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A new class of restricted type spaces

Part of: Linear function spaces and their duals Inequalities

Published online by Cambridge University Press:  11 April 2011

Salvador Rodríguez-López  and
Javier Soria
Salvador Rodríguez-López
Affiliation:
Department of Applied Mathematics and Analysis, University of Barcelona, 08007 Barcelona, Spain (soria@ub.edu)
Javier Soria
Affiliation:
Department of Applied Mathematics and Analysis, University of Barcelona, 08007 Barcelona, Spain (soria@ub.edu)
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Abstract

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We find new properties for the space R(X), introduced by Soria in the study of the best constant for the Hardy operator minus the identity. In particular, we characterize when R(X) coincides with the minimal Lorentz space Λ(X). The condition that R(X) ≠ {0} is also described in terms of the embedding (L1, ∞ ∩ L∞) ⊂ X. Finally, we also show the existence of a minimal rearrangement-invariant Banach function space (RIBFS) X among those for which R(X) ≠ {0} (which is the RIBFS envelope of the quasi-Banach space L1, ∞ ∩ L∞).

Keywords

rearrangement-invariant spacesrestricted typeLorentz spaces

MSC classification

Secondary: 26D10: Inequalities involving derivatives and differential and integral operators 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
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 749 - 759
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Ariño, M. A. and Muckenhoupt, B., Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for non-increasing functions, Trans. Am. Math. Soc. 320(2) (1990), 727–735.Google Scholar
2.Bennett, C. and Sharpley, R., Interpolation of operators, Pure and Applied Mathematics, Volume 129 (Academic Press, 1988).Google Scholar
3.Carro, M. J., Pick, L., Soria, J. and Stepanov, V. D., On embeddings between classical Lorentz spaces, Math. Inequal. Applic. 4(3) (2001), 397–428.Google Scholar
4.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, Volume 187, Number 877 (American Mathematical Society, Providence, RI, 2007).Google Scholar
5.Carro, M. J. and Soria, J., Weighted Lorentz spaces and the Hardy operator, J. Funct. Analysis 112(2) (1993), 480–494.CrossRefGoogle Scholar
6.Carro, M. J. and Soria, J., Boundedness of some integral operators, Can. J. Math. 45(6) (1993), 1155–1166.CrossRefGoogle Scholar
7.Kruglyak, N. and Setterqvist, E., Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions, Proc. Am. Math. Soc. 136(7) (2008), 2505–2513.Google Scholar
8.Lorentz, G. G., Some new functional spaces, Annals Math. (2) 51(1) (1950), 37–55.Google Scholar
9.Soria, J., Lorentz spaces of weak-type, Q. J. Math. 49(193) (1998), 93–103.CrossRefGoogle Scholar
10.Soria, J., Optimal bounds of restricted type for the Hardy operator minus the identity on the cone of radially decreasing functions, Studia Math. 197 (2010), 69–79.CrossRefGoogle Scholar
11.Stepanov, V. D., The weighted Hardy's inequality for non-increasing functions, Trans. Am. Math. Soc. 338(1) (1993), 173–186.Google Scholar