On weighted weak type inequalities

for modified Hardy operators

Authors:
F. J. Martín-Reyes and P. Ortega

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1739-1746

MSC (1991):
Primary 26D15

DOI:
https://doi.org/10.1090/S0002-9939-98-04247-6

MathSciNet review:
1443843

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Abstract | References | Similar Articles | Additional Information

Abstract: We characterize the pairs of weights for which the modified Hardy operator applies into weak- where is a monotone function and .

**[AM]**K. F. Andersen and B. Muckenhoupt,*Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions*, Studia Math.**72**(1982), 9-26. MR**83k:42018****[B]**J. Bradley,*Hardy inequalities with mixed norms*, Canad. Math. Bull.**21**(1978), 405-408. MR**80a:26005****[F]**E. V. Ferreyra,*Weighted Lorentz norm inequalities for integral operators*, Studia Math**96**(1990), 125-134. MR**91c:26025****[M]**F. J. Martín-Reyes,*New proofs of weighted inequalities for the one sided Hardy-Littlewood maximal functions*, Proc. Amer. Math. Soc.**117**(1993), 691-698. MR**93d:42016****[MOS]**F. J. Martín-Reyes, P. Ortega Salvador and M. D. Sarrión Gavilán,*Boundedness of operators of Hardy type in spaces and weighted mixed inequalities for singular integral operators*, Proc. Roy. Soc. Edinburgh Sect. A**127**(1997), 157-170. CMP**97:08****[MT]**F. J. Martín-Reyes and A. de la Torre,*Some weighted inequalities for general one-sided maximal operators*, Studia Math.**122**(1997), 1-14. CMP**97:06****[Ma]**V. G. Mazja,*Sobolev Spaces*, Springer-Verlag Berlin Heidelberg New York Tokyo, 1985. MR**87g:46056****[Mu]**B. Muckenhoupt,*Hardy's inequality with weights*, Studia Math.**44**(1972), 31-38. MR**47:418****[Q]**Lai Qinsheng,*A note on the weighted norm inequality for the one-sided maximal operator*, Proc. Amer. Math. Soc.**124**(1996), 527-537. MR**96d:42029****[S]**E. T. Sawyer,*Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator*, Trans. Amer. Math. Soc.**281**(1984), 329-337. MR**85f:26013**

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Additional Information

**F. J. Martín-Reyes**

Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain

Email:
martin@anamat.cie.uma.es

**P. Ortega**

Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain

Email:
ortega@anamat.cie.uma.es

DOI:
https://doi.org/10.1090/S0002-9939-98-04247-6

Keywords:
Hardy operators,
weights,
inequalities

Received by editor(s):
September 8, 1995

Received by editor(s) in revised form:
December 1, 1996

Additional Notes:
This research has been partially supported by D.G.I.C.Y.T. grant (PB94-1496) and Junta de Andalucía

Communicated by:
J. Marshall Ash

Article copyright:
© Copyright 1998
American Mathematical Society