Embeddings and Duality Theorems for Weak Classical Lorentz Spaces

Gogatishvili, Amiran ; Pick, Luboš

Canadian mathematical bulletin, Tome 49 (2006) no. 1, pp. 82-95

Résumé

We characterize the weight functions $u,v,w$ on $\left( 0,\infty\right)$ such that $${{\left( \int\limits_{0}^{\infty }{{{f}^{*}}{{\left( t \right)}^{q}}w\left( t \right)}\,dt \right)}^{1/q}}\le C\,\,\underset{t\in \left( 0,\infty\right)}{\mathop{\sup }}\,{{f}_{u}}^{**}\left( t \right)v\left( t \right),$$ where $${{f}_{u}}^{**}\left( t \right):={{\left( \int\limits_{0}^{t}{u\left( s \right)}\,ds \right)}^{-1}}\int\limits_{0}^{t}{{{f}^{*}}}\left( s \right)u\left( s \right)\,ds.$$ As an application we present a new simple characterization of the associate space to the space ${{\Gamma }^{\infty }}\left( v \right)$ , determined by the norm $${{\left\| f \right\|}_{\Gamma \infty \left( v \right)}}=\,\underset{t\in \left( 0,\infty\right)}{\mathop{\sup }}\,{{f}^{**}}\left( t \right)v\left( t \right),$$ where $${{f}^{**}}\left( t \right):=\frac{1}{t}\int\limits_{0}^{t}{{{f}^{*}}}\left( s \right)\,ds.$$

Détail

DOI : 10.4153/CMB-2006-008-3

Mots-clés : 26D10, 46E20, Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality