# Embeddings and associated spaces of Copson-Lorentz spaces

Research paper by **Martin Křepela**

Indexed on: **12 Dec '16**Published on: **12 Dec '16**Published in: **arXiv - Mathematics - Functional Analysis**

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#### 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 $m,p\in(0,\infty)$.