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

Abstract

Abstract:In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

Product generalized local Morrey spaces and commutators of multi-sublinear operators generated by multilinear Calderón-Zygmund operators and local Campanato functions

“…We also refer to [1] for the latest research on the theory of Morrey spaces associated with Harmonic Analysis. In recent years, more and more researches focus on function spaces based on Morrey spaces to fill in some gaps in the theory of Morrey type spaces (see, for example, [9,10,11,12,13,18]). Moreover, these spaces are useful in harmonic analysis and PDEs.…”

“…Recall that the concept of the generalized local (central) Morrey space LM {x0} p,ϕ has been introduced in [2] and studied in [9,10]. .…”

“…For the properties and applications of generalized local (central) Morrey spaces LM {x0} p,ϕ , see also [2,9,10].…”

“…We point out that the condition (1.8) in the case of m = 1 was first introduced by Soria and Weiss in [19] . The condition (1.8) is satisfied by many interesting operators in harmonic analysis, such as the m-linear Calderón-Zygmund operators, m-sublinear Carleson's maximal operator, m-sublinear Hardy-Littlewood maximal operator, C. Fefferman's singular multipliers, R. Fefferman's m-linear singular integrals, Ricci-Stein's m-linear oscillatory singular integrals, the m-linear Bochner-Riesz means and so on (see [2,9,10,19] for details).…”

Vanishing generalized Morrey spaces and commutators of Marcinkiewicz integrals with rough kernel associated with Schrödinger operator

“…Here, 1 < p < ∞ and 0 < λ < n and the quantity of (1.1) is the (p, λ)-Morrey norm, denoted by f L p,λ . In recent years, more and more researches focus on function spaces based on Morrey spaces to fill in some gaps in the theory of Morrey type spaces (see, for example, [7,8,9,10,11,16]). Moreover, these spaces are proved useful in harmonic analysis and PDEs.…”