Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-12T08:54:31.322Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted Generalized Hardy Inequalities for Nonincreasing Functions

Published online by Cambridge University Press:  20 November 2018

Kenneth F. Andersen
Kenneth F. Andersen*
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The nonnegative weight function pairs u, v for which the operator maps the nonnegative nonincreasing functions in LP(v) boundedly into weak Lq(u) are characterized. This result is used, in particular, both to generalize and to provide an alternate proof of certain strong type inequalities recently obtained by Ariño and Muckenhouptfor the Hardy averaging operator restricted to nonnegative nonincreasing functions.

Keywords

26D1542B25Weighted inequalitiesweak type inequalityHardy's inequality
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 6 , 01 December 1991 , pp. 1121 - 1135
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Andersen, K.F., Weighted inequalities for the Stieltjes transformation and Hubert's double series, Proc. Roy. Soc. Edinburgh 86A(1980), 75–84.Google Scholar
2. Andersen, K.F. and Muckenhoupt, B., Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72(1982), 9–26.Google Scholar
3. Ariño, M. and Muckenhoupt, B., Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for noninc re asing functions, Trans. Amer. Math. Soc. 320(1990), 727–735.Google Scholar
4. Bennett, C. and Sharpley, R., Interpolation of Operators. Academic Press, Orlando, 1988.Google Scholar
5. Boyd, D.W., The Hilbert transform on rearrangement-invariant spaces, Canad. J. Math. 19(1967), 599–616.Google Scholar
6. Braverman, M. Sh., On a class of operators, preprint.Google Scholar
7. Lai, S., Weighted norm inequalities for general operators on monotone functions, preprint.Google Scholar
8. Lai, S., Weak and strong type inequalities with weights for general operators on monotone functions, preprint.Google Scholar
9. Maźja, V.G., Sobolev Spaces. Springer-Verlag, Berlin, 1985.Google Scholar
10. Neugebauer, C. J., Weightednorm inequalitiesfor averaging operators ofmonotone functions, Publicationes Mathématiques, to appear.Google Scholar
11. Neugebauer, C. J., Some classical operators on Lorentz spaces, preprint.Google Scholar
12. Sawyer, E.T., Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96(1990), 145–158.Google Scholar
13. Stepanov, V.D., The weighted Hardy's inequality for nonincreasing functions, Trans. Amer. Math. Soc. (to appear).Google Scholar