Interpolation properties of Besov spaces defined on metric spaces

Let X = (X, d, mu) be a doubling metric measure space. For 0 < alpha < 1, 1 <= p, q < infinity, we define semi-norms parallel to f parallel to B-p,q(alpha)(X) = (integral(infinity)(0)(integral(X) (sic)(B(x,t)) vertical bar f(x) - f(y)vertical bar(p) d mu(y) d mu(x))(q/p) dt/t(alpha q+1))(1/q). When q = infinity the usual change from integral to supremum is made in the definition. The Besov space B-p,q(alpha) (X) is the set of those functions f in L-loc(p) (X) for which the semi-norm parallel to f parallel to B-p,q(alpha) (x) is finite. We will show that if a doubling metric measure space (X, d, mu) supports a (1, p)-Poincare inequality, then the Besov space B-p,q(alpha)(X) coincides with the real interpolation space (L-p(X), K S-1,S-p(X))(alpha,q), where K S-1,S-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. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim