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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 Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality
MSC Classifications: 26D10, 46E20 show english descriptions Inequalities involving derivatives and differential and integral operators
Hilbert spaces of continuous, differentiable or analytic functions
26D10 - Inequalities involving derivatives and differential and integral operators
46E20 - Hilbert spaces of continuous, differentiable or analytic functions
 

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