Fractional integration operators of variable order: continuity and compactness properties

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).