these estimates are sharp.

We investigate the energy of measures (both positive and signed) on compact Riemannian manifolds. A formula is given relating the energy integral of a positive measure with the projections of the measure onto the eigenspaces of the Laplacian. This formula is analogous to the classical formula comparing the energy of a measure in Euclidean space with a weighted L² norm of its Fourier transform. We show that the boundedness of a modified energy integral for signed measures gives bounds on the Hausdorff...

Soit $𝒰$ (resp. ${𝒰}_0$) l’ensemble des compacts d’unicité (resp. d’unicité au sens large) du tore $\mathbf{T}$. On montre qu’un borélien de $\mathbf{T}$ dont tout sous-compact est dans ${𝒰}_0$ est nécessairement contenu dans une réunion dénombrable de compacts de ${𝒰}_0$, et on montre que cette propriété n’est plus vraie quand on remplace ${𝒰}_0$ par $𝒰$.Comme conséquence on obtient qu’un borélien qui est d’unicité est nécessairement maigre. On en déduit aussi l’existence d’un compact d’unicité qui ne peut être recouvert par une suite de compacts...

On étudie les ensembles de Sidon d’un groupe abélien localement compact et métrisable $\Gamma$. Après avoir démontré des résultats sur la réunion, l’élargissement et la stabilité de ces ensembles lacunaires, on détaille le résultat fondamental de ce travail : lorsque le dual $G$ de $\Gamma$ est connexe, toute partie compacte d’intérieur non vide de $G$ est associée à tout ensemble de Sidon de $\Lambda$. Autrement dit, étant donné un compact $K$ d’intérieur non vide de $G$, toute fonction bornée à valeurs complexes définie sur...

On connaît le lien intime qui existe entre les équations fonctionnelles des fonctions $L$ et les formules sommatoires dont le prototype est donné par celle de Poisson. Ce lien fait intervenir la transformation intégrale de Fourier et ses généralisations. Ici, nous réexaminons la signification harmonique (ainsi qu’hilbertienne et distributionnelle) des équations fonctionnelles ayant la forme la plus simple, à savoir, celle s’appliquant pour la fonction dzêta de Riemann et les séries $L$ de Dirichlet...

The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map $t{r}_{\Gamma }:B_{p₁,q}^s(ℝⁿ,w_{\varkappa }^{\Gamma })\to L_{p₂}\left(\Gamma \right)$, where Γ is a d-set with 0 < d < n and $w_{\varkappa }^{\Gamma }$ a weight of type $w_{\varkappa }^{\Gamma }\left(x\right)dist{(x,\Gamma )}^{\varkappa }$ near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.

We investigate weighted inequalities for fractional maximal operators and fractional integral operators.We work within the innovative framework of “entropy bounds” introduced by Treil–Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.

We study the tridimensional Navier-Stokes equation when the value of the vertical viscosity is zero, in a critical space (invariant by the scaling). We shall prove local in time existence of the solution, respectively global in time when the initial data is small compared with the horizontal viscosity.