Embeddings and associated spaces of Copson—Lorentz spaces

Let m, p, q ∈ (0, ∞) and let u, v, w be nonnegative weights. We characterize validity of the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left(\int_{0}^{\infty} w(t)\left(f^{*}(t)\right)^{q} \mathrm{d} t\right)^{\frac{1}{q}} \leq C\left(\int_{0}^{\infty} v(t)\left(\int_{t}^{\infty} u(s)\left(f^{*}(s)\right)^{m} \mathrm{d} s\right)^{\frac{p}{m}} \mathrm{d} t\right)^{\frac{1}{p}}$$\end{document} for all measurable functions f defined on ℝn and provide equivalent estimates of the optimal constant C > 0 in terms of the weights and exponents. The obtained conditions characterize the embedding of the Copson—Lorentz space CLm,p(u, v), generated by the functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\|f\|_{C L^{m, p}(u, v)}:=\left(\int_{0}^{\infty} v(t)\left(\int_{t}^{\infty} u(s)\left(f^{*}(s)\right)^{m} \ \mathrm{d} s\right)^{\frac{p}{m}} \mathrm{d} t\right)^{\frac{1}{p}},$$\end{document} into the Lorentz space Λq(w). Moreover, the results are applied to describe the associated space of the Copson—Lorentz space CLm,p(u, v) for the full range of exponents m, p ∈ (0, ∞).