Dans cet article, on considère les opérateurs différentiels $T=b\left(x\right)D\left(a\right(x\left)D\right)$, où $a\left(x\right)$ et $b\left(x\right)$ sont deux fonctions mesurables, bornées et accrétives, et $D=-i\frac{d}{dx}$. Les résultats principaux portent sur les propriétés fonctionnelles de $T$, de sa racine carrée, avec applications à l’équation elliptique ${\partial }_t^{2}u-Tu=0$ sur $ℝ\times [0,+\infty [$. On démontre que $T^{1/2}{D}^{-1}$ est un opérateur de Calderón-Zygmund qui dépend analytiquement du couple $(a,b)$. Les estimations ponctuelles optimales sur le noyau du semi-groupe $\exp (-t{L}^{1/2})$ et le calcul fonctionnel permettent de développer une théorie...

In this paper we use the Calderón-Zygmund operator theory to prove a Calderón type reproducing formula associated with a para-accretive function. Using our Calderón-type reproducing formula we introduce a new class of the Besov and Triebel-Lizorkin spaces and prove a Tb theorem for these new spaces.

We study Calderón-Zygmund operators acting on generalized Carleson measure spaces $CM{O}_r^{\alpha ,q}$ and show a necessary and sufficient condition for their boundedness. The spaces $CM{O}_r^{\alpha ,q}$ are a generalization of BMO, and can be regarded as the duals of homogeneous Triebel-Lizorkin spaces as well.

We study sufficient conditions on the weight w, in terms of membership in the $A_p$ classes, for the spline wavelet systems to be unconditional bases of the weighted space $H^p\left(w\right)$. The main tool to obtain these results is a very simple theory of regular Calderón-Zygmund operators.

We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures $\mu$ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures $\mu$ for which the Sobolev space $W^{2,p}$ can be imbedded into $L^p\left(\mu \right)$. The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the theory of singular...

Le problème de Painlevé consiste à trouver une caractérisation géométrique des sous-ensembles du plan complexe qui sont effaçables pour les fonctions holomorphes bornées. Ce problème d’analyse complexe a connu ces dernières années des avancées étonnantes, essentiellement grâce au dévelopement de techniques fines d’analyse réelle et de théorie de la mesure géométrique. Dans cet exposé, nous allons présenter et discuter une solution proposée par X. Tolsa en termes de courbure de Menger au problème...

We investigate the construction of Carleson measures from families of multilinear integral operators applied to tuples of $L^{\infty }$ and BMO functions. We show that if the family $R_t$ of multilinear operators has cancellation in each variable, then for BMO functions b₁, ..., bₘ, the measure $|R_t(b₁,...,bₘ)\left(x\right)|²dxdt/t$ is Carleson. However, if the family of multilinear operators has cancellation in all variables combined, this result is still valid if $b_j$ are $L^{\infty }$ functions, but it may fail if $b_j$ are unbounded BMO functions, as we indicate...

We study weighted mixed norm spaces of harmonic functions defined on smoothly bounded domains in ${ℝ}^n$. Our principal result is a characterization of Carleson measures for these spaces. First, we obtain an equivalence of norms on these spaces. Then we give a necessary and sufficient condition for the embedding of the weighted harmonic mixed norm space into the corresponding mixed norm space.

The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation...

Carleson's Theorem from 1965 states that the partial Fourier sums of a square integrable function converge pointwise. We prove an equivalent statement on the real line, following the method developed by the author and C. Thiele. This theorem, and the proof presented, is at the center of an emerging theory which complements the statement and proof of Carleson's theorem. An outline of these variations is also given.

It is shown that the operator below maps $L^p$ into itself for 1 < p < ∞. $Cf\left(x\right):=su{p}_{a,b}|p.v.\int f(x-y)e^{i(ay²+by)}dy/y|$. The supremum over b alone gives the famous theorem of L. Carleson [2] on the pointwise convergence of Fourier series. The supremum over a alone is an observation of E. M. Stein [12]. The method of proof builds upon Stein’s observation and an approach to Carleson’s theorem jointly developed by the author and C. M. Thiele [7].