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Maximal operators on spaces of homogeneous type
Author(s):
Gladis
Pradolini;
Oscar
Salinas
Journal:
Proc. Amer. Math. Soc.
132
(2004),
435-441.
MSC (2000):
Primary 42B25
Posted:
June 30, 2003
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Abstract:
We avoid the assumption given in the work of C. Pérez and R. Wheeden (2001) to prove boundedness properties of certain maximal functions in a general setting of the spaces of homogeneous type with infinite measure. In addition, an example shows that the result can be false if the space has finite measure.
References:
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- [A]
- Aimar, H.:``Singular integrals and approximate identities on spaces of homogeneous type", Trans. Amer. Math. Soc., Vol. 292, No.1 (1985), pp. 135-153 MR 86m:42022
- [BS]
- Bernardis, A. and Salinas, O.:``Two-weighted inequalities for certain maximal fractional operators on spaces of homogeneous type", Revista de la Unión Matemática Argentina, Vol. 41, 3, (1999) MR 2001e:42024
- [CP1]
- Cruz-Uribe, D. and Pérez, C.:``Two weight extrapolation via the maximal operator", J. Funct. Anal. 174 (2000), no. 1, 1-17. MR 2001g:42040
- [CP2]
- Cruz-Uribe, D. and Pérez, C.: ``Sharp two-weight, weak-type norm inequalities for singular integral operators", Math. Res. Lett. 6 (1999), pp. 417-428 MR 2000k:42020
- [MS]
- Macías, R. and Segovia, C.:``Lipschitz functions on spaces of homogeneous type", Advances in Math. 33 (1979) pp. 257-270 MR 81c:32017a
- [MST]
- Macías, R., Segovia, C. and Torrea, J.:``Singular integral operators with non-necessarily bounded kernels on spaces of homogeneous type", Advances in Math., Vol. 93, No. 1 (1992) MR 93h:42018
- [P1]
- Pérez, C.: ``On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted
 -spaces with different weights", Proc. London Math. Soc. (3) 71, 1995, pp. 135-157 MR 96k:42023 - [P2]
- Pérez, C.: ``Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function", Journal of Fourier Analysis and Applications, Vol. 3, No. 6, 1997, pp. 743-756 MR 99f:42042
- [P3]
- Pérez, C.: ``Sharp weighted inequalities for the vector-valued maximal function", Trans. Amer. Math. Soc., Vol. 352, No. 7 (2000), pp. 3265-3288 MR 2000j:42033
- [P4]
- Pérez, C.: ``Two weighted inequalities for potential and fractional type maximal operators", Indiana Univ. Math. J. 43, 1994, pp. 663-683 MR 95m:42028
- [PW]
- Pérez, C. and Wheeden, R.: ``Uncertainty principle estimates for vector fields", Journal of Functional Analysis, 181, 2001, pp. 146-188 MR 2002h:42035
- [RR]
- Rao, M. and Ren, Z.: ``Theory of Orlicz spaces", Marcel Dekker, Inc., New York, 1991 MR 92e:46059
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Additional Information:
Gladis
Pradolini
Affiliation:
Department of Mathematics, Universidad Nacional del Litoral, Instituto de Mate- mática Aplicada del Litoral (IMAL), Güemes 3450, 3000 Santa Fe, Argentina
Email:
gprado@math.unl.edu.ar
Oscar
Salinas
Affiliation:
Department of Mathematics, Universidad Nacional del Litoral, Instituto de Mate- mática Aplicada del Litoral (IMAL), Güemes 3450, 3000 Santa Fe, Argentina
Email:
salinas@ceride.gov.ar
DOI:
10.1090/S0002-9939-03-07079-5
PII:
S 0002-9939(03)07079-5
Keywords:
Maximal operator,
spaces of homogeneous type
Received by editor(s):
July 29, 2002
Received by editor(s) in revised form:
September 30, 2002
Posted:
June 30, 2003
Additional Notes:
The authors were supported by Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina and Universidad Nacional del Litoral
Communicated by:
Andreas Seeger
Copyright of article:
Copyright
2003,
American Mathematical Society
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