Variable Weak Hardy Spaces Associated with Operators on Metric Measure Spaces of Homogeneous Type

Suppose that (X, d, ) is a metric measure space of homogeneous type and suppose that L is a one-to-one operator of type on L^2 (X), with 0, /2), which has a bounded holomorphic functional calculus, and whose heat kernel satisfies the Davies–Gaffney estimates. Suppose that p () X (0, 1 is a variable exponent function with the globally log-Hölder continuous condition. In this paper, we introduce the variable weak Hardy space WH₋^p () (X) associated with L, and establish the molecular characterization of WH₋^p () (X) via the atomic decomposition of variable weak tent spaces. Particularly, we obtain the atomic characterization of WH₋^p () (X) when L is non-negative and self-adjoint. Furthermore, if L is a non-negative self-adjoint operator whose heat kernel has a Gaussian upper bound, we obtain the non-tangential and the radial maximal function characterizations of WH₋^p () (X) via establishing its atomic decomposition.