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.
The boundednees of multilinear commutators of Calderón-Zygmund singular integrals on Lebesgue spaces with variable exponent is obtained. The multilinear commutators of generalized Hardy-Littlewood maximal operator are also considered.
We discuss recent progress on issues surrounding the Brascamp–Lieb inequalities.
A stronger version of almost uniform convergence in von Neumann algebras is introduced. This "bundle convergence" is additive and the limit is unique. Some extensions of classical limit theorems are obtained.
We investigate the relations between the Campanato, Morrey and Hölder spaces on spaces of homogeneous type and extend the results of Campanato, Mayers, and Macías and Segovia. The results are new even for the ℝⁿ case. Let (X,d,μ) be a space of homogeneous type and (X,δ,μ) its normalized space in the sense of Macías and Segovia. We also study the relations of these function spaces for (X,d,μ) and for (X,δ,μ). Using these relations, we can show that theorems for the Campanato, Morrey or Hölder spaces...
In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index s > 2 to obtain local solutions for any initial data, and global solutions for small initial data.
In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.
The boundedness properties of commutators for operators are of central importance in Mathematical Analysis, and some of these commutators arise in a natural way from interpolation theory. Our aim is to present a general abstract method to prove the boundedness of the commutator for linear operators and certain unbounded operators that appear in interpolation theory, previously known and a priori unrelated for both real and complex interpolation methods, and also to show how the abstract result...
We first show that a linear operator which is bounded on with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space if and only if this operator is uniformly bounded on all -atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to .