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

Abstract Let be a space of homogeneous type with and a variable exponent function satisfying , where . Assume that is a nonnegative self‐adjoint operator on 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 associated with . Second, as applications, when , is a bounded Lipschitz domain, is a second‐order divergence form elliptic operator with the Robin boundary condition, the authors establish a new atomic characterization of the associated with , and then the authors further show that, for , where and , respectively, denote the variable Hardy spaces on associated with the corresponding elliptic operators having the Dirichlet and the Neumann boundary conditions, and ( and ), respectively, denote “supported type” and “restricted type” Hardy spaces on (its local version), and is a critical index depending on the operators , , and . Third, the authors prove the boundedness of the Riesz transform on the variable Lebesgue space , and from to , or to . Finally, the authors further establish the global regularity estimates in both and for the inhomogeneous Robin problem of on .