Characterizations of Variable Exponent Hardy Spaces via Riesz Transforms

TL;DR

This paper characterizes variable exponent Hardy spaces using Riesz transforms, linking boundary behavior of harmonic functions with these spaces under certain conditions on the variable exponent.

Contribution

It provides new characterizations of variable exponent Hardy spaces via Riesz transforms, extending previous work to broader exponent ranges.

Findings

Characterization via first order Riesz transforms for $p_- ext{ in }(rac{n-1}n, fty)$.

Characterization via compositions of Riesz transforms for $p_- ext{ in }(0,rac{n-1}n)$.

Established relations between harmonic boundary values and variable Hardy spaces.

Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying that there exists a constant $p_0\in(0,p_-)$, where $p_-:=\mathop{\mathrm {ess\,inf}}_{x\in \mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(\cdot)/p_0}(\mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(\cdot)}(\mathbb R^n)$ via the first order Riesz transforms when $p_-\in (\frac{n-1}n,\infty)$, and via compositions of all the first order Riesz transforms when $p_-\in(0,\frac{n-1}n)$.