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).     

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.     

Hausdorff type operators on the power weighted Hardy spaces

In this paper, we study the high‐dimensional Hausdorff type operators and establish their boundedness on the power weighted Hardy spaces for . As a consequence, we obtain that the Hausdorff type operator is bounded on if Φ is the Gauss function, or the Poisson function.     

Boundedness of Marcinkiewicz integral on the weighted Herz-type Hardy spaces

In this paper, we discuss the boundedness of Marcinkiewicz integral μΩ with homogeneous kernel on the weighted Herz-type Hardy spaces, and prove that μΩ is bounded from H K αq ,p ( w1 ; w2 ) into Kαq ,p (w 1; w2).     

On Marcinkiewicz Integral Operators with Rough Kernels

This paper is devoted to the study on the L^p-mapping properties of Marcinkiewicz integral operators with rough kernels along “polynomial curves” on The boundedness of the Marcinkiewicz integrals for some fixed 1 < p < ∞ are obtained under some size conditions, which essentially improve or extend some well-known results.     

Strong maximal operator and singular integral operators in weighted Morrey spaces on product domains

We introduce the Morrey spaces on product domains and extend the boundedness of strong maximal operator and singular integral operators on product domains to Morrey spaces.     

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces.     

Two-weighted estimates for some integral transforms in Lebesgue spaces with mixed norm and imbedding theorems

Two-weighted inequalities are proved for anisotropic potentials. These estimates are used to obtain refinements of the well-known imbedding theorems in the scale of weighted Lebesgue spaces.     

Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces Lφ(Rd). We give a necessary condition for the continuity of M on Lφ(Rd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces Lp(⋅)(Rd) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also Mq is continuous for some q>1, where .     

Wavelet characterization of weighted spaces

We give a characterization of weighted Hardy spaces H ^p (w), valid for a rather large collection of wavelets, 0 <p ≤ 1,and weights w in the Muckenhoupt class A _∞ We improve the previously known results and adopt a systematic point of view based upon the theory of vector-valued Calderón-Zygmund operators. Some consequences of this characterization are also given, like the criterion for a wavelet to give an unconditional basis and a criterion for membership into the space from the size of the wavelet coefficients.     

Weighted estimates for integral operators on local type spaces

We prove the weighted boundedness for a family of integral operators on Lebesgue spaces and local type spaces. To this end we show that can be controlled by the Calderón operator and a local maximal operator. This approach allows us to characterize the power weighted boundedness on Lebesgue spaces.     

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.     

On the existence of principal values for the cauchy integral on weighted lebesgue spaces for non-doubling measures

Let T be a Calderón-Zygmund operator in a “non-homogeneous” space ( , d, μ), where, in particular, the measure μ may be non-doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil, and A. Volberg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T_*, which is the supremum of the truncated operators associated with T. Specifically, for1<p<∞, we obtain sufficient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L^p weighted inequality holds for T_*. The main tool to deal with this problem is the theory of vector-valued inequalities for T_* and some related operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderón-Zygmund operators in non-homogeneous spaces, developed in our previous paper [6]. For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C_* to characterize the existence of principal values for functions in weighted L^p.     

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.     

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.     

Conditions for boundedness into Hardy spaces

We obtain the boundedness from a product of Lebesgue or Hardy spaces into Hardy spaces under suitable cancellation conditions for a large class of multilinear operators that includes the Coifman–Meyer class, sums of products of linear Calderón–Zygmund operators and combinations of these two types.     

Convolutions, multipliers and maximal regularity on vector-valued Hardy spaces

We extend the recently developed L^p-theory for the maximal regularity of the abstract Cauchy problem and the related Fourier multiplier techniques to the real-variable Hardy space H¹. Some results for H^p, 0 < p < 1, are also proved.     

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.     

Boundedness of parammetric Marcinkiewicz integrals on weighted Hardy spaces

In this paper, several new results on the boundedness of parammetric Marcinkiewicz integrals on the weighted Hardy spaces and the weak weighted Hardy spaces are established.