Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces (B) over dot(p,q)(s) and Triebel-Lizorkin spaces (F) over dot(p,q)(s) for all s is an element of (0, 1) and p, q is an element of (n/(n + s), infinity] both in R-n and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve (F) over dot(n/s,q)(s) on R-n for all s is an element of (0, 1) and q is an element of (n/(n + s), infinity]. A metric measure space version of the above morphism property is also established. (C) 2010 Elsevier Inc. All rights reserved.