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

In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator $T: L^p(ℝ^+) → L^p(ℝ^+)$ defined by $(Tf)(x) ≔ v(x) ʃ_{0}^{∞} u(t)f(t)dt$ when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3]