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

Abstract

Let $p(\cdot ):{\mathbb{R}}^n\to (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder 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))_{θ,\infty} =W\!H^{p(\cdot)/(1-θ)}(\mathbb R^n),\quad θ\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 $W{L}^{p(\cdot )}({\mathbb{R}}^n)$.

22 pages; This paper was submitted on January 1 of 2017 to a journal