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

Bibliographie

[1] [1] Bennett, C. and Sharpley, R., Interpolation of Operators. Pure and Applied Mathematics 129, Academic Press, Boston, 1988. Google Scholar

[2] [2] Bradley, J. S., Hardy inequalities with mixed norms. Canad. Math. Bull. 49(1978), 405–408. Google Scholar

[3] [3] Carro, M., García del Amo, A. and Soria, J., Weak-type weights and normable Lorentz spaces. Proc. Amer.Math. Soc. 49(1996), 849–857. Google Scholar

[4] [4] Carro, M. and Soria, J., Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112(1993), 480–494. Google Scholar

[5] [5] Carro, M. and Soria, J., Boundedness of some integral operators. Canad. J. Math. 45(1993), 1155–1166. Google Scholar

[6] [6] Carro, M. and Soria, J., The Hardy-Littlewood maximal function and weighted Lorentz spaces. J. London Math. Soc. 49(1997), 146–158. Google Scholar

[7] [7] Carro, M., Soria, J., Pick, L. and Stepanov, V., On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 49(2001), 397–428. Google Scholar

[8] [8] Gogatishvili, A. and Pick, L., Discretization and anti-discretization of rearrangement-invariant norms. Publ. Mat. 49(2003), 311–358. Google Scholar

[9] [9] Gol’dman, M. L., On integral inequalities on a cone of functions with monotonicity properties. Soviet Math. Dokl. 44(1992), 581–587. Google Scholar

[10] [10] Gol’dman, M. L., On integral inequalities on the set of functions with some properties of monotonicity. In: Function spaces, Differential Operators and Nonlinear Analysis, Teubner Texte Zur Math. 133, Teubner, Stuttgart, 1993, pp. 274–279. Google Scholar

[11] [11] Gol’dman, M. L., Heinig, H. P. and Stepanov, V. D., On the principle of duality in Lorentz spaces. Canad. J. Math. 49(1996), 959–979. Google Scholar

[12] [12] Grosse-Erdmann, K.-G., The Blocking Technique, Weighted Mean Operators and Hardy's Inequality. Lecture Notes in Mathematics 1679, Springer-Verlag, Berlin, 1998. Google Scholar

[13] [13] Lorentz, G. G., On the theory of spaces Λ . Pacific J. Math. 49(1951), 411–429. Google Scholar

[14] [14] Opic, B. and Kufner, A., Hardy-type inequalities. Pitman Research Notes in Mathematics 219, Longman Sci., Harlow, 1990. Google Scholar

[15] [15] Sawyer, E., Boundedness of classical operators on classical Lorentz spaces. Studia Math. 49(1990), 145–158. Google Scholar

[16] [16] Sinnamon, G., Spaces defined by level functions and their duals. Studia Math. 49(1994), 19–52. Google Scholar

[17] [17] Sinnamon, G. and Stepanov, V. D., The weighted Hardy inequality: new proofs and the case p = 1 . J. London Math. Soc. 49(1996), 89–101. Google Scholar

[18] [18] Stepanov, V. D., The weighted Hardy's inequality for nonincreasing functions. Trans. Amer. Math. Soc. 49(1993), 173–186. Google Scholar