Operators of harmonic analysis in weighted spaces with non-standard growth

Last years there was increasing an interest to the so-called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues.     

含参数加权极大Lebesgue空间中次线性算子的有界性质

引进一类含参数加权极大Lebesgue空间并得到满足一定尺寸条件的次线性算子在该类空间中的有界性质.特别地,还考虑了该类空间上次线性算子与BMO函数生成交换子的相应有界性质.     

Corrections to the paper “The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces”

In this paper the author proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with variable exponent. As an application he proved the boundedness of certain sublinear operators on the weighted variable Lebesgue space. The proof of the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent does not contain any mistakes. But in the proof of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in my paper, which was published as R.A.Bandaliev, The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces, Czech. Math. J. 60 (2010), 327–337.     

The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces

The main purpose of this paper is to prove the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space.     

Integral operators with homogeneous kernels in grand Lebesgue spaces

Sufficient conditions on the kernel and the grandizer that ensure the boundedness of integral operators with homogeneous kernels in grand Lebesgue spaces on ? ⁿ as well as an upper bound for their norms are obtained. For some classes of grandizers, necessary conditions and lower bounds for the norm of these operators are also obtained. In the case of a radial kernel, stronger estimates are established in terms of one-dimensional grand norms of spherical means of the function. A sufficient condition for the boundedness of the operator with homogeneous kernel in classical Lebesgue spaces with arbitrary radial weight is obtained. As an application, boundedness in grand spaces of the one-dimensional operator of fractional Riemann–Liouville integration and of a multidimensional Hilbert-type operator is studied.     

Continuity and Schatten–von Neumann Properties for Localization Operators on Modulation Spaces

We use sharp convolution estimates for weighted Lebesgue and modulation spaces to obtain an extension of the celebrated Cordero-Gröchenig theorems on boundedness and Schatten–von Neumann properties of localization operators on modulation spaces. We also give a new proof of the Weyl connection based on the kernel theorem for Gelfand–Shilov spaces.     

Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-H¨older continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.     

R~n上加权弱Hardy空间中的Calderón-Zygmund型算子

作者引进了某些 Ｃａｌｄｅｒóｎ－Ｚｙｇｍｕｎｄ型算子,并且讨论了它们在加权 Ｌｅｂｅｓｇｕｅ空间、加权弱Ｌｅｂｅｓｇｕｅ空间、加权Ｈａｒｄｙ空间和加权弱Ｈａｒｄｙ空间上的有界性．作者也考察了一些结果的尖锐性．     

Sawyer's duality principle for grand Lebesgue spaces

The aim of this paper is to extend Sawyer's duality principle from the cone of decreasing functions of the Lebesgue space to the cone of decreasing functions of the grand Lebesgue space and to prove the boundedness of classical Hardy operators in the grand Lebesgue spaces.     

Integral operators on fully measurable weighted grand Lebesgue spaces

In this paper we introduce the weighted version of fully measurable grand Lebesgue spaces and obtain characterizations for the boundedness of maximal operator, Hilbert transform and the Hardy averaging operator on these spaces.     

带非卷积核的分数次振荡积分算子及其交换子的加权有界性质——献给陆善镇教授75华诞

振荡积分算子的有界性质是调和分析研究的中心内容之一.本文建立一类由Ricci和Stein定义的带非卷积核的分数次振荡积分算子在加权Lebesgue空间中的有界性质.特别地,结合复分析和数学归纳等方法得到该类算子和有界平均振幅（BMO）函数生成交换子的加权有界性质.     

The boundedness of multidimensional hardy operators in weighted variable Lebesgue spaces

The main purpose of this paper is to prove a two-weight criterion for the multidimensional Hardy-type operator in weighted Lebesgue spaces with variable exponent. As an application, we prove the boundedness of Riesz potential and fractional maximal operators on the weighted variable Lebesgue space.     

Weighted estimates for parametrized Littlewood-Paley operators

In this paper, we obtain some boundedness on the weighted Lebesgue spaces and the weighted Hardy spaces for the parametrized Littlewood-Paley operators with the kernel Ω satisfying the logarithmic type Lipschitz conditions.     

Calderón-Zygmund-Type Operators on Weighted Weak Hardy Spaces over ℝn

We introduce certain Calderón-Zygmund-type operators and discuss their boundedness on spaces such as weighted Lebesgue spaces, weighted weak Lebesgue spaces, weighted Hardy spaces and weighted weak Hardy spaces. The sharpness of some results is also investigated. Received December 1, 1998, Revised May 2, 1999, Accepted May 19, 1999     

Beyond Local Maximal Operators

We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces are derived as consequences of our main result. We also apply our boundedness results by studying both generalized neighbourhood capacities and the Lebesgue differentiation of fractional weighted Sobolev 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.     

Boundedness and compactness in weighted Lebesgue spaces of integral operators with variable integration limits

Considering the integral operators with nonnegative kernels and variable integration limits, we obtain criteria of boundedness and compactness in weighted Lebesgue spaces under some conditions on the kernels that are weaker than those studied before.     

Boundedness and compactness of a class of convolution integral operators of fractional integration type

For a class of convolution integral operators whose kernels may have integrable singularities, boundedness and compactness criteria in weighted Lebesgue spaces are obtained.     

广义Morrey空间上的奇异积分多线性交换子

本文得到了具有混合齐次变量核的奇异积分算子的多线性交换子在广义Morrey空间和加权 Lebesgue空间上的有界性．     

Weighted Estimates for the Integral Operators with Monotone Kernel on a Half-Axis

We give criteria for the boundedness of integral operators with nonnegative monotone kernel in the weighted Lebesgue spaces on a half-axis.