A littlewood-paley inequality for the Carleson operator

The Carleson operator is closely related to the maximal partial sum operator for Fourier series. We study generalizations of this operator in one and several variables.     

A new proof for the estimates of Calderón-Zygmund type singular integrals

In this paper, a new proof for the estimates of Calderón-Zygmund type singular integrals will be presented. Received: 30 October 2005     

Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals

We present an extrapolation theory that allows us to obtain, from weighted L^p inequalities on pairs of functions for p fixed and all A_∞ weights, estimates for the same pairs on very general rearrangement invariant quasi-Banach function spaces with A_∞ weights and also modular inequalities with A_∞ weights. Vector-valued inequalities are obtained automatically, without the need of a Banach-valued theory. This provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators. In particular, we obtain weighted, and vector-valued, extensions of the classical theorems of Boyd and Lorentz-Shimogaki. The key is to develop appropriate versions of Rubio de Francia's algorithm.     

A weak estimate of Calderón-Zygmund operators

We obtain a weak type theorem of Calderón-Zygmund operators in the Hardy space.     

Criteria of strong type two-weighted inequalities for fractional maximal functions

A strong type two-weight problem is solved for fractional maximal functions defined in homogeneous type general spaces. A similar problem is also solved for one-sided fractional maximal functions.     

On modular inequalities in variable L p spaces

We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type modular inequality in variable L^p spaces if and only if the variable exponent p(x) ∼ const. Received: 15 September 2004     

Characterization of BMO in terms of rearrangement-invariant Banach function spaces

The atomic decomposition of Hardy spaces by atoms defined by rearrangement-invariant Banach function spaces is proved in this paper. Using this decomposition, we obtain the characterizations of BMO and Lipschitz spaces by rearrangement-invariant Banach function spaces. We also provide the sharp function characterization of the rearrangement-invariant Banach function spaces.     

Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces

The necessary and sufficient conditions are derived in order that a strong type weighted inequality be fulfilled in Orlicz classes for scalar and vector-valued maximal functions defined on homogeneous type spaces. A weak type problem with weights is solved for vector-valued maximal functions.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Boundedness of Marcinkiewicz Integral

A weighted norm inequality for the Marcinkiewicz integral operator is proved when belongs to . We also give the weighted L^p-boundedness for a class of Marcinkiewicz integral operators with rough kernels and related to the Littlewood-Paley -function and the area integral S, respectively.     

A necessary condition for Calderón-Zygmund singular integral operators

Calderón-Zygmund singular integral operators have been extensively studied for almost half a century. This paper provides a context for and proof of the following result: If a Calderón-Zygmund convolution singular integral operator is bounded on the Hardy space H¹ (Rⁿ), then the homogeneous of degree zero kernel is in the Hardy space H¹(S^n–1) on the sphere.     

Marcinkiewicz integral on hardy spaces

In this paper we prove that the Marcinkiewicz integral is an operator of type (H ¹,L ¹) and of type (H ^1,,L ^1,). As a corollary of the results above, we obtain again the the weak type (1,1) boundedness of , but the smoothness condition assumed on is weaker than Stein's condition.The research was supported partly by Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.The author was supported partly by NSF of China (Grant No. 19971010).The author was supported partly by NSF of China (Grant No. 19131080).     

Marcinkiewicz integral on weighted Hardy spaces

In this paper we give sufficient conditions to imply the $H^{1}_{w}-L^{1}_{w}$ boundedness of the Marcinkiewicz integral operator $\mu_\Omega$, where w is a Muckenhoupt weight. We also prove that, under the stronger condition $\Omega \in {\rm Lip}_\alpha$, the operator $\mu_\Omega$ is bounded from $H^{p}_{w}$ to $L^{p}_{w}$ for $\max\{n/(n+1/2), n/(n+\alpha)\}$ < p < 1.     

The boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution

In this paper, we will consider the boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution. In order to get our result, we need to give some characterizations of the Hardy space associated with twisted convolution. Including Lusin area integral, Littlewood-Paley g-function.     

Inequalities related toH p smoothness of Sobolev type

By exploiting a class of maximal functions and Littlewood-Paley theory, a list of embedding inequalities onH ^p-Sobolev spaces andH ^p boundedness results for Riesz and Bessel potentials are obtained at one stroke.This work was supported in part by the Chung-Ang University Academic Research Special Grants, 1997.     

Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. II

This paper continues the investigation of weight problems in Orlicz classes for maximal functions and singular integrals defined on homogeneous type spaces considered in [1].     

Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. I

Criteria of various weak and strong type weighted inequalities are established for singular integrals and maximal functions defined on homogeneous type spaces in the Orlicz classes.     

Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.Research of the first author was supported in part by NSERC grant A5149.Research of the second author was supported in part by NSF grant DMS93-02991.     

Two-weightedL1 p -inequalities for singular integral operators on Heisenberg groups

Some sufficient conditions are found for a pair of weight functions, providing the validity of two-weighted inequalities for singular integrals defined on Heisenberg groups.