Commutators of singular integrals on generalized $L^p$ spaces with variable exponent

Abstract

Abstract. A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized L p spaces with variable exponent.

Remarks on eigenvalue problems involving the p(x)-Laplacian

“…In recent years many researchers (see e.g. [3][4][5][6][7]10,15,19,22,24,25,33]) have studied the basic properties of variable exponent Lebesgue-Sobolev spaces and the boundedness of some classical operators in the variable exponent spaces, such as Hardy-Littlewood maximal operators, singular integrals, commutators and fractional integrals. These research results reflect the characteristics of the variable exponent problems very well.…”

The maximal operator in Lebesgue spaces with variable exponent near 1

“…Recently, L. Diening has derived some new results on the boundedness of the maximal operator in more general Orlicz-Musielak spaces [6] and P. Harjulehto, P. Hästö and M. Pere have investigated the maximal operator on variable exponent Lebesgue spaces on metric measure spaces [14]. Progress with the maximal operators has in turn led to investigations of potential-type operators and singular integrals (e.g., L. Diening and M. Růžička [7], D. Edmunds, V. Kokilashvili and A. Meskhi [8], A. Karlovich and A. Lerner [18], and V. Kokilasvili and S. Samko [19,20]). …”

Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

“…Indeed, Coifman et al [1] characterized the -boundedness of [ , ], where the list is not exhaust. The boundedness in variable function spaces of many classical operators from harmonic analysis has been obtained; see [18,19,[41][42][43][44]. Motivated by these works, we will consider analogous results in [8] to variable exponent situation.…”