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1. CMB 2006 (vol 49 pp. 82)

Gogatishvili, Amiran; Pick, Luboš
 Embeddings and Duality Theorem for Weak Classical Lorentz Spaces We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$\left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),$$ where $$f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds.$$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$\|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),$$ where $$f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.$$ Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequalityCategories:26D10, 46E20
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