Embeddings and associated spaces of Copson-Lorentz spaces

TL;DR

This paper characterizes when certain inequalities involving Copson-Lorentz spaces hold, providing conditions for embeddings into Lorentz spaces and describing associated spaces for all positive exponents.

Contribution

It offers a complete characterization of embeddings and associated spaces of Copson-Lorentz spaces with explicit conditions and estimates for the optimal constants.

Findings

Derived necessary and sufficient conditions for the inequality validity.

Provided explicit estimates for the optimal constant C.

Described the associated space of Copson-Lorentz spaces for all exponents.

Abstract

Let $m,p,q\in(0,\infty)$ and let $u,v,w$ be nonnegative weights. We characterize validity of the inequality \[ \left(\int_0^\infty w(t) (f^*(t))^q \, dt \right)^\frac 1q \le C \left(\int_0^\infty v(t) \left(\int_t^\infty u(s) (f^*(s))^m \,ds \right)^\frac pm \! dt \right)^\frac 1p \] for all measurable functions $f$ defined on $\mathbb{R}^n$ 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 $CL^{m,p}(u,v)$, generated by the functional \[ \|f\|_{{CL^{m,p}(u,v)}} := \left(\int_0^\infty v(t) \left(\int_t^\infty u(s) (f^*(s))^m \,ds \right)^\frac pm \! dt \right)^\frac 1p, \] into the Lorentz space $\Lambda^q(w)$. Moreover, the results are applied to describe the associated space of the Copson-Lorentz space ${CL^{m,p}(u,v)}$ for the full range of exponents $m,p\in(0,\infty)$.