Variable Weak Hardy Spaces $W\text{\hspace{0.17em}⁣}{H}_L^{p(\cdot )}({\mathbb{R}}^n)$ Associated with Operators Satisfying Davies-Gaffney Estimates

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 $\omega \in [0,\pi \mathrm{/}2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W\text{\hspace{0.17em}⁣}{H}_L^{p(\cdot )}({\mathbb{R}}^n)$ 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 $W\text{\hspace{0.17em}⁣}{T}^{p(\cdot )}({\mathbb{R}}^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W\text{\hspace{0.17em}⁣}{H}_L^{p(\cdot )}({\mathbb{R}}^n)$. 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 $\mathrm{\nabla }{L}^{-1\mathrm{/}2}$ is bounded from $W\text{\hspace{0.17em}⁣}{H}_L^{p(\cdot )}({\mathbb{R}}^n)$ to the variable weak Hardy space $W\text{\hspace{0.17em}⁣}{H}^{p(\cdot )}({\mathbb{R}}^n)$. 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.