Extrapolation from A∞ weights and applications

We generalize the A_p extrapolation theorem of Rubio de Francia to A_∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good-λ inequalities for proving L^p norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calderón-Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman's inequality relating singular integrals and the maximal operator, of the Fefferman-Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt-Wheeden inequality relating the fractional integral operator and the fractional maximal operator.     

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.     

Sufficient conditions for one-sided operators

In this article we give sufficient conditions on a pair of weight (w, v) for some one-sided operators to be bounded from L^p (v^p) to L^p (w^p). The operators we deal with include the one-sided fractional maximal operator and the one-sided singular integrals. For the first operator, necessary and sufficient conditions are known (see [8, 6]). These conditions usually amount to checking the boundedness of the operator on functions that are powers of the weights and are hard to check. Our conditions are of A_p type and are therefore easy to verify. Similar results for two-sided operators were obtained by C. Pérez in [9] and [10].     

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     

Singular Radon transforms along odd curves on the Heisenberg group

In this article we study the singular integral operators along the curve on the Heisenberg group. It is a variable coefficient extension of the singular integrals along the odd curves on the Euclidean space ℝ². The proof is based on the generalized Calderon-Zygmund theory on the space of homogeneous type.     

Decay of the Fourier transform of surfaces with vanishing curvature

We prove L ^p -bounds on the Fourier transform of measures μ supported on two dimensional surfaces. Our method allows to consider surfaces whose Gauss curvature vanishes on a one-dimensional submanifold. Under a certain non-degeneracy condition, we prove that \({\hat{\mu}\in L^{4+\beta}}\) , β > 0, and we give a logarithmically divergent bound on the L ⁴-norm. We use this latter bound to estimate almost singular integrals involving the dispersion relation, \({e(p)= \sum_1^3 [1-\cos p_j]}\) , of the discrete Laplace operator on the cubic lattice. We briefly explain our motivation for this bound originating in the theory of random Schrödinger operators.     

Bilinear oscillatory integrals and boundedness for new bilinear multipliers

We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.     

Fractional Integral Operators on Anisotropic Hardy Spaces

In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and . First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H ^p to H ^q , or from H ^p to L ^q , where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals or Calderón-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function. Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

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     

A new approach to L p estimates for Calderón-Zygmund type singular integrals

In this paper we give a new proof for L^p estimates of the Calderón-Zygmund type singular integrals. Our approach is completely free from harmonic analysis. This work was partially supported by the innovation foundation of Shanghai university under Grant A10-0101-08-905.     

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces Lp₁,…,pN(Rn₁×?×RnN;X) where X is a UMD Banach space satisfying Pisier's property (α). These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.     

The estimates of Bessel integrals with oscillatory factors

In this paper, we establish some sharp estimates of Bessel integrals with oscillatory factors . As an application, we obtain the boundedness of the oscillatory singular integral operators with variable kernels.     

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.     

Rough singular integrals and maximal operators with mixed homogeneity along compound curves

The parabolic singular integrals along certain compound curves as well as the related maximal operators are considered. Under rather weakened size conditions on the integral kernels both on the unit sphere and in the radial direction, the ‐mapping properties for such operators are established. Some previous results are greatly extended and improved.     

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.     

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].     

Hypersingular integral operators along surfaces

In this note, we estimate the boundedness for singular integral operators along curves and surfaces with highly singular kernels.     

Some Rough Operators on Product Spaces

In this survey report, we shall mainly summarize some recent progress, interesting problems and typical methods used in the theory related to rough Marcinkiewicz integrals and rough singular integrals on product spaces. In addition, we give new proofs for some known results.     

Some Rough Operators on Product Spaces

In this survey report, we shall mainly summarize some recent progress, interesting problems and typical methods used in the theory related to rough Marcinkiewicz integrals and rough singular integrals on product spaces. In addition, we give new proofs for some known results.     

Boundedness of singular integrals of variable rough Calderón-Zygmund kernels along surfaces

Letn2. The authors establish theL ²( ⁿ )-boundedness of singular integrals with variable rough Calderón-Zygmund kernels associated to surfaces satisfying some conditions.The research is supported in part by the NNSF and the SEDF of China.