Fractional integration operators of variable order: continuity and compactness properties

TL;DR

This paper studies the continuity and compactness of variable order fractional integration operators on Lebesgue spaces, revealing conditions for their boundedness and providing bounds for their entropy numbers.

Contribution

It characterizes when the Riemann-Liouville operator of variable order is continuous and compact on L_p spaces, including the case p=1, and derives bounds for entropy numbers.

Findings

R^a is always continuous on L_p for p>1

Continuity on L_1 requires additional conditions on a(.)

Provides order-optimal bounds for entropy numbers e_n(R^a)

Abstract

Let a:[0,1] -> R be a Lebesgue-almost everywhere positive function. We consider the Riemann-Liouville operator R^a of variable order a(.) 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^a is always continuous in L_p[0,1] if p>1. Surprisingly, this becomes false for p=1. In order R^a to be continuous in L_1[0,1], the function a(.) has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of R^a. We characterize functions a(.) for which R^a is a compact operator and for certain classes of functions a(.) we provide order-optimal bounds for the dyadic entropy numbers e_n(R^a).