Anisotropic Singular Integrals in Product Spaces

TL;DR

This paper introduces a class of anisotropic singular integrals adapted to product spaces with expansive dilations, establishing their boundedness on weighted Lebesgue and Hardy spaces, extending classical results to anisotropic settings.

Contribution

The authors define anisotropic singular integrals with kernels adapted to product space dilations and prove their boundedness on weighted Lebesgue and Hardy spaces, a novel extension in anisotropic analysis.

Findings

Boundedness of anisotropic singular integrals on weighted $L^q$ spaces.

Boundedness on weighted Hardy spaces for $p o 0$.

Results hold even when weights are trivial ($w=1$).

Abstract

Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$ and $\vec A\equiv(A_1, A_2)$. Denote by ${\mathcal A}_\infty(\rnm; \vec A)$ the class of Muckenhoupt weights associated with $\vec A$. The authors introduce a class of anisotropic singular integrals on $\mathbb R^n\times\mathbb R^m$, whose kernels are adapted to $\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\mathbb R^n\times\mathbb R^m)$ with $q\in(1, \infty)$ and $w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)$ or on $H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)$ with $p\in(0, 1]$ and $w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)$. These results are also new even when $w=1$.