The characterizations of Hardy–Sobolev spaces by fractional square functions related to Schrödinger operators

Abstract

Let L = −∆ + V be a Schrödinger operator, where the potential V satisfies the reverse Hölder condition. In this paper, via the heat semigroup e and the Poisson semigroup e √ , we introduce several classes of fractional square functions associated with L including the Litttlewood–Paley g-function, the area integral and the g λ-function, respectively. By the regularities of semigroup, we establish several square function characterizations of the Hardy space and the Hardy–Sobolev space related to the Schrödinger operator.