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Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent functionsatisfying the globally log-H\"older continuous condition and $L$ a one to oneoperator of type $\omega$ in $L^2({\mathbb R}^n)$, with $\omega\in[0,\,\pi/2)$,which has a bounded holomorphic functional calculus and satisfies theDavies-Gaffney estimates. In this article, the authors introduce the variableweak Hardy space $W\!H_L^{p(\cdot)}(\mathbb R^n)$ associated with $L$ via thecorresponding square function. Its molecular characterization is thenestablished by means of the atomic decomposition of the variable weak tentspace $W\!T^{p(\cdot)}(\mathbb R^n)$ which is also obtained in this article. Inparticular, when $L$ is non-negative and self-adjoint, the authors obtain theatomic characterization of $W\!H_L^{p(\cdot)}(\mathbb R^n)$. As an applicationof the molecular characterization, when $L$ is the second-order divergence formelliptic operator with complex bounded measurable coefficient, the authorsprove 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 andself-adjoint with the kernels of $\{e^{-tL}\}_{t>0}$ satisfying the Gauss upperbound estimates, the atomic characterization of $W\!H_L^{p(\cdot)}(\mathbbR^n)$ is further used to characterize the space via non-tangential maximalfunctions.

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