Variable Weak Hardy Spaces $W\!H_L^{p(\cdot)}({\mathbb R}^n)$ Associated with Operators Satisfying Davies-Gaffney Estimates

TL;DR

This paper introduces variable weak Hardy spaces associated with operators satisfying Davies-Gaffney estimates, providing atomic and molecular characterizations, and establishing boundedness of Riesz transforms in these spaces.

Contribution

It develops the theory of variable weak Hardy spaces linked to operators with Davies-Gaffney estimates, including atomic/molecular characterizations and boundedness results for Riesz transforms.

Findings

Established molecular characterization via atomic decomposition.

Proved Riesz transform boundedness from $W ext!H_L^{p( ext{cdot})}$ to $W ext!H^{p( ext{cdot})}$.

Characterized spaces using non-tangential maximal functions under certain conditions.

Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-H\"older continuous condition and $L$ a one to one operator of type $\omega$ in $L^2({\mathbb R}^n)$, with $\omega\in[0,\,\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W\!H_L^{p(\cdot)}(\mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W\!T^{p(\cdot)}(\mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W\!H_L^{p(\cdot)}(\mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $\nabla L^{-1/2}$ is bounded from $W\!H_L^{p(\cdot)}(\mathbb R^n)$ to the variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of $\{e^{-tL}\}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W\!H_L^{p(\cdot)}(\mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.