Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-22T14:03:30.105Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundedness of operators of Hardy type in ΛP,q spaces and weighted mixed inequalities for singular integral operators*

Published online by Cambridge University Press:  14 November 2011

F. J. Martín-Reyes ,
P. Ortega Salvador  and
M. D. Sarrión Gavilán
F. J. Martín-Reyes
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
P. Ortega Salvador
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
M. D. Sarrión Gavilán
Affiliation:
Departamento de Economía Aplicada (Estadística y Econometría), Facultad de Ciencias Económicas, Universidad de Málaga, 29071 Málaga, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider certain n-dimensional operators of Hardy type and we study their boundedness in These spaces were introduced by M. J. Carro and J. Soria and include weighted Lp, q spaces and classical Lorentz spaces. As an application, we obtain mixed weak-type inequalities for Calderón—Zygmund singular integrals, improving results due to K. Andersen and B. Muckenhoupt.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 1 , 1997 , pp. 157 - 170
Copyright
Copyright © Royal Society of Edinburgh 1997

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.)

References

1Andersen, K. F. and Muckenhoupt, B.. Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions. Studia Math. 72 (1982), 9–26.CrossRefGoogle Scholar
2Bradley, J.. Hardy inequalities with mixed norms. Canad. Math. Bull. 21 (1978), 405–08.CrossRefGoogle Scholar
3Carro, M. J. and Soria, J.. Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112 (1993), 480–94.CrossRefGoogle Scholar
4Ferreyra, E. V.. Weighted Lorentz norm inequalities for integral operators. Studia Math. 96 (1990), 125–34.CrossRefGoogle Scholar
5García-Cuerva, J. and Francia, J. L. Rubio de. Weighted norm inequalities and related topics (Amsterdam: North–Holland, 1985).Google Scholar
6Martín-Reyes, F. J.. New proofs of weighted inequalities for the one sided Hardy–Littlewood maximal functions. Proc. Amer. Math. Soc. 117 (1993), 691–8.CrossRefGoogle Scholar
7Muckenhoupt, B.. Hardy's inequality with weights. Studia Math. 34 (1972), 31–8.CrossRefGoogle Scholar
8Muckenhoupt, B.. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207–26.CrossRefGoogle Scholar
9Opic, B. and Kufner, A.. Hardy-type inequalities (Harlow: Longman Scientific and Technical, 1990).Google Scholar
10Sawyer, E. T.. Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans. Amer. Math. Soc. 281 (1984), 329–37.CrossRefGoogle Scholar
11Sawyer, E. T.. Boundedness of classical operators in classical Lorentz spaces. Studia Math. 96 (1990), 145–58.CrossRefGoogle Scholar
12Talenti, G.. Osservazioni sopra una classe di disuguagliance. Rend. Sem. Mat. Fis. Milano 39 (1969), 171–85.CrossRefGoogle Scholar