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Transactions of the American Mathematical Society
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
Published electronically: December 20, 2005
MathSciNet review: 2197436
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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

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


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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: http://dx.doi.org/10.1090/S0002-9947-05-04097-3
PII: S 0002-9947(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.