广义分数次积分算子在加权Herz型Hardy空间的有界性

利用加权Herz型Hardy空间的原子分解理论,讨论了广义分数次积分算子Tl从加权Lp空间到加权Lq空间,以及从加权Herz型Hardy空间到加权Herz空间的有界性问题.将已有的分数次积分算子的结论推广到广义分数次积分算子的情形.     

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.     

Composition operators acting on holomorphic Sobolev spaces

We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

     

Equivalence of Sobolev Spaces

We repair the proof of equivalence of certain L²-Sobolev spaces on manifolds with bounded curvature of all orders from [4]. The results are extended to generalized compatible Dirac operators, fractional order Sobolev spaces and weighted Sobolev spaces. A certain way of doing coordinate free computations is presented.     

A Characterization of Boundedness of Fractional Maximal Operator with Variable Kernel on Herz-Morrey Spaces

A significant number of studies have been carried out on the generalized Lebesgue spaces L~(p(x)), Sobolev spaces W~(1,p(x)) and Herz spaces. In this paper, we demonstrated a characterization of boundedness of the fractional maximal operator with variable kernel on Herz-Morrey spaces.     

Boundedness of generalized fractional integrals in weighted Herz-type spaces

In this paper, the authors investigate the boundedness of the generalized fractional integrals of Pérez on the weighted Herz spaces, the weighted weak Herz spaces and the weighted Herz-type Hardy spaces for general weights.     

Weighted Estimates of Variable Kernel Fractional Integral and Its Commutators on Vanishing Generalized Morrey Spaces with Variable Exponent

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces. And the corresponding commutators generated by BMO function are also considered.     

Morrey spaces and fractional integral operators

The present paper is devoted to the boundedness of fractional integral operators in Morrey spaces defined on quasimetric measure spaces. In particular, Sobolev, trace and weighted inequalities with power weights for potential operators are established. In the case when measure satisfies the doubling condition the derived conditions are simultaneously necessary and sufficient for appropriate inequalities.     

Boundedness of commutators of generalized fractional integral operators on weighted herz-type hardy spaces

In this paper, we study the boundedness of higher order commutators of generalized fractional integral operators on weighted Lp spaces and Herz-type Hardy spaces.     

Boundedness of generalized fractional integrals in weighted Herz-type spaces

In this paper, the authors investigate the boundedness of the generalized fractional integrals of Pérez on the weighted Herz spaces, the weighted weak Herz spaces and the weighted Herz-type Hardy spaces for general weights. This project was partially supported by the SEDF and the NNSF of China.     

Decay for nonlinear Klein-Gordon equations

We study the asymptotic behavior of the semilinear Klein-Gordon equation with nonlinearity of fractional order. By the aid of a suitable generalization of the weighted Sobolev spaces we define the weighted Sobolev spaces on the upper branch of the unit hyperboloid. In these spaces of fractional order we obtain a weighted Sobolev embedding and a nonlinear estimate. Using these, we establish the decay estimate of the solution for large time provided the power of nonlinearity is greater than a critical value.     

The finite Hilbert transform and weighted Sobolev spaces

The boundedness of the finite Hilbert transform operator on certain weighted L_p spaces is well known. We extend this result to give the boundedness of that operator on certain weighted Sobolev spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

θ-型C-Z算子在加权变指数Morrey空间上的有界性

在满足一定的正则性假设条件下,建立了θ-型Calderón-Zygmund算子T_θ在一类变指数Lebesgue空间上的加权有界性.进一步得到了T_θ在加权变指数Herz空间和Herz-Morrey空间上的有界性.另外,还证明了相应的交换子[b,T_θ]在广义加权变指数Morrey空间上是有界的.     

Singular Integral Operators in Generalized Morrey Spaces on Curves in the Complex Plane

We study the boundedness of the Cauchy singular integral operators on curves in complex plane in generalized Morrey spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of singular integral operators in weighted generalized Morrey spaces.     

GLOBALLY ATTRACTING SOLUTIONS TO IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS OF SOBOLEV TYPE

We study a generalized Cauchy problem associated with a class of impulsive fractional differential inclusions of Sobolev type in Banach spaces. Our aim is to prove the existence of a compact set of globally attracting solutions to the problem in question. An application to fractional partial differential equations subject to impulsive effects is given to illustrate our results.     

Generalization of the notion of grand Lebesgue space

We introduce families of weighted grand Lebesgue spaces which generalize weighted grand Lebesgue spaces (known also as Iwaniec-Sbordone spaces). The generalization admits a possibility of expanding usual (weighted) Lebesgue spaces to grand spaces by various ways by means of additional functional parameter. For such generalized grand spaces we prove a theorem on the boundedness of linear operators under the information of their boundedness in ordinary weighted Lebesgue spaces. By means of this theorem we prove boundedness of the Hardy-Littlewood maximal operator and the Calderon-Zygmund singular operators in the weighted grand spaces.     

Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups

In this paper, we study the boundedness of the fractional integral operator I _α on Carnot group G in the generalized Morrey spaces M _{p, φ} (G). We shall give a characterization for the strong and weak type boundedness of I _α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.     

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications.     

Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators

The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed point theorem. Finally, an example is given to illustrate our theoretical results.     

Hardy—Steklov operators and Sobolev-type embedding inequalities

We characterize weighted inequalities corresponding to the embedding of a class of absolutely continuous functions into a fractional-order Sobolev space. As auxiliary results of the paper, which are also of independent interest, we obtain several new types of necessary and sufficient conditions for the boundedness of the Hardy–Steklov operator (integral operator with two variable limits) in weighted Lebesgue spaces.