Embeddings between weighted Copson and Cesàro function spaces

In this paper, characterizations of the embeddings between weighted Copson function spaces Copp1,q1(u1,v1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)$$\end{document} and weighted Cesàro function spaces Cesp2,q2(u2,v2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)$$\end{document} are given. In particular, two-sided estimates of the optimal constant c in the inequality (∫0∞(∫0tf(τ)p2v2(τ)dτ)q2/p2u2(t)dt)1/q2≤c(∫0∞(∫t∞f(τ)p1v1(τ)dτ)q1/p1u1(t)dt)1/q1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$\end{document} where p1, p2, q1, q2 ∈ (0,∞), p2 ≤ q2 and u1, u2, v1, v2 are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p1 and p2 and possibly different inner weights v1 and v2 are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.