Let f ∈ L∞ and g ∈ L2 be supported on [0,1]. Then the principal value integral below exists in L1.p.v.   ∫ f(x + y) g(x - y) dy / y.

Calderón-Zygmund operators are generalizations of the singular integral operators introduced by Calderón and Zygmund in the fifties [CZ]. These singular integrals are principal value convolutions of the formTf(x) = límε→0 ∫|x-y|>ε K(x-y) f(y) dy = p.v.K * f(x),where f belongs to some class of test functions.

The main purpose of this paper is to prove the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space.

The boundednees of multilinear commutators of Calderón-Zygmund singular integrals on Lebesgue spaces with variable exponent is obtained. The multilinear commutators of generalized Hardy-Littlewood maximal operator are also considered.

We first show that a linear operator which is bounded on $L{²}_w$ with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space $h{¹}_w$ if and only if this operator is uniformly bounded on all $h{¹}_w$-atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to $h{¹}_w$.

The most important results of standard Calderón-Zygmund theory have recently been extended to very general non-homogeneous contexts. In this survey paper we describe these extensions and their striking applications to removability problems for bounded analytic functions. We also discuss some of the techniques that allow us to dispense with the doubling condition in dealing with singular integrals. Special attention is paid to the Cauchy Integral.[Proceedings of the 6th International Conference on...

We deal with the Hardy-Lorentz spaces $H^{p,q}\left(ℝⁿ\right)$ where 0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.

The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator $L:=d^2/d{x}^2-2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in $L^1\left(d\gamma \right)$, where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.

In this paper we study the Hilbert transform and maximal function related to a curve in R2.

For an L²-bounded Calderón-Zygmund Operator T acting on $L²\left({ℝ}^d\right)$, and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_T{\left|\right|w\left|\right|}_{A₂}$. The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can be exactly...

Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$ which are the local versions of $H^p\left({ℝ}^n\right)$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$ and discuss the $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$-boundedness of Calderón-Zygmund operators. Similar results can also be obtained for the non-homogeneous...

This article is concerned with the question of whether Marcinkiewicz multipliers on ${ℝ}^{2n}$ give rise to bilinear multipliers on ℝⁿ × ℝⁿ. We show that this is not always the case. Moreover, we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.