Difference characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type (Q2674043)

The nowadays well-known inhomogeneous spaces \(A^s_{p,q} (\mathbb R^n)\), \(A \in \{B,F \}\), \(s\in \mathbb R\) and \(0<p,q \le \infty\) and their homogeneous counterparts \(\dot{A}^s_{p,q} (\mathbb R^n)\) have been extended and modified in many directions. In particular the underlying Euclidean space \(\mathbb R^n\) has been replaced by so-called spaces of homogeneous type \((X, d, \mu)\) where \(X\) is a set, \(d\) is a quasi-metric and \(\mu\) is a measure (satisfying the doubling condition). The resulting spaces \(A^s_{p,q} (X)\) have been studied in detail. The paper under review contributes to this topic, characterizing some of these spaces in terms of first differences of functions.