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 when regarded as operator from L_p0,1] into L_q0,1]. We prove that its Kolmogorov numbers tend to zero faster than exp(−c_αn^1/2). 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. In memoria of Eduard (University of the West Indies) who passed away in October 2004.