Remainder estimates for the approximation numbers of weighted Hardy operators acting onL ²

Abstract

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by$(Tf)(x) = v(x)\smallint _a^x u(t)f(t)dt$. In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv,$\begin{gathered} \mathop {\lim \sup }\limits_{n \to \infty } n^{1/2} \left| {\frac{1}{\pi }\int {_a^b \left| {u(t)v(t)} \right|dt - na_n (T)} } \right| \hfill \\ \leqslant c(\left\| {u'} \right\|_{2/3} + \left\| {v'} \right\|_{2/3} )(\left\| u \right\|_2 + \left\| v \right\|_2 ) + \frac{3}{\pi }\left\| {uv} \right\|_1 , \hfill \\ \end{gathered}$ where ∥w∥_p=(∫ ^b_a |w(t)|^p dt)^1/p.