Anisotropic Singular Integrals in Product Spaces

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 $L_w^q({\mathbb{R}}^n\times {\mathbb{R}}^m)$ with $q\in (1,\infty )$ and $w\in {\mathcal{A}}_q({\mathbb{R}}^n\times {\mathbb{R}}^m;\vec{A})$ or on $H_w^p({\mathbb{R}}^n\times {\mathbb{R}}^m;\vec{A})$ 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 $w=1$.

Sci. China Math., to appear

References in corpus (1)

• Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with flag singular integrals

Cited by in corpus (3)

• Local Hardy Spaces of Musielak-Orlicz Type and Their Applications
• Hardy Spaces $H_L^p({\mathbb{R}}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates
• Atomic and Littlewood-Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications