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)

 
 

 

Limiting weak-type behavior for singular integral and maximal operators


Author: Prabhu Janakiraman
Journal: Trans. Amer. Math. Soc. 358 (2006), 1937-1952
MSC (2000): Primary 42B20, 42B25
DOI: https://doi.org/10.1090/S0002-9947-05-04097-3
Published electronically: December 20, 2005
MathSciNet review: 2197436
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operator $ T$ in $ \mathbb{R}^n$: for $ f\in L^1(\mathbb{R}^n)$, $ f\geq 0$,

$\displaystyle \lim_{\lambda\rightarrow 0} \lambda\hspace{1mm}m\{x\in\mathbb{R}^... ...a\} = \frac{1}{n} \int_{S^{n-1}}\vert\Omega(x)\vert d\sigma(x)\Vert f\Vert _1.$

For the maximal operator $ M$, the corresponding result is

$\displaystyle \lim_{\lambda\rightarrow 0} \lambda\hspace{1mm}m\{x\in\mathbb{R}^n: \vert Mf(x)\vert>\lambda\} = \Vert f\Vert _1.$


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

  • 1. A. Baernstein II, Some sharp inequalities for conjugate functions, Indiana University Mathematics Journal 27 (1978), 833-852. MR 0503717 (80g:30022)
  • 2. R. Bañuelos, Martingale transforms and related singular integrals, Transactions of the American Mathematical Society 293 (1986), 547-564. MR 0816309 (87h:60095)
  • 3. R. Bañuelos and G. Wang, Sharp inequalities for Martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Mathematical Journal 80, No. 3, (1995), 575-600. MR 1370109 (96k:60108)
  • 4. R. Bañuelos and G. Wang, Davis's inequality for orthogonal martingales under differential subordination, Michigan Math. J. 47 (2000), no. 1, 109-124. MR 1755259 (2001g:60100)
  • 5. B. Davis, On the weak type $ (1,1)$ inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 241-249. MR 0348381 (50:879)
  • 6. B. Davis, On the distribution of conjugate functions of nonnegative measures, Duke Math. J. 40 (1973), 695-700. MR 0324297 (48:2649)
  • 7. J. Duoandikoetxea, Fourier Analysis, American Mathematical Society, Providence, R.I., 2001. MR 1800316 (2001k:42001)
  • 8. J. Duoandikoetxea and J.L. Rubio de Francia, Estimations independantes de la dimension pour les transformees de Riesz, C. R. Acad. Sci. 300, Serie I (1985), 193-196. MR 0780616 (86e:42028)
  • 9. A. Fiorenza, A note on the spherical maximal function, Rend. Accad. Sci. Fis. Mat. Napoli (4) 54 (1987), 77-83. MR 1005720 (91f:42016)
  • 10. T. Iwaniec and G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25-57. MR 1390681 (97k:42033)
  • 11. P. Janakiraman, Weak-type estimates for singular integrals and Riesz transform, Indiana University Mathematics Journal 53 (2004), no. 2, 533-555. MR 2060044 (2005e:42048)
  • 12. S. Korry, Fixed points of the Hardy-Littlewood maximal operator, Collect. Math. 52 (2001), no. 3, 289-294. MR 1885223 (2003d:42032)
  • 13. S.K. Pichorides, On the best value of the constants in the theorems of M. Riesz, Zygmund, and Kolmogorov, Studia Math. 44 (1972), 165-179. MR 0312140 (47:702)
  • 14. G. Pisier, Riesz transforms: a simpler analytic proof of P.-A. Meyer's inequality, Séminaire de Probabilités, XXII, 485-501, Lecture Notes in Math., 1321, Springer, Berlin, 1988. MR 0960544 (89m:60178)
  • 15. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. MR 0290095 (44:7280)
  • 16. E.M. Stein, Some results in harmonic analysis in $ \mathbb{R}^n$ for $ n\rightarrow \infty$, Bull. Amer. Math. Soc. 9 (1983), 71-73. MR 0699317 (84g:42019)
  • 17. E.M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of the International Congress of Mathematicians, 1986, Berkeley, CA. MR 0934224 (89d:42028)
  • 18. E.M. Stein with assistance of T.S. Murphy, Harmonic Analysis: Real-Variable methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. MR 1232192 (95c:42002)
  • 19. E.M. Stein and J.-O. Strömberg, Behavior of maximal functions in $ \mathbb{R}^n$ for large $ n$, Ark. Mat. 21 (1983), no. 2, 259-269. MR 0727348 (86a:42027)
  • 20. E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, 1971. MR 0304972 (46:4102)
  • 21. E.M. Stein and G. Weiss, An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959), 263-284. MR 0107163 (21:5888)

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

Prabhu Janakiraman
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906
Address at time of publication: Department of Mathematics, University of Illinois–Champaign, Urbana, Illinois 61801
Email: pjanakir@math.purdue.edu, pjanakir@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9947-05-04097-3
Received by editor(s): September 25, 2003
Published electronically: December 20, 2005
Additional Notes: This paper is part of the author’s thesis work under the direction of Professor Rodrigo Bañuelos. The research was partly supported by an NSF grant.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society