In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces ${\mathscr{H}}_{q,\alpha }^{p,+}\left(\omega \right)$ for $0<p\le 1$, $0<\alpha <\infty$ and $1<q<\infty$. Specifically, we show that, for suitable values of $p,q,\gamma ,\alpha$ and $s$, if $\omega \in A_s^{+}$ (Sawyer’s classes of weights) then the one-sided fractional integral $I_{\gamma }^{+}$ can be extended to a bounded operator from ${\mathscr{H}}_{q,\alpha }^{p,+}\left(\omega \right)$ to ${\mathscr{H}}_{q,\alpha +\gamma }^{p,+}\left(\omega \right)$. The result is a consequence of the pointwise inequality $$N_{q,\alpha +\gamma }^{+}\left(I_{\gamma }^{+}F;x\right)\le C_{\alpha ,\gamma }{N}_{q,\alpha }^{+}\left(F;x\right),$$ where $N_{q,\alpha }^{+}(F;x)$ denotes the Calderón maximal function.

Let w be in the Muckenhoupt $A_{\infty }$ weight class. We show that the Riesz transforms are bounded on the weighted Carleson measure space $CM{O}_w^p$, the dual of the weighted Hardy space $H_w^p$, 0 < p ≤ 1.

The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set $z{\zeta }^{-1}:z,\zeta \in ,z\ne \zeta$, where is a star-shaped Lipschitz curve, $=exp\left(iz\right):z=x+iA\left(x\right),A^{\text{'}}\in L^{\infty }[-\pi ,\pi ],A(-\pi )=A\left(\pi \right)$. Under suitable conditions on F and z, the operators are given by (1) $TF\left(z\right)=p.v.{\int }_{(}z{\eta }^{-1})F\left(\eta \right)(d\eta /\eta ).$ We identify a class of kernels of the stated type that give rise to bounded operators on $L^2(,|d\left|\right)$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.

Some boundedness results are established for sublinear operators on the homogeneous Herz spaces. As applications, some new theorems about the boundedness on homogeneous Herz spaces for commutators of singular integral operators are obtained.

In this paper, the boundedness properties for some Toeplitz type operators associated to the Riesz potential and general integral operators from Lebesgue spaces to Orlicz spaces are proved. The general integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

The main purpose of this paper is to investigate the behavior of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on Euclidean space and their known extensions for doubling...

Let $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${ℝ}^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p\&gt;2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {\left(ℒ\right)}^{-1/2}$ on the $L^p$ space. As an application, for $1\&lt;p\&lt;3+ϵ$, we establish the $L^p$ boundedness of Riesz transforms on Lipschitz domains for operators with $VMO$ coefficients. The range of $p$ is sharp. The closely related boundedness of ...

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...