Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weighted rearrangement inequalities for local sharp maximal functions

Author: Andrei K. Lerner
Journal: Trans. Amer. Math. Soc. 357 (2005), 2445-2465
MSC (2000): Primary 42B20, 42B25
Published electronically: October 28, 2004
MathSciNet review: 2140445
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.

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

  • [ABKP] J. Alvarez, R.J. Bagby, D.S. Kurtz and C. Pérez, Weighted estimates for commutators of linear operators, Studia Math. 104(1993), no. 2, 195-209. MR 94k:47044
  • [AP] J. Alvarez and C. Pérez, Estimates with $A\sb \infty$weights for various singular integral operators, Boll. Un. Mat. Ital. A (7) 8(1994), no. 1, 123-133. MR 95f:42027
  • [AKMP] I.U. Asekritova, N.Ya. Krugljak, L. Maligranda and L.-E. Persson, Distribution and rearrangement estimates of the maximal function and interpolation, Studia Math. 124(1997), no. 2, 107-132. MR 98g:46032
  • [BK] R.J. Bagby and D.S. Kurtz, Covering lemmas and the sharp function, Proc. Amer. Math. Soc. 93(1985), 291-296. MR 86f:42011
  • [BDS] C. Bennett, R. DeVore and R. Sharpley, Weak- $L^{\infty}$ and $BMO$, Ann. of Math. 113(1981), 601-611. MR 82h:46047
  • [BS] C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, New York, 1988. MR 89e:46001
  • [CF1] R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia. Math. 15(1974), 241-250. MR 50:10670
  • [CF2] A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math. 57(1976), 97-101. MR 54:8132
  • [CP1] D. Cruz-Uribe and C. Pérez, Two-weight extrapolation via the maximal operator, J. Funct. Anal. 174(2000), 1-17. MR 2001g:42040
  • [CP2] D. Cruz-Uribe and C. Pérez, Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators, Indiana Univ. Math. J. 49 (2000), no. 2, 697-721. MR 2001i:42021
  • [CR] K.M. Chong and N.M. Rice, Equimeasurable rearrangements of functions, Queen's Papers in Pure and Appl. Math. 28, Queen's University, Kingston, Ont., 1971. MR 51:8357
  • [FS1] C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 44:2026
  • [FS2] C. Fefferman and E.M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 56:6263
  • [Ja] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263-270. MR 80j:42034
  • [JT] B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985), 231-270. MR 86k:42034
  • [Jo] F. John, Quasi-isometric mappings, Seminari 1962 - 1963 di Analisi, Algebra, Geometria e Topologia, Rome, 1965. MR 32:8315
  • [JN] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14(1961), 415-426. MR 24:A1348
  • [Ku] D.S. Kurtz, Operator estimates using the sharp function, Pacific J. Math. 139(1989), no. 2, 267-277. MR 90g:42036
  • [L1] A.K. Lerner, On weighted estimates of non-increasing rearrangements, East J. Approx. 4(1998), 277-290. MR 99k:42043
  • [L2] Z. Ercan, On the Hahn decomposition theorem, Real Anal. Exchange 28 (2002/03), no. 2, 611–615. MR 2010341,
  • [L3] A.K. Lerner, Weighted norm inequalities for the local sharp maximal function, J. Fourier Anal. Appl. 10 (2004), no. 5, 465-474.
  • [MMNO] J. Mateu, P. Mattila, A. Nicolau and J. Orobitg, BMO for nondoubling measures, Duke Math. J. 102(2000), no.3, 533-565. MR 2001e:26019
  • [Pe] C. Pérez, Weighted norm inequalities for singular integral operators, J. London Math. Soc. 49(1994), 296-308. MR 94m:42037
  • [St] E.M. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton, 1993. MR 95c:42002
  • [Str] J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511-544. MR 81f:42021
  • [W1] J.M. Wilson, Weighted inequalities for the dyadic square function without dyadic $A\sb{\infty}$, Duke Math. J. 55 (1987), 19-49. MR 88d:42034
  • [W2] J.M. Wilson, A sharp inequality for the square function, Duke Math. J. 55(1987), 879-887. MR 89a:42029
  • [W3] J.M. Wilson, $L\sp p$ weighted norm inequalities for the square function, $0<p<2$, Ill. J. Math. 33 (1989), no.3, 361-366. MR 90g:42037
  • [W4] J.M. Wilson, Weighted norm inequalities for the continuous square functions, Trans. Amer. Math. Soc. 314(1989), 661-692. MR 91e:42025

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B20, 42B25

Retrieve articles in all journals with MSC (2000): 42B20, 42B25

Additional Information

Andrei K. Lerner
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel

Keywords: Weighted rearrangements, sharp maximal functions, singular integrals
Received by editor(s): September 10, 2003
Received by editor(s) in revised form: December 4, 2003
Published electronically: October 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society