Fractional integration operators of variable order: continuity and compactness properties |
| |
Authors: | Werner Linde |
| |
Institution: | Friedrich‐Schiller‐Universit?t Jena, Institut für Stochastik, 07743 Jena, Germany |
| |
Abstract: | Let be a Lebesgue‐almost everywhere positive function. We consider the Riemann‐Liouville operator of variable order defined by as an operator from to . Our first aim is to study its continuity properties. For example, we show that is always bounded (continuous) in provided that . Surprisingly, this becomes false for . In order to be bounded in L10, 1], the function has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of . We characterize functions for which is a compact operator and for certain classes of functions we provide order‐optimal bounds for the dyadic entropy numbers . |
| |
Keywords: | Riemann‐Liouville operator integration of variable order compactness properties entropy numbers Primary: 26A33 Secondary: 47B06 47B07 |
|
|