Maximal Function Characterizations of Variable Hardy Spaces Associated with Non-negative Self-adjoint Operators Satisfying Gaussian Estimates

Abstract

Let $p(\cdot ):{\mathbb{R}}^n\to (0,1]$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition and $L$ a non-negative self-adjoint operator on $L^2({\mathbb{R}}^n)$ whose heat kernels satisfying the Gaussian upper bound estimates. Let $H_L^{p(\cdot )}({\mathbb{R}}^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels $\{e^{-t^{2}L}{\}}_{t\in (0,\infty )}$. In this article, the authors first establish the atomic characterization of $H_L^{p(\cdot )}({\mathbb{R}}^n)$; using this, the authors then obtain its non-tangential maximal function characterization which, when $p(\cdot )$ is a constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of $H_L^{p(\cdot )}({\mathbb{R}}^n)$ under an additional assumption that the heat kernels of $L$ have the Hölder regularity.

32 pages, submitted. arXiv admin note: text overlap with arXiv:1512.05950