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<p,q<infty, were studied. In the present paper the authors determine the compactness further in terms of (the asymptotic behaviour of) its entropy and approximation numbers, e_n(T_{alpha,v}) and a_n(T_{alpha,v}). This is in some sense parallel to the comprehensive literature on related questions for the Hardy operator and also complements related results on Weyl estimates and Schatten-von Neumann norms of singular values for p=q=2, as well as some further results by Solomyak [see J. Newman and M. Solomyak, Integral Equations Operator Theory {bf 20} (1994), no.~3, 335--349; [msn] MR1299892 (96a:47087) [/msn]; M. Solomyak, in {it Complex analysis, operators, and related topics}, 371--383, Birkhäuser, Basel, 2000; [msn] MR1771775 (2001d:47077) [/msn]]. The main result, Theorem 1.1, gives (two-sided) estimates for sup_n n^alpha e_n(T_{alpha,v}), limsup_{ntoinfty} n^alpha e_n(T_{alpha,v}), liminf_{ntoinfty} n^alpha e_n(T_{alpha,v}), and parallel results for a_n(T_{alpha,v}), involving special norms of the weight v. In the unweighted case vequiv 1, Theorem 2.2 characterizes the sharp asymptotic behaviour of the entropy and approximation numbers of T_alpha = T_{alpha, 1} in the form e_n(T_alpha) sim n^{-alpha}, and a corresponding expression for a_n(T_alpha).