Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents

TL;DR

This paper characterizes variable exponent Hardy spaces using intrinsic square functions and establishes duality with variable Campanato spaces, advancing the understanding of harmonic analysis in variable exponent settings.

Contribution

It introduces intrinsic square function characterizations for variable exponent Hardy spaces and explores their duality with variable Campanato spaces, extending classical harmonic analysis tools.

Findings

Characterizations of $H^{p( ext{cdot})}$ via intrinsic square functions.

Duality between $H^{p( ext{cdot})}$ and variable Campanato spaces.

Establishment of $p( ext{cdot})$-Carleson measure characterization.

Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a measurable function satisfying some decay condition and some locally log-H\"older continuity. In this article, via first establishing characterizations of the variable exponent Hardy space $H^{p(\cdot)}(\mathbb R^n)$ in terms of the Littlewood-Paley $g$-function, the Lusin area function and the $g_\lambda^\ast$-function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_\lambda^\ast$-function. The $p(\cdot)$-Carleson measure characterization for the dual space of $H^{p(\cdot)}(\mathbb R^n)$, the variable exponent Campanato space $\mathcal{L}_{1,p(\cdot),s}(\mathbb R^n)$, in terms of the intrinsic function is also presented.