Improved estimates for the approximation numbers of Hardy-type operators

Consider the Hardy-type integral operator T : L-p (a,b) L-p (a, b), -infinity less than or equal to a < b less than or equal to infinity, which is defined by (Tf)(x) = v(x) integral(a)(x) u(t)f(t) dt. In the papers by Edmunds et a]. (J. London Math. Soc. (2) 37 (1988) 471) and Evans et a]. (Studia Math. 130 (2) (1998) 171) upper and lower estimates and asymptotic results were obtained for the approximation numbers a(n) (T) of T. In case p = 2 for "nice" u and v these results were improved in Edmunds et al. (J. Anal. Math. 85 (2001) 225). In this paper, we extend these results for 1 < p < infinity by using a new technique. We will show that under suitable conditions on u and v, lim(n-->infinity)sup n(1/2)\lambda(p)(-1/p) integral(a)(b) \u(t)v(t)\ dt - na(n) (T)\ less than or equal to c(\\u'\\(p'/p'+1)) + \\upsilon'\\(p/(p+1)))(\\u\\(p') + \\upsilon\\(p)) + 3alpha(p) \\uv\\(1), where \\w\\(p) = (integral(a)(b)\w(t)\(p) dt)(1/p) and lambda(p) is the first eigenvalue of the p-Laplacian eigenvalue f", problem on (0, 1). (C) 2002 Elsevier Science (USA). All rights reserved.