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

Let (X, d, mu) be a metric measure space of homogeneous type and L be a one-to-one operator of type omega on L-2(X) for omega is an element of [0, pi/2). In this article, under the assumptions that L has a bounded H-infinity-functional calculus on L-2(X) and satisfies (p(L), q(L)) off-diagonal estimates on balls, where p(L) is an element of [1, 2) and q(L) is an element of (2, infinity], the authors establish a characterization of the Sobolev space W-L(alpha,p)(X), defined via L-alpha/2, of order alpha is an element of (0, 2] for p is an element of (p(L), q(L)) by means of a quadratic function S-alpha,S- L. As an application, the authors show that for the degenerate elliptic operator L-w := -w(-1) div (A del) and the Schrodinger type operator (L) over tilde (w) := L-w+a vertical bar.vertical bar(-2) with a is an element of(0, infinity) on the weighted Euclidean space (R-n, vertical bar.vertical bar, w(x) dx) with A being real symmetric, if n >= 3, w is an element of A(q)(R-n) boolean AND RHr (R-n) with q is an element of [1, 2], r is an element of(n/n-2, infinity], p is an element of (1, infinity) and alpha is an element of (0, min {2 + n (1 - 1/r - q), n/p (1 - 1/r)}) with q + 1/r < 1 + 2/n, then, for all f is an element of D(L-w) boolean AND D(<(L)over tilde>(w)), parallel to(L) over tilde (alpha/2)(w)(f)parallel to(Lpw(Rn))) similar to parallel to L-w(alpha/2)(f)parallel to(Lpw(Rn)), where the implicit equivalent positive constants are independent of f, A(q)(R-n) denotes the class of Muckenhoupt weights, RHr(R-n) the reverse Holder class, and D(L-w) and D((L) over tilde (w)) the domains of L-w and (L) over tilde (w), respectively. Copyright (C) 2016 John Wiley & Sons, Ltd.