Variable Weak Hardy Spaces $W\!H_L^{p(\cdot)}({\mathbb R}^n)$ Associated with Operators Satisfying Davies-Gaffney Estimates (1805.07778v1)
Abstract: Let $p(\cdot):\ \mathbb Rn\to(0,1]$ be a variable exponent function satisfying the globally log-H\"older continuous condition and $L$ a one to one operator of type $\omega$ in $L2({\mathbb R}n)$, 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 $W!H_L{p(\cdot)}(\mathbb Rn)$ 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 $W!T{p(\cdot)}(\mathbb Rn)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W!H_L{p(\cdot)}(\mathbb Rn)$. 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 $W!H_L{p(\cdot)}(\mathbb Rn)$ to the variable weak Hardy space $W!H{p(\cdot)}(\mathbb Rn)$. 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 $W!H_L{p(\cdot)}(\mathbb Rn)$ is further used to characterize the space via non-tangential maximal functions.