Littlewood-Paley square functions and the local Hardy space for Multi-Norm Structures on $\mathbb{R}^{d}$

TL;DR

This paper develops a local Hardy space framework for multi-norm structures on , extending Littlewood-Paley theory and square functions to general dilation cases beyond the previously studied 2-dilation scenario.

Contribution

It introduces a new local multi-norm Hardy space ^1_E(), providing multiple characterizations including atomic decomposition, for general dilation matrices, advancing the analysis of multi-norm Fourier multipliers.

Findings

Established $L^1$-equivalence of square functions.

Defined the local multi-norm Hardy space ^1_E().

Provided atomic and other characterizations of the space.

Abstract

Multi-norm singular integrals and Fourier multipliers were introduced in [29], and one application of these notions was a precise description of the composition of convolution operators with Calder\'on-Zygmund kernels adapted to $n$ different families of dilations. The description of the resulting operators was given in terms of differential inequalities specified by a matrix $\mathbf E$, and in terms of dyadic decompositions of the kernels and multipliers. In this paper we extend the analysis of multi-norm structures on $\mathbb{R}^d$ by studying the induced Littlewood-Paley decomposition of the frequency space and various associated square functions. After establishing their $L^1$-equivalence, we use these square functions to define a local multi-norm Hardy space $\mathbf{h}^{1}_{\mathbf{E}}(\mathbb{R}^d)$. We give several equivalent descriptions of this space, including an atomic characterization. There has been recent work, limited to the $2$-dilation case, by other authors. The general $n$-dilation case treated here is considerably harder and requires new ideas and a more systematic approach.