Embeddings and associated spaces of Copson-Lorentz spaces
  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 $m,p\in(0,\infty)$. 
Martin K\v{r}epela
math.FA