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Pointwise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness and their applications. (English) Zbl 1496.46033

The nowadays well-known homogeneous spaces \(\dot{A}^s_{p,q} (\mathbb R^n)\) with \(A \in \{B,F \}\), \(s\in \mathbb R\) and \(0<p,q \le \infty\) have been modified in several ways. The smoothness \(s\), characterized by \(\{ 2^{js} \}^\infty_{j=0}\), is generalized by suitable sequences \(\{\sigma_j \}^\infty_{j=0}\) of positive numbers. Furthermore, \(\mathbb R^n\) is replaced by metric spaces \((X, d, \mu)\) with the metric \(d\) and the Borel measure \(\mu\) on the set \(X\), where the smoothness is expressed by so-called Hajłasz gradients. More recently, there is some type of discretization, called hyperbolic filling. The paper deals with spaces based on these ingredients and their relations, especially to \(\dot{A}^\sigma_{p,q} (\mathbb R^n)\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E36 Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces
42B35 Function spaces arising in harmonic analysis
30L99 Analysis on metric spaces
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