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

Let ( X , d , μ ) be a metric measure space of homogeneous type and L be a one‐to‐one operator of type ω onL 2 ( X ) for ω ∈[0,  π /2). In this article, under the assumptions that L has a bounded H ∞ ‐functional calculus onL 2 ( X ) and satisfies ( p L ,  q L ) off‐diagonal estimates on balls, where p L ∈[1, 2) and q L ∈(2,  ∞ ], the authors establish a characterization of the Sobolev spaceW ̇L α , p ( X ) , defined via L α /2 , of order α ∈(0, 2] for p ∈( p L ,  q L ) by means of a quadratic function S α ,  L . As an application, the authors show that for the degenerate elliptic operator L w : =−  w  − 1 div( A ∇) and the Schrödinger type operatorL ˜w : = L w + a | · | − 2with a ∈(0,  ∞ ) on the weighted Euclidean space ( R n , | · | , w ( x ) d x ) with A being real symmetric, if n ⩾3, w ∈ A q ( R n ) ∩ R H r ( R n ) with q ∈[1, 2], r ∈n n − 2 , ∞, p ∈(1,  ∞ ) and α ∈ 0 , min 2 + n 1 − 1 r − q ,n p1 − 1 rwith q + 1 r < 1 + 2 n , then, for all f ∈ D ( L w ) ∩ DL ˜w, ∥L ˜w α / 2 ( f ) ∥L w p ( R n ) ) ∼ ∥ L w α / 2 ( f ) ∥L w p ( R n ) , where the implicit equivalent positive constants are independent of f ,A q ( R n ) denotes the class of Muckenhoupt weights, R H r ( R n ) the reverse Hölder class, and D ( L w ) and DL ˜wthe domains of L w andL ˜w , respectively. Copyright © 2016 John Wiley & Sons, Ltd.