A non-doubling Trudinger inequality

Interpolation properties of Besov spaces defined on metric spaces

Let X = (X, d, μ)be a doubling metric measure space. For 0 < α < 1, 1 ≤p, q < ∞, we define semi-norms 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)-Poincare 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 definiti…