Compactness properties of certain integral operators related to fractional integration

Suppose 1≤p,q≤∞ and α > (1/p−1/q)+. Then we investigate compactness properties of the integral operator [inline-graphic not available: see fulltext] when regarded as operator from Lp[0,1] into Lq[0,1]. We prove that its Kolmogorov numbers tend to zero faster than exp(−cαn1/2). This extends former results of Laptev in the case p=q=2 and of the authors for p=2 and q=∞. As application we investigate compactness properties of related integral operators as, for example, of the difference between the fractional integration operators of Riemann–Liouville and Weyl type. It is shown that both types of fractional integration operators possess the same degree of compactness. In some cases this allows to determine the strong asymptotic behavior of the Kolmogorov numbers of Riemann–Liouville operators.