The space for nondoubling measures in terms of a grand maximal operator
Author:
Xavier Tolsa
Journal:
Trans. Amer. Math. Soc. 355 (2003), 315348
MSC (2000):
Primary 42B20; Secondary 42B30
Published electronically:
September 11, 2002
MathSciNet review:
1928090
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Abstract: Let be a Radon measure on , which may be nondoubling. The only condition that must satisfy is the size condition , for some fixed . Recently, some spaces of type and were introduced by the author. These new spaces have properties similar to those of the classical spaces and defined for doubling measures, and they have proved to be useful for studying the boundedness of CalderónZygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space in terms of a maximal operator is given. It is shown that belongs to if and only if , and , as in the usual doubling situation.
 [Ca]
L. Carleson. Two remarks on and , Advances in Math. 22 (1976), 269277. MR 57:16602
 [Co]
R. R. Coifman. A real variable characterization of , Studia Math. 51 (1974) 269274. MR 50:10784
 [CW]
R. R. Coifman and G. Weiss. Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569645. MR 56:6264
 [GR]
J. GarcíaCuerva and J. L. Rubio de Francia. Weighted norm inequalities and related topics, NorthHolland Math. Studies 116, 1985. MR 87d:42023
 [Jo]
J.L. Journé. CalderónZygmund operators, pseudodifferential operators and the Cauchy integral of Calderón. Lecture Notes in Math. 994, SpringerVerlag, 1983. MR 85i:42021
 [La]
R. H. Latter. A characterization of in terms of atoms, Studia Math. 62 1978, 92101. MR 58:2198
 [MS1]
R. A. Macías and C. Segovia. Lipschitz functions on spaces of homogeneous type, Advances in Math. 33 (1979), 257270. MR 81c:32017a
 [MS2]
R. A. Macías and C. Segovia. A decomposition into atoms of distributions on spaces of homogeneous type, Advances in Math. 33 (1979), 271309. MR 81c:32017b
 [MMNO]
J. Mateu, P. Mattila, A. Nicolau, and J. Orobitg. for nondoubling measures. Duke Math. J. 102 (2000), 533565. MR 2001e:26019
 [NTV1]
F. Nazarov, S. Treil, and A. Volberg. Cauchy integral and CalderónZygmund operators on nonhomogeneous spaces, Internat. Math. Res. Not. 15 (1997), 703726. MR 99e:42028
 [NTV2]
F. Nazarov, S. Treil, and A. Volberg. Weak type estimates and Cotlar inequalities for CalderónZygmund operators in nonhomogeneous spaces, Internat. Math. Res. Not. 9 (1998), 463487. MR 99f:42035
 [NTV3]
F. Nazarov, S. Treil, and A. Volberg. theorem on nonhomogeneous spaces. Preprint (1999).
 [NTV4]
F. Nazarov, S. Treil, and A. Volberg. Accretive systems on nonhomogeneous spaces, Duke Math J. 113 (2002), 259312.
 [OP]
J. Orobitg and C. Pérez. weights for nondoubling measures in and applications. Trans. Amer. Math. Soc. 354 (2002), 20132033.
 [St]
E. M. Stein. Harmonic analysis. RealVariable methods, orthogonality, and oscillatory integrals. Princeton Univ. Press. Princeton, N.J., 1993. MR 95c:42002
 [To1]
X. Tolsa. boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98:2 (1999), 269304. MR 2000d:31001
 [To2]
X. Tolsa. A theorem for non doubling measures with atoms. Proc. London Math. Soc. 82 (2001), 195228.
 [To3]
X. Tolsa. , , and CalderónZygmund operators for non doubling measures. Math. Ann. 319 (2001), 89149. MR 2002c:42029
 [To4]
X. Tolsa. A proof of the weak inequality for singular integrals with non doubling measures based on a CalderónZygmund decomposition. Publ. Mat. 45 (2001), 163174. MR 2002d:42019
 [To5]
X. Tolsa. Characterización of the atomic space for non doubling measures in terms of a grand maximal operator. Chalmers Univ. of Technology, Mathematics, Preprint 2000:2. http://www.math.chalmers.se/Math/Research/Preprints/.
 [Ve]
J. Verdera, On the theorem for the Cauchy integral, Arkiv Mat. 38 (2000), 183199. MR 2001e:30074
 [Uc]
A. Uchiyama. A maximal function characterization of on the space of homogeneous type, Trans. Amer. Math. Soc. 262:2, (1980), 579592. MR 81j:46085
 [Ca]
 L. Carleson. Two remarks on and , Advances in Math. 22 (1976), 269277. MR 57:16602
 [Co]
 R. R. Coifman. A real variable characterization of , Studia Math. 51 (1974) 269274. MR 50:10784
 [CW]
 R. R. Coifman and G. Weiss. Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569645. MR 56:6264
 [GR]
 J. GarcíaCuerva and J. L. Rubio de Francia. Weighted norm inequalities and related topics, NorthHolland Math. Studies 116, 1985. MR 87d:42023
 [Jo]
 J.L. Journé. CalderónZygmund operators, pseudodifferential operators and the Cauchy integral of Calderón. Lecture Notes in Math. 994, SpringerVerlag, 1983. MR 85i:42021
 [La]
 R. H. Latter. A characterization of in terms of atoms, Studia Math. 62 1978, 92101. MR 58:2198
 [MS1]
 R. A. Macías and C. Segovia. Lipschitz functions on spaces of homogeneous type, Advances in Math. 33 (1979), 257270. MR 81c:32017a
 [MS2]
 R. A. Macías and C. Segovia. A decomposition into atoms of distributions on spaces of homogeneous type, Advances in Math. 33 (1979), 271309. MR 81c:32017b
 [MMNO]
 J. Mateu, P. Mattila, A. Nicolau, and J. Orobitg. for nondoubling measures. Duke Math. J. 102 (2000), 533565. MR 2001e:26019
 [NTV1]
 F. Nazarov, S. Treil, and A. Volberg. Cauchy integral and CalderónZygmund operators on nonhomogeneous spaces, Internat. Math. Res. Not. 15 (1997), 703726. MR 99e:42028
 [NTV2]
 F. Nazarov, S. Treil, and A. Volberg. Weak type estimates and Cotlar inequalities for CalderónZygmund operators in nonhomogeneous spaces, Internat. Math. Res. Not. 9 (1998), 463487. MR 99f:42035
 [NTV3]
 F. Nazarov, S. Treil, and A. Volberg. theorem on nonhomogeneous spaces. Preprint (1999).
 [NTV4]
 F. Nazarov, S. Treil, and A. Volberg. Accretive systems on nonhomogeneous spaces, Duke Math J. 113 (2002), 259312.
 [OP]
 J. Orobitg and C. Pérez. weights for nondoubling measures in and applications. Trans. Amer. Math. Soc. 354 (2002), 20132033.
 [St]
 E. M. Stein. Harmonic analysis. RealVariable methods, orthogonality, and oscillatory integrals. Princeton Univ. Press. Princeton, N.J., 1993. MR 95c:42002
 [To1]
 X. Tolsa. boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98:2 (1999), 269304. MR 2000d:31001
 [To2]
 X. Tolsa. A theorem for non doubling measures with atoms. Proc. London Math. Soc. 82 (2001), 195228.
 [To3]
 X. Tolsa. , , and CalderónZygmund operators for non doubling measures. Math. Ann. 319 (2001), 89149. MR 2002c:42029
 [To4]
 X. Tolsa. A proof of the weak inequality for singular integrals with non doubling measures based on a CalderónZygmund decomposition. Publ. Mat. 45 (2001), 163174. MR 2002d:42019
 [To5]
 X. Tolsa. Characterización of the atomic space for non doubling measures in terms of a grand maximal operator. Chalmers Univ. of Technology, Mathematics, Preprint 2000:2. http://www.math.chalmers.se/Math/Research/Preprints/.
 [Ve]
 J. Verdera, On the theorem for the Cauchy integral, Arkiv Mat. 38 (2000), 183199. MR 2001e:30074
 [Uc]
 A. Uchiyama. A maximal function characterization of on the space of homogeneous type, Trans. Amer. Math. Soc. 262:2, (1980), 579592. MR 81j:46085
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Additional Information
Xavier Tolsa
Affiliation:
Département de Mathématique, Bâtiment 425, Université de ParisSud, 91405 OrsayCedex, France
Address at time of publication:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
Email:
xtolsa@mat.uab.es
DOI:
http://dx.doi.org/10.1090/S0002994702031318
PII:
S 00029947(02)031318
Keywords:
BMO,
atomic spaces,
Hardy spaces,
Calder\'onZygmund operators,
nondoubling measures,
maximal functions,
grand maximal operator
Received by editor(s):
October 31, 2000
Published electronically:
September 11, 2002
Additional Notes:
Supported by a postdoctoral grant from the European Commission for the TMR Network “Harmonic Analysis”. Also partially supported by grants DGICYT PB961183 and CIRIT 1998SGR00052 (Spain)
Article copyright:
© Copyright 2002
American Mathematical Society
