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SHARP CONSTANTS BETWEEN EQUIVALENT NORMS IN WEIGHTED LORENTZ SPACES

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  22 January 2010

SORINA BARZA  and
JAVIER SORIA
SORINA BARZA
Affiliation:
Department of Mathematics, Karlstad University, SE-65188 Karlstad, Sweden (email: sorina.barza@kau.se)
JAVIER SORIA*
Affiliation:
Department of Applied Mathematics and Analysis, University of Barcelona, E-08007 Barcelona, Spain (email: soria@ub.edu)
*
✉For correspondence; e-mail: soria@ub.edu
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Abstract

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For an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

Keywords

equivalent normslevel functionweighted Lorentz spacessharp constantsweights

MSC classification

Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 1 , February 2010 , pp. 19 - 27
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This research was partially supported by grants MTM2007-60500 and 2005SGR00556.

References

[1]Ariño, M. A. and Muckenhoupt, B., ‘Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions’, Trans. Amer. Math. Soc. 320 (1990), 727–735.Google Scholar
[2]Barza, S., Kolyada, V. and Soria, J., ‘Sharp constants related to the triangle inequality in Lorentz spaces’, Trans. Amer. Math. Soc. 361 (2009), 5555–5574.CrossRefGoogle Scholar
[3]Bennett, C. and Sharpley, R., Interpolation of Operators (Academic Press, New York, 1988).Google Scholar
[4]Carro, M. J., Pick, L., Soria, J. and Stepanov, V. D., ‘On embeddings between classical Lorentz spaces’, Math. Inequal. Appl. 4 (2001), 397–428.Google Scholar
[5]Carro, M. J., Raposo, J. A. and Soria, J., Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities, Memoirs of the American Mathematical Society, 187 (American Mathematical Society, Providence, RI, 2007).CrossRefGoogle Scholar
[6]Carro, M. J. and Soria, J., ‘Weighted Lorentz spaces and the Hardy operator’, J. Funct. Anal. 112 (1993), 480–494.CrossRefGoogle Scholar
[7]Cerdà, J. and Martín, J., ‘Weighted Hardy inequalities and Hardy transforms of weights’, Studia Math. 139 (2000), 189–196.Google Scholar
[8]Halperin, I., ‘Function spaces’, Canad. J. Math. 5 (1953), 273–288.Google Scholar
[9]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, 2nd edn (Cambridge University Press, Cambridge, 1952).Google Scholar
[10]Kamińska, A. and Parrish, A. M., ‘Convexity and concavity constants in Lorentz and Marcinkiewicz spaces’, J. Math. Anal. Appl. 343 (2008), 337–351.CrossRefGoogle Scholar
[11]Korenovskiĭ, A. A., ‘The exact continuation of a reverse Hölder inequality and Muckenhoupt’s condition’, Mat. Zametki 52(6) (1992), 32–44 (Engl. Transl. Math. Notes 52(5–6) (1992), 1192–1201).Google Scholar
[12]Lorentz, G. G., ‘Some new functional spaces’, Ann. of Math. (2) 51 (1950), 37–55.CrossRefGoogle Scholar
[13]Lorentz, G. G., ‘On the theory of spaces Λ’, Pacific J. Math. 1 (1951), 411–429.Google Scholar
[14]Muckenhoupt, B., ‘Weighted norm inequalities for the Hardy maximal function’, Trans. Amer. Math. Soc. 165 (1972), 207–227.CrossRefGoogle Scholar
[15]Sawyer, E., ‘Boundedness of classical operators on classical Lorentz spaces’, Studia Math. 96 (1990), 145–158.CrossRefGoogle Scholar