estimates for maximal averages along one-variable vector fields in

Author:
Michael Bateman

Journal:
Proc. Amer. Math. Soc. **137** (2009), 955-963

MSC (2000):
Primary 42B25

DOI:
https://doi.org/10.1090/S0002-9939-08-09583-X

Published electronically:
September 5, 2008

MathSciNet review:
2457435

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let be a vector field defined on the unit square such that for some measurable . Let be a small parameter, and let be the collection of rectangles of a fixed width such that much of the vector field inside is pointed in (approximately) the same direction as . We show that the operator defined by

(1) |

is bounded on for with constants comparable to .

**1.**Carbery, Anthony, Andreas Seeger, Stephen Wainger, and James Wright.*Classes of singular integral operators along variable lines*, J. Geom. Anal., vol. 9, no. 4, 1999, pp. 583-605. MR**1757580 (2001g:42026)****2.**Cordoba, Antonio, and Robert Fefferman.*On differentiation of integrals*, Proc. Natl. Acad. Sci. USA, vol. 74, no. 6, June 1977, pp. 2211-2213. MR**0476977 (57:16522)****3.**Cordoba, Antonio, and Robert Fefferman.*A geometric proof of the strong maximal theorem*, Annals of Math., 2nd Series, vol. 102, no. 1, July 1975, pp. 95-100. MR**0379785 (52:690)****4.**Karagulyan, Grigor.*On unboundedness of maximal operators for directional Hilbert transforms*, Proc. Amer. Math. Soc., vol. 135, no. 10, 2007, pp. 3133-3141 (electronic). MR**2322743 (2008e:42044)****5.**Kim, Joonil.*Sharp bound of maximal Hilbert transforms over arbitrary sets of directions*. (English summary), J. Math. Anal. Appl., vol. 335, no. 1, 2007, pp. 56-63. MR**2340304****6.**Lacey, Michael, and Xiaochun Li.*On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields*. Available at`http://arxiv.org/abs/0704.0808`.**7.**Lacey, Michael, and Xiaochun Li.*On a Lipschitz Variant of the Kakeya Maximal Function*. Available at`http://arxiv.org/abs/math/0601213`.**8.**Strömberg, Jan-Olov.*Maximal functions associated to rectangles with uniformly distributed directions*, Annals of Math., 2nd Series, vol. 107, no. 2, 1978, pp. 399-402. MR**0481883 (58:1978)**

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Additional Information

**Michael Bateman**

Affiliation:
Department of Mathematics, Indiana University, Rawles Hall, 831 East 3rd Street, Bloomington, Indiana 47405

Email:
mdbatema@indiana.edu

DOI:
https://doi.org/10.1090/S0002-9939-08-09583-X

Received by editor(s):
January 1, 2018

Received by editor(s) in revised form:
January 1, 2008

Published electronically:
September 5, 2008

Additional Notes:
This work was supported in part by NSF Grant DMS0653763.

Communicated by:
Michael T. Lacey

Article copyright:
© Copyright 2008
American Mathematical Society