Remainder estimates for the approximation numbers of weighted hardy operators acting on L2

We consider the weighted Hardy integral operator T : L-2 (a, b) --> L-2 (a, b), -infinity less than or equal to a < b less than or equal to infinity, defined by (Tf) (x) = v(x) integral (x)(a) u(t)f(t)dt. In [EEH1] and [EEH2], under certain conditions on u and v, upper and lower estimates and asymptotic results were obtained for the approximation numbers a, (T) of T. In this paper, we show that under suitable conditions on u and v, lim(n --> infinity) sup n(1/2) \ (1)/(pi)integral (b)(a)\u(t)v(t)\ dt - na(n) (T)\ less than or equal to c(\ \u'\ \ (2/3) + \ \v'\ \ (2/3))(\ \u \ \ (2) + \ \v \ \ (2)) + (3)/(pi)\ \ uv \ \ (1), where \ \w \ \ (p) = (integral (b)(a)\w(t)\ (P)dt)(1/p).