Real-variable characterizations of Musielak-Orlicz-Hardy spaces associated with Schrödinger operators on domains

Abstract

Let n >= 3, Omega be a strongly Lipschitz domain of R-n and L-Omega :=-Delta + V a Schrodinger operator on L-2(Omega) with the Dirichlet boundary condition, where Delta is the Laplace operator and the nonnegative potential V belongs to the reverse Holder class RHq0(R-n) for some q(0) > n/2. Assume that the growth function phi : R-n x [0, infinity) -> [0, infinity) satisfies that phi(x, .) is an Orlicz function, phi(., t) is an element of A(infinity)(R-n) (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index i(phi) is an element of (n/n+delta, 1], where delta := min {mu(0), 2- n/q(0)} and mu(0) is an element of (0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Delta on Omega. In this article, the authors first show that the heat kernels of L-Omega satisfy the Gaussian upper bound estimates and the Holder continuity. The authors then introduce the 'geometrical' Musielak-Orlicz-Hardy space H-phi,H-LRn,H-r(Omega) via H-phi,H-LRn (R-n), the Hardy space associated with L-Rn :=-Delta + Von R-n, and establish its several equivalent characterizations, respectively, in terms of the non-tangential or the vertical maximal functions or the Lusin area functions associated with L-Omega. All the results essentially improve the known results even on Hardy spaces H-LRn,r(p)(Omega) with p is an element of (n / (n + delta),1] (in this case, phi(x, t) := t(p) for all x is an element of Omega and t is an element of [0, infinity)). Copyright (C) 2016 John Wiley & Sons, Ltd.