Spectral Theory of Riesz Potentials on Quasi-Metric Spaces

Abstract

This paper deals with spectral assertions of Riesz potentials in some classes of quasi‐metric spaces. In addition we survey briefly a few related subjects: integral operators, local means and function spaces, euclidean charts of quasi‐metric spaces, relations to fractal geometry.

Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces

“…[31] imitating the notation of d-sets in R n . We refer to [12] where Besov and Triebel-Lizorkin spaces with classical smoothness on d-spaces were studied in detail.…”

New characterizations and applications of inhomogeneous Besov and Triebel–Lizorkin spaces on homogeneous type spaces and fractals

“…More applications can be found in [35] and [36]; see also [6] and [33]. We now make some conventions.…”

“…The main ideas come from [33]. See [36] and [35] for more applications. Let us recall some basic facts of [6] and [33].…”

“…Lemma 7.1 was proved by Edmunds and Triebel [6]. This inequality plays a key role in applications of estimates of entropy numbers to estimates of the eigenvalues for differential operators; see [33], [6], [36] and [35].…”

“…We will do this in the setting of general homogeneous type spaces. We have given some applications of these estimates for entropy numbers to estimates of the eigenvalues of some fractal differential operators on d-sets in [36] and Riesz potentials on quasi-metric spaces in [35]. Another main purpose of this paper is to show that the fractional integrals and derivatives can be used as a lifting tool in these function spaces on homogeneous type spaces.…”