A PDE Characterization of Anisotropic Hardy Spaces

TL;DR

This paper characterizes anisotropic Hardy spaces using PDEs, linking them to parabolic Hardy spaces and providing a classification of dilations, thus advancing the understanding of their structure and equivalence.

Contribution

It introduces a PDE-based differential characterization of anisotropic Hardy spaces and classifies dilations for their equivalence under linear transformations.

Findings

Identification of $H_A^p$ with a parabolic Hardy space

Definition of $H_A^p$ via a parabolic differential equation

Classification of dilations for equivalent anisotropic Hardy spaces

Abstract

We obtain a differential characterization for the anisotropic Hardy space $H_A^p$ by identifying it with a parabolic Hardy space associated with a general continuous group. This allows $H_A^p$ to be defined using a parabolic differential equation of Calderon and Torchinsky. We also provide a classification of dilations corresponding to equivalent anisotropic Hardy spaces with respect to linear transformations.