Boundedness of a class of multilinear operators and their iterated commutators on Morrey-Banach function spaces

TL;DR

This paper establishes the boundedness of a broad class of multilinear and linear operators, including their iterated commutators, on generalized Morrey-Banach function spaces, extending classical results and applying sparse domination and $W_{r}$ property conditions.

Contribution

It introduces new boundedness results for multilinear and linear operators and their commutators on Morrey-Banach spaces under generalized conditions, broadening the classical theory.

Findings

Boundedness of multilinear $ ext{ω}$-Calderón-Zygmund operators established.

Boundedness of multilinear maximal singular integrals proved.

Extension of boundedness to Morrey-Lorentz and variable exponent Morrey spaces.

Abstract

This paper investigates the boundedness of a broad class of operators within the framework of generalized Morrey-Banach function spaces. This class includes multilinear operators such as multilinear $\omega$-Calder\'{o}n-Zygmund operators, multilinear maximal singular integral operators, multilinear pseudo-differential operators, and multilinear square functions, as well as linear operators such as rough singular integral operators, nonintegral operators, and Stein's square functions. The boundedness of this class of operators and their commutators on Morrey-Banach spaces is established provided they satisfy either a pointwise sparse domination assumption or the $W_{r}$ property (Lerner, Lorist and Ombrosi, Math. Z. 2024), which significantly generalize the classical theory of Morrey spaces. As an application, the boundedness of iterated commutators of this class of multilinear operators is extended to Morrey-Lorentz spaces and Morrey spaces with variable exponents.