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Characterizations of Sobolev spaces associated to operators satisfying off‐diagonal estimates on balls
Author(s) -
Zhang Junqiang,
Chang DerChen,
Yang Dachun
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4206
Subject(s) - mathematics , sobolev space , diagonal , bounded function , type (biology) , operator (biology) , quadratic form (statistics) , measure (data warehouse) , euclidean space , combinatorics , order (exchange) , space (punctuation) , pure mathematics , elliptic operator , mathematical analysis , characterization (materials science) , philosophy , materials science , repressor , database , ecology , linguistics , chemistry , computer science , biology , biochemistry , geometry , transcription factor , finance , economics , gene , nanotechnology
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.