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.     

Weighted sharp boundedness for multilinear commutators

We obtain sharp estimates for some multilinear commutators related to certain sublinear integral operators. These operators include the Littlewood-Paley operator and Marcinkiewicz operator. As an application, we obtain weighted L ^p (p > 1) inequalities and an L log L-type estimate for multilinear commutators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1419–1431, October, 2007.     

Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderón-Zygmund singular integrals

This paper concerns with multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. The Cotlar-type inequality of maximal vector-valued multilinear singular integrals operator is obtained. On the other hand, pointwise estimates for sharp maximal function of two kinds of maximal vector-valued multilinear singular integrals and maximal vector-valued commutators are also established. By the weighted estimates of a class of new variant maximal operator, Cotlar's inequality and the sharp maximal function estimates, multiple weighted strong estimates and weak estimates for maximal vector-valued singular integrals of multilinear operators and those for maximal vector-valued commutator of multilinear singular integrals are obtained.     

Embeddings of Lorentz spaces of vector-valued martingales

We obtain new embedding theorems for Lorentz spaces of vector-valued martingales, thus generalizing the classical martingale inequalities. In contrast to earlier methods, we use martingale transformations defined by sequences of operators and identify the operator S ^(p)(f) for a martingale f ranging in a Banach space X with the maximal operator for some ℓ ^p (X)-valued martingale transform. The obtained inequalities are closely related to geometric properties of the Banach space in question.     

Vector-valued weak martingale Hardy spaces and atomic decompositions

Some atomic decomposition theorems are proved in vector-valued weak martingale Hardy spaces w _p Σ_α(X), w _p Q _α(X) and wD _α(X). As applications of atomic decompositions, a sufficient condition for sublinear operators defined on some vector-valued weak martingale Hardy spaces to be bounded is given. In particular, some weak versions of martingale inequalities for the operators f*, S ^(p)(f) and σ^(p)(f) are obtained. This research was supported by the National Science Foundation of China (No. 10371093).     

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.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Two‐weight norm inequalities for multilinear potential type integral operators

We give a condition which is sufficient for the two‐weight (p, q) inequalities for multilinear potential type integral operators, where 1 < p ≤ q < ∞. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Domination spaces and factorization of linear and multilinear summing operators

It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1,?. . .?, pn)-dominated multilinear operators and dominated (p1,?. . .?, pn; σ)-continuous multilinear operators.     

L^p（R^n） Boundedness for the Maximal Operator of Multilinear Singular Integrals with Calderon-Zygmund Kernels

In this paper, the authors consider a class of maximal multilinear singular integral operators and maximal multilinear oscillatory singular integral operators with standard Calderón–Zygmund kernels, and obtain their boundedness on L ^p (ℝ ⁿ ) for 1 < p < ∞. Research supported by Professor Xu Yuesheng’s Research Grant in the program of "One hundred Distinguished Young Scientists" of the Chinese Academy of Sciences     

Boundedness of vector-valued operators on weighted Herz spaces

In this paper, the authors establish the weighted weak and strong type norm inequalities in the set of weighted Herz-type spaces for a vector-valued analogue of the Hardy-Littlewood maximal operator; and using this, the authors obtain the weightd inequalies for a wide class of sublinear singular operators defined onR ⁿ which include the Calderón-Zygmund operators as special cases. The fractional versions of these results are also given.     

Remarks on Vector-valued BMOA and Vector-valued Multipliers

In this paper we consider the vector-valued interpretation of the space BMOA defined in terms of Carleson measures and analyze the relationship with the one defined in terms of oscillation. We study the space of multipliers between H ^p and BMOA in the vector-valued setting. This leads us to the consideration of some geometric properties depending upon the validity of certain inequalities due to Littlewood and Paley on the g-function for vector-valued functions.     

Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels

This paper is concerned with certain multilinear commutators of BMO functions and multilinear singular integral operators with non-smooth kernels. By the sharp maximal functions estimates, the weighted norm inequalities for this kind of commutators are established.     

Weighted inequalities for multilinear commutators of some sublinear operators

We establish sharp estimates for some multilinear commutators related to the Littlewood-Paley and Marcinkiewicz operators. As an application, we obtain the weighted norm inequalities and L log L type estimate for the multilinear commutators.      

Weighted Inequalities for Multilinear Potential Operators and their Commutators

We prove weighted strong inequalities for the multilinear potential operator T_f{\cal T}_{\phi} and its commutator, where the kernel ϕ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type estimates. Moreover we prove weighted weak type inequalities for the multilinear maximal operator M_j,L^B\mathcal{M}_{\varphi,L^{B}} associated to an essentially nondecreasing function φ and to the Orlicz space L ^B for a given Young function B. This result allows us to obtain a weighted weak type inequality for the operator T_f{\cal T}_{\phi}.     

Littlewood–Paley Operators on Spaces with Variable Exponent on Homogeneous Groups

In this paper,on homogeneous groups,we study the Littlewood–Paley operators in variable exponent spaces.First,we prove that the weighted Littlewood–Paley operators are controlled by the weighted Hardy–Littlewood maximal function,and obtain the vector-valued inequalities of the Littlewood–Paley operators,including the Lusin function,Littlewood–Paley g function and gλ* function.Second,we prove the boundedness of multilinear Littlewood–Paley gψ,λ* function.     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For L in some class of elliptic operators, we study weighted norm Lp inequalities for singular “non-integral” operators arising from L; those are the operators φ(L) for bounded holomorphic functions φ, the Riesz transforms ∇L−1/2 (or (−Δ)^1/2L−1/2) and its inverse L1/2(−Δ)^−1/2, some quadratic functionals gL and GL of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal Lp-regularity. For each, we obtain sharp or nearly sharp ranges of p using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions.     

Extrapolation of Weighted norm inequalities for multivariable operators and applications

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderón-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on ℓ²-valued extensions of bounded linear operators is also obtained.     

Diagonal multilinear operators on Köthe sequence spaces

We analyse the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of (E, p)-summing multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.     

BOUNDEDNESS OF VECTOR-VALUED OPERATORS ON WEIGHTED HERZ SPACES

In this paper, the authors establish the weighted weak and strong type norm inequalities in the set of weighted Herz-type spaces for a vector-valued analogue of the Hardy-Littlewood maximal operator; and using this, the authors obtain the weightd inequalies for a wide class of sublinear singular operators defined onR ⁿ which include the Calderón-Zygmund operators as special cases. The fractional versions of these results are also given. Yang Dachun was partially supported by the NNSF and the SEDF of China and the Grant-in-Aid for Scientific research(11304009), Japan Society for the Promotion of Science.