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

Suppose that (X,d,μ) is a metric measure space of homogeneous type and suppose that L is a one-to-one operator of type ω on L²(X), with ω ∈ [0,π/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 WHLp(·)(X) associated with L, and establish the molecular characterization of WHLp(·)(X) via the atomic decomposition of variable weak tent spaces. Particularly, we obtain the atomic characterization of WHLp(·)(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 WHLp(·)(X) via establishing its atomic decomposition.