# Interpolation between $H^{p(\cdot)}(\mathbb R^n)$ and $L^\infty(\mathbb R^n)$: Real Method

## TL;DR

This paper establishes a real interpolation theorem between variable Hardy spaces and $L^
abla$ spaces, characterizing the structure of variable weak Hardy spaces and their relation to variable Lebesgue spaces.

## Contribution

It introduces a decomposition for variable weak Hardy space distributions and proves a new interpolation theorem linking variable Hardy and $L^
abla$ spaces.

## Key findings

- Interpolation formula for $(H^{p(
abla)},L^
abla)$ spaces.
- Equivalence of variable weak Hardy and Lebesgue spaces under certain conditions.
- Characterization of variable weak Hardy spaces as variable Lebesgue spaces.

## Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-H\"older continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into "good" and "bad" parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(\cdot)}(\mathbb R^n)$ and the space $L^{\infty}(\mathbb R^n)$: \begin{equation*} (H^{p(\cdot)}(\mathbb R^n),L^{\infty}(\mathbb R^n))_{\theta,\infty} =W\!H^{p(\cdot)/(1-\theta)}(\mathbb R^n),\quad \theta\in(0,1), \end{equation*} where $W\!H^{p(\cdot)/(1-\theta)}(\mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb R^n)$ with $p_-:=\mathop\mathrm{ess\,inf}_{x\in\rn}p(x)\in(1,\infty)$ is proved to coincide with the variable Lebesgue space $W\!L^{p(\cdot)}(\mathbb R^n)$.