Asymptotic estimates for the approximation and entropy numbers of the one-weight Riemann-Liouville operator

For alpha>0, let T_{alpha,v}f(x) = v(x) int_0^x (x-y)^{alpha-1} f(y) dy,quad x>0, be the weighted Riemann-Liouville operator. In [D. V. Prokhorov, J. London Math. Soc. (2) {bf 61} (2000), no.~2, 617--628; [msn] MR1760684 (2001b:47082) [/msn]; D. V. Prokhorov and V. D. Stepanov, Tr. Mat. Inst. Steklova {bf 243} (2003), Funkts. Prostran., Priblizh., Differ. Uravn., 289--312; [msn] MR2054439 (2005d:47089) [/msn]] criteria for the boundedness and compactness of T_{alpha,v} colon L_p(0,infty) rightarrow L_q(0,infty), quad 1