Limiting weak–type behavior for singular integral and maximal operators
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- by Prabhu Janakiraman PDF
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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 $f\in L^1(\mathbb {R}^n)$, $f\geq 0$, \[ \lim _{\lambda \rightarrow 0} \lambda \hspace {1mm}m\{x\in \mathbb {R}^n: |Tf(x)|>\lambda \} = \frac {1}{n} \int _{S^{n-1}}|\Omega (x)|d\sigma (x)\|f\|_1.\] For the maximal operator $M$, the corresponding result is \[ \lim _{\lambda \rightarrow 0} \lambda \hspace {1mm}m\{x\in \mathbb {R}^n: |Mf(x)|>\lambda \} = \|f\|_1.\]References
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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
- 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.
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2197436