Approximation numbers and Kolmogorov widths of Hardy-type operators in a non-homogeneous case

Let I = [a, b] subset of R, let 1 < q <= p < infinity, let u and v be positive functions with u is an element of L-p' (I) and v is an element of L-q (I), and let T : L-p (I) -> L-q (I) be the Hardy-type operator given by (Tf)(x) = v(x) integral(x)(a) f (t)u(t) dt, x is an element of I. Given any n is an element of N, let s(n) stand for either the n-th approximation number of T or the n-th Kolmogorov width of T. We show that lim(n ->infinity) ns(n) = C-pq (integral(I) (uv)(1/r) dt)(r), r = 1/p' + 1/q, where c(pq) is an explicit constant depending only on p and q. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.