Herz-type Hardy spaces with variable exponents associated with operators satisfying Davies-Gaffney estimates

Let L be a one-to-one operator of type w, with w is an element of [0, pi/2], satisfying the Davies-Gaffney estimates. For alpha is an element of (0, infinity) and p is an element of (0, infinity) and under the condition that q(.) : R-n -> [1, infinity) satisfies the globally log-Holder continuity condition, we introduce the Herz-type Hardy space with variable exponents associated to L and establish its molecular decomposition. The atomic characterization and maximal function characterizations of the space are proved under the assumption that L is a non-negative self-adjoint operator on L-2(R-n) whose heat kernels satisfy the Gaussian upper bound estimates. All the results are new even for the constant case.