Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Singular integrals along $ N$ directions in $ \mathbb{R}^2$

Author: Ciprian Demeter
Journal: Proc. Amer. Math. Soc. 138 (2010), 4433-4442
MSC (2010): Primary 42B20; Secondary 42B25
Published electronically: June 11, 2010
MathSciNet review: 2680067
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Abstract: We prove optimal bounds in $ L^2(\mathbb{R}^2)$ for the maximal operator obtained by taking a singular integral along $ N$ arbitrary directions in the plane. We also give a new proof for the optimal $ L^2$ bound for the single scale Kakeya maximal function in the plane.

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Ciprian Demeter
Affiliation: Department of Mathematics, Indiana University, 831 East 3rd Street, Bloomington, Indiana 47405

Keywords: Kakeya maximal function, singular integrals
Received by editor(s): January 12, 2010
Received by editor(s) in revised form: February 12, 2010
Published electronically: June 11, 2010
Additional Notes: The author is supported by a Sloan Research Fellowship and by NSF Grants DMS-0742740 and 0901208
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.