Characterizations of Sobolev spaces associated to operators satisfying off‐diagonal estimates on balls

Abstract

Let be a metric measure space of homogeneous type and L be a one‐to‐one operator of type ω on for ω∈[0, π/2). In this article, under the assumptions that L has a bounded H∞‐functional calculus on and satisfies (pL, qL) off‐diagonal estimates on balls, where pL∈[1, 2) and qL∈(2, ∞], the authors establish a characterization of the Sobolev space , defined via Lα/2, of order α∈(0, 2] for p∈(pL, qL) by means of a quadratic function Sα, L. As an application, the authors show that for the degenerate elliptic operator Lw: =− w − 1div(A∇) and the Schrödinger type operator with a∈(0, ∞) on the weighted Euclidean space with A being real symmetric, if n⩾3, with q∈[1, 2], , p∈(1, ∞) and with , then, for all , , where the implicit equivalent positive constants are independent of f, denotes the class of Muckenhoupt weights, the reverse Hölder class, and D(Lw) and the domains of Lw and , respectively. Copyright © 2016 John Wiley & Sons, Ltd.