Two-sided estimates for the approximation numbers of Hardy-type operators in ${\displaystyle L^{\infty }}$ and L¹

In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator ${\displaystyle T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)}$ defined by ${\displaystyle \left(Tf\right)\left(x\right)≔v\left(x\right){ʃ}_0^{\infty }u\left(t\right)f\left(t\right)dt}$ when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].