Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates

Abstract

Let p(⋅):Rn→(0,1] be a variable exponent function satisfying the globally log-Hölder continuous condition and L a non-negative self-adjoint operator on L2(Rn) whose heat kernels satisfying the Gaussian upper bound estimates. Let HLp(⋅)(Rn) be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels {e−t2L}t∈(0,∞). In this article, the authors first establish the atomic characterization of HLp(⋅)(Rn); using this, the authors then obtain its non-tangential maximal function characterization which, when p(⋅) is a constant in (0,1], coincides with a recent result by L. Song and L. Yan (2016) and further induces the radial maximal function characterization of HLp(⋅)(Rn) under an additional assumption that the heat kernels of L have the Hölder regularity.