Interpolation properties of Besov spaces defined on metric spaces

Abstract

AbstractLet X = (X, d, μ)be a doubling metric measure space. For 0 < α < 1, 1 ≤p, q < ∞, we define semi‐norms equation image When q = ∞ the usual change from integral to supremum is made in the definition. The Besov space Bp, qα (X) is the set of those functions f in Llocp(X) for which the semi‐norm ‖f ‖ is finite. We will show that if a doubling metric measure space (X, d, μ) supports a (1, p)‐Poincaré inequality, then the Besov space Bp, qα (X) coincides with the real interpolation space (Lp (X), KS1, p(X))α, q, where KS1, p(X) is the Sobolev space defined by Korevaar and Schoen [15]. This results in (sharp) imbedding theorems. We further show that our definition of a Besov space is equivalent with the definition given by Bourdon and Pajot [3], and establish a trace theorem (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)