Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We characterize the pairs of weights $(w,v)$ for which the modified Hardy operator $Tf(x)=g(x)\int _{0}^{x}f$ applies $L^{p}(v)$ into weak-$L^{q}(w)$ where $g$ is a monotone function and $1\le q<p<\infty $.


References [Enhancements On Off] (What's this?)

  • [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 $\Lambda ^{p,q}$ 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 26D15

Retrieve articles in all journals with MSC (1991): 26D15


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

American Mathematical Society