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Interpolation properties of Besov spaces defined on metric spaces

✍ Scribed by Amiran Gogatishvili; Pekka Koskela; Nageswari Shanmugalingam

Publisher
John Wiley and Sons
Year
2010
Tongue
English
Weight
198 KB
Volume
283
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let 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 B~p, q~^α^ (X) is the set of those functions f in L~loc~^p^(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 B~p, q~^α^ (X) coincides with the real interpolation space (L^p^ (X), KS^1, p^(X))~α, q~, where KS^1, 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)

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