Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized local Morrey spaces

Abstract

In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel T on the generalized local Morrey spaces LM

Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces

“…Note that, in the case of P = I Corollary 15 has been proved in [2,30,31]. Also, in the case of P = I and s = ∞ Corollary 15 has been proved in [2,29,30,31,34]. Now, we give the applications of Theorem 7 and Theorem 12 for the parabolic Marcinkiewicz operator.…”

“…We introduce the parabolic local Campanato space LC {x0} p,λ,P following the known ideas of defining local Campanato space (see [2,30] etc). Apparently, (4.1) is equivalent to the following condition:…”

“…However, the space LC Theorem 8. [2,30] Let v 1 , v 2 and ω be weigths on (0, ∞) and v 1 (t) be bounded outside a neighbourhood of the origin. The inequality…”

“…In this case, T P Ω defined as in (1.2) is the classical singular integral operator with rough kernel of convolution type whose boundedness in various function spaces has been wellstudied by many authors (see [2,24], [30,31], [38], and so on). And also, in this case, [b, T…”

On maximal and potential operators with rough kernels in variable exponent spaces

“…The main operators of harmonic analysis, maximal, singular and of potential type, with the so called rough kernel have been widely studied, see e.g. [5,6,7,8,9,12,16,17,23,24,25,26,29,34,39] and references therein. The study of operators with rough kernels was based on the usage of the rotation method, which goes back to [15].…”