Potential Operators,Maximal Functions,and Generalizations of A ∞

We derive weighted norm estimates which relate integral operators of potential type (fractional integrals) to corresponding maximal operators (fractional maximal operators). We also derive norm estimates for the maximal operators. The conditions that we impose on the weights involve A conditions of content type which are weaker than the usual A condition. The analysis is carried out in the context of spaces of homogeneous type.     

Endp oint Estimates for Commutators of Fractional Integrals Asso ciated to Op erators with Heat Kernel Bounds

Let L be the infinitesimal generator of an analytic semigroup on L~2(R~n)with pointwise upper bounds on heat kernel,and denote by L~(-α/2)the fractional integrals of L.For a BMO function b(x),we show a weak type Llog L estimate of the commutators [b,L~(-α/2)](f)(x) = b(x)L~(-α/2)(f)(x)-L~(-α/2)(bf)(x).We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.     

Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting

We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on IR+. which are closely related to the best constants of the weak type (1,1) estimates for such operators. Meanwhile, the corresponding results for Hardy-Littlewood maximal operator and fractional maximal operator in Bessel setting are also obtained.     

Regular Maximal Monotone Operators

The purpose of this paper is to introduce a class of maximal monotone operators on Banach spaces that contains all maximal monotone operators on reflexive spaces, all subdifferential operators of proper, lsc, convex functions, and, more generally, all maximal monotone operators that verify the simplest possible sum theorem. Dually strongly maximal monotone operators are also contained in this class. We shall prove that if T is an operator in this class, then (the norm closure of its domain) is convex, the interior of co(dom(T)) (the convex hull of the domain of T) is exactly the set of all points of at which T is locally bounded, and T is maximal monotone locally, as well as other results.     

Weighted endpoint estimates for maximal commutators associated with the sections

We prove endpoint estimates for maximal commutators for a class of singular integral operators related to the real analysis of the Monge Ampere equation.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates of rough maximal functions along surfaces with applications

In this paper, we study the L^p mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the L^p boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(S^n-1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.     

各向异性极大算子及它们的应用

本文证明了一类与异性伸缩相关的极大算子在Ｌ＾ｐ（Ｒ＾ｎ）上是有界的，其中１〈ｐ≤∝，ｎ≥２。进一步，作为应用我们建立了算子Ｔｆ（ｘ）＝ｓｕｐε〉０│Ｔεｆ（ｘ）│Ｌ＾ｐ有界性，其中Ｔε是带有变化核的奇异积分。     

Commuting Dual Toeplitz Operators on the Polydisk

On the polydisk, the commutativity of dual Toeplitz operators is studied. We obtain characterizations of commuting dual Toeplitz operators, essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators.     

Generalized weighted Morrey spaces and classical operators

We introduce the notion of generalized weighted Morrey spaces and investigate the boundedness of some operators in these spaces, such as the Hardy–Littlewood maximal operator, generalized fractional maximal operators, generalized fractional integral operators, and singular integral operators. We also study their boundedness in the vector‐valued setting.     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Alternative characterisations of Lorentz-Karamata spaces

We present new formulae providing equivalent quasi-norms on Lorentz-Karamata spaces. Our results are based on properties of certain averaging operators on the cone of non-negative and non-increasing functions in convenient weighted Lebesgue spaces. We also illustrate connections between our results and mapping properties of such classical operators as the fractional maximal operator and the Riesz potential (and their variants) on the Lorentz-Karamata spaces.     

Extreme preservers of maximal column rank inequalities of matrix sums over semirings

We characterize linear operators that preserve sets of matrix ordered pairs which satisfy extreme properties with respect to maximal column rank inequalities of matrix sums over semirings. This work was supported by the research grant of the Cheju National University in 2007.     

Two-weight norm inequalities for fractional maximal functions and fractional integral operators on weighted Morrey spaces

In this paper, we consider weighted norm inequalities for fractional maximal operators and fractional integral operators. For suitable weights, we prove the two-weight norm inequalities for both operators on weighted Morrey spaces.     

Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals

In this paper, we study weighted inequalities for multilinear fractional maximal operators and fractional integrals. We prove sharp weighted Lebesgue space estimates for both operators when the vector of weights belongs to . In addition we prove sharp two weight mixed estimates for multilinear operators in the spirit of the linear estimates given in 3 .     

WEIGHTED ESTIMATES FOR COMMUTATORS OF POTENTIAL OPERATORS ON SPACES OF HOMOGENEOUS TYPE

We derive some strong type and weak type weighted norm estimates which relate the commutators of potential integral operators to the corresponding maximal operators in the context of spaces of homogeneous type.     

有界局部紧Vilenkin群上的广义Calderón-Zygmund算子(英文)

记Ｇ为一有界局部紧的Ｖｉｌｅｎｋｉｎ群．我们引进了Ｇ上的一类Ｃａｌｄｅｒｏｎ－Ｚｙｇｍｕｎｄ型算子,并且证明了它们的加权ＬＰ（Ｇ）有界性．另外还给出了这些结果的一些应用．     

M ‐elliptic pseudo‐differential operators on L p (ℝn )

We construct the minimal and maximal extensions in L ^p (?ⁿ ), 1 < p < ∞, for M ‐elliptic pseudo‐differential operators initiated by Garello and Morando. We prove that they are equal and determine the domains of the minimal, and hence maximal, extensions of M ‐elliptic pseudo‐differential operators. For M ‐elliptic pseudodifferential operators with constant coefficients, the spectra and essential spectra are computed. An application to quantization is given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Calderón–Zygmund Operators in the Bessel Setting for All Possible Type Indices

We show that many harmonic analysis operators in the Bessel setting,including maximal operators,Littlewood–Paley–Stein type square functions,multipliers of Laplace or Laplace–Stieltjes transform type and Riesz transforms are,or can be viewed as,Calderón–Zygmund operators for all possible values of type parameter λ in this context.This extends results existing in the literature,but being justified only for a restricted range of λ.     

M -Hypoelliptic pseudo-differential operators on Lp (ℝ n )

Following Wong's point of view, we construct the minimal and maximal extension in L^p (? ⁿ ), 1 < p < ∞ for M-hypoelliptic pseudo-differential operators, which have been introduced and studied by Garello and Morando. We give some facts about the domain of minimal and maximal extensions of M-hypoelliptic pseudo-differential operators. For M-hypoelliptic pseudo-differential operators with constant coefficients, the spectrum and essential spectrum are computed.