Fractional integration operators of variable order: continuity and compactness properties
2014 (English)In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 287, no 8-9, 980-1000 p.Article in journal (Refereed) Published
Let alpha : [0, 1] -greater than R be a Lebesgue-almost everywhere positive function. We consider the Riemann-Liouville operator of variable order defined by (R-alpha(.) f) (t) := 1/Gamma(alpha(t)) integral(t)(0)(t - s)(alpha(t)-1) f(s) ds, t is an element of [0, 1], as an operator from L-p[0, 1] to L-q[0, 1]. Our first aim is to study its continuity properties. For example, we show that R-alpha(.) is always bounded (continuous) in L-p[0, 1] provided that 1 less than p less than= infinity. Surprisingly, this becomes false for p = 1. In order R-alpha(.) to be bounded in L-1 [0, 1], the function alpha(.) has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of R-alpha(.). We characterize functions alpha(.) for which R-alpha(.) is a compact operator and for certain classes of functions alpha(.) we provide order-optimal bounds for the dyadic entropy numbers e(n)(R-alpha(.)).
Place, publisher, year, edition, pages
Wiley-VCH Verlagsgesellschaft, 2014. Vol. 287, no 8-9, 980-1000 p.
Riemann-Liouville operator; integration of variable order; compactness properties; entropy numbers
IdentifiersURN: urn:nbn:se:liu:diva-109274DOI: 10.1002/mana.201200337ISI: 000338487200009OAI: oai:DiVA.org:liu-109274DiVA: diva2:737140