Variable Hardy spaces on spaces of homogeneous type and applications to variable Hardy spaces associated with elliptic operators having Robin boundary conditions on Lipschitz domains

Let $(X,d,\mu )$ be a space of homogeneous type with and $p(·):X\to (0,\infty )$ a variable exponent function satisfying , where $p_{-}:=\mathrm{ess}{\inf }_{x\in X}p\left(x\right)$. Assume that $L$ is a nonnegative self-adjoint operator on $L^2\left(X\right)$ whose heat kernels satisfy the Gaussian upper bound estimates. In this paper, the authors first establish the atomic, nontangential, and radial maximal function characterizations for the variable Hardy space $H_L^{p(·)}\left(X\right)$ associated with $L$. Second, as applications, when $n\ge 2$, $\Omega \subset R^n$ is a bounded Lipschitz domain, $L_R$ is a second-order divergence form elliptic operator with the Robin boundary condition, the authors establish a new atomic characterization of the $H_{L_R}^{p(·)}\left(\Omega \right)$ associated with $L_R$, and then the authors further show that, for $p_{-}\in (\frac{n}{n+{\delta }_0},1]$,