Hostname: page-component-7dd5485656-zklqj Total loading time: 0 Render date: 2025-10-31T03:23:14.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Iterating Bilinear Hardy Inequalities

Part of: Inequalities Linear function spaces and their duals Integral, integro-differential, and pseudodifferential operators

Published online by Cambridge University Press:  30 January 2017

Martin Křepela
Martin Křepela*
Affiliation:
Karlstad University, Faculty of Health, Science and Technology, Department of Mathematics and Computer Science, 651 88 Karlstad, Sweden (martin.krepela@kau.se) Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Praha 8, Czech Republic
Get access Rights & Permissions [Opens in a new window]

Abstract

An iteration technique for characterizing boundedness of certain types of multilinear operators is presented, reducing the problem to a corresponding linear-operator case. The method gives a simple proof of a characterization of validity of the weighted bilinear Hardy inequality

for all non-negative f, g on (a, b), for 1 < p 1, p 2, q < ∞. More equivalent characterizing conditions are presented.

The same technique is applied to various further problems, in particular those involving multilinear integral operators of Hardy type.

Keywords

Hardy operatorsbilinear operatorsweightsoperator inequalities

MSC classification

Secondary: 26D10: Inequalities involving derivatives and differential and integral operators 47G10: Integral operators 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 955 - 971
Copyright
Copyright © Edinburgh Mathematical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Aguilar Cañestro, M. I., Ortega Salvador, P. and Ramírez Torreblanca, C., Weighted bilinear Hardy inequalities, J. Math. Analysis Applic. 387(1) (2012), 320–334.Google Scholar
2. Bennett, C. and Sharpley, R., Interpolation of operators, Pure and Applied Mathematics, Volume 129 (Academic Press, 1988).Google Scholar
3. Bradley, J., Hardy inequalities with mixed norms, Can. Math. Bull. 21(4) (1978), 405–408.Google Scholar
4. Carro, M., Pick, L., Soria, J. and Stepanov, V. D., On embeddings between classical Lorentz spaces, Math. Inequal. Applic. 4 (2001), 397–428.Google Scholar
5. Gogatishvili, A. and Stepanov, V. D., Reduction theorems for operators on the cones of monotone functions, J. Math. Analysis Applic. 405 (2013), 156–172.CrossRefGoogle Scholar
6. Gogatishvili, A., Johansson, M., Okpoti, C. A. and Persson, L.-E., Characterisation of embeddings in Lorentz spaces, Bull. Austral. Math. Soc. 76(1) (2007), 69–92.CrossRefGoogle Scholar
7. Gogatishvili, A., Kufner, A. and Persson, L.-E., Some new scales of weight characterizations of the class Bp , Acta Math. Hungar. 123 (2009), 365–377.CrossRefGoogle Scholar
8. Gogatishvili, A., Persson, L.-E., Stepanov, V. D. and Wall, P., Some scales of equivalent conditions to characterize the Stieltjes inequality: the case q < p , Math. Nachr. 287 (2014), 242–253.Google Scholar
9. Křepela, M., Convolution inequalities in weighted Lorentz spaces, Math. Inequal. Applic. 17(4) (2014), 1201–1223.Google Scholar
10. Křepela, M., Convolution in rearrangement-invariant spaces defined in terms of oscillation and the maximal function, Z. Analysis Anwend. 33(4) (2014), 369–383.Google Scholar
11. Křepela, M., Convolution in weighted Lorentz spaces of type Γ , Math. Scand. 119(1) (2016), DOI: 10.7146/math.scand.a-24187.Google Scholar
12. Křepela, M., Bilinear weighted Hardy inequality for nonincreasing functions, Publ. Mat., 61(1) (2017), 3–50.Google Scholar
13. Kufner, A. and Persson, L.-E., Weighted inequalities of Hardy type (World Scientific, 2003).Google Scholar
14. Maz’ja, V. G., Sobolev spaces (Springer, 1985).CrossRefGoogle Scholar
15. Muckenhoupt, B., Hardy's inequality with weights, Studia Math. 44 (1972), 31–38.Google Scholar
16. O’Neil, R., Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963), 129–142.Google Scholar
17. Stepanov, V. D., The weighted Hardy's inequality for nonincreasing functions, Trans. Am. Math. Soc. 338 (1993), 173–186.Google Scholar
18. Yap, L. Y. H., Some remarks on convolution operators and L(p, q) spaces, Duke Math. J. 36 (1969), 647–658.Google Scholar