On weighted inequalities for singular integrals

Authors:
H. Aimar, L. Forzani and F. J. Martín-Reyes

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2057-2064

MSC (1991):
Primary 42B25

DOI:
https://doi.org/10.1090/S0002-9939-97-03787-8

MathSciNet review:
1376747

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in then the one-sided condition, , is a sufficient condition for the singular integral to be bounded in , , or from into weak- if . This one-sided condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in . The two-sided version of this result is also obtained: Muckenhoupts condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in or in .

**[AS]**K.F. Andersen and E. T. Sawyer,*Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators*, Trans. Amer. Math. Soc.**308**(1988), 547-557. MR**89h:26006****[CF]**R. Coifman and C. Fefferman,*Weighted norm inequalities for maximal functions and singular integrals*, Studia Math.**51**(1974), 241-250. MR**50:10670****[HMW]**R.A. Hunt, B. Muckenhoupt and R.L. Wheeden,*Weighted norm inequalities for the conjugate function and the Hilbert transform*, Trans. Amer. Math. Soc.**176**(1973), 261-274. MR**47:701****[MOT]**F. J. Martín-Reyes, P. Ortega Salvador and A. de la Torre,*Weighted inequalities for one-sided maximal functions*, Trans. Amer. Math. Soc.**319**(1990), 517-534. MR**90j:42047****[MPT]**F.J. Martín-Reyes, L. Pick and A. de la Torre,*condition*, Canadian Journal of Mathematics**45**(1993), 1231-1244. MR**94m:42042****[MT]**F. J. Martín-Reyes and A. de la Torre,*Two weight norm inequalities for fractional one-sided maximal operators*, Proc. Amer. Math. Soc.**117**(1993), 483-489. MR**94b:42010****[S]**E. Sawyer,*Weighted inequalities for the one sided Hardy-Littlewood maximal functions*, Trans. Amer. Math. Soc.**297**(1986), 53-61. MR**87k:42018****[T]**A. Torchinsky,*Real variable methods in Harmonic Analysis*, Academic Press, Inc., 1986. MR**88e:42001**

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Additional Information

**H. Aimar**

Affiliation:
Dept. Matematica, FIQ, Prop.CAI+D, INTEC, Gëmes 3450, 3000 Santa Fe, Argentina

Email:
haimar@fiqus.unl.edu.ar

**F. J. Martín-Reyes**

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

Email:
martin_reyes@ccuma.uma.es

DOI:
https://doi.org/10.1090/S0002-9939-97-03787-8

Keywords:
Singular integrals,
Calderon-Zygmund operators,
weights

Received by editor(s):
March 15, 1995

Received by editor(s) in revised form:
January 30, 1996

Additional Notes:
The research of the third author has been partially supported by D.G.I.C.Y.T. grant (PB91-0413) and Junta de Andalucía.

Communicated by:
J. Marshall Ash

Article copyright:
© Copyright 1997
American Mathematical Society