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# Embeddings and Duality Theorem for Weak Classical Lorentz Spaces

Published:2006-03-01
Printed: Mar 2006
• Amiran Gogatishvili
• Luboš Pick
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## Abstract

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 inequality
 MSC Classifications: 26D10 - Inequalities involving derivatives and differential and integral operators 46E20 - Hilbert spaces of continuous, differentiable or analytic functions

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