Fractional Integrals on the Line

Abstract

In this chapter, boundedness and compactness problems are investigated for various fractional integrals defined on the real line. Our main objective is to give complete descriptions of those pairs of weight functions for which these fractional integrals generate operators which are bounded or compact from one weighted Banach function space into another. This problem was studied earlier by many authors, for instance, for fractional Riemann-Liouville operators R _± when ± ≥ 1. Here the problem is studied in a more general setting. Transparent, easy to verify criteria are presented for a wider range of fractional integral orders. At the end of the chapter, some applications to nonlinear Volterra-type integral equations are given.