Multilinear Strongly Singular Integral Operators with Generalized Kernels on RD-Spaces

TL;DR

This paper introduces a new class of multilinear strongly singular integral operators with generalized kernels on RD-spaces, establishing their boundedness on various function spaces and providing new endpoint and weak-type estimates.

Contribution

It generalizes boundedness results of multilinear singular integrals to RD-spaces and introduces novel endpoint and weak-type estimates, extending classical Euclidean space results.

Findings

Boundedness on weighted Lebesgue spaces established

Two types of endpoint estimates obtained

Weak-type results on weighted Lebesgue spaces are new even in Euclidean spaces

Abstract

In this article, we introduce a class of multilinear strongly singular integral operators with generalized kernels on the RD-space. The boundedness of these operators on weighted Lebesgue spaces is established. Moreover, two types of endpoint estimates and their boundedness on generalized weighted Morrey spaces are obtained. Our results further generalize the relevant conclusions on generalized kernels in Euclidean spaces. Moreover, the weak-type results on weighted Lebesgue spaces are brand new even in the situation of Euclidean spaces. In addition, when the generalized kernels degenerate into classical kernels, our research results also extend the relevant known results. It is worth mentioning that our RD spaces are more general than theirs.