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$ L^p$ estimates for maximal averages along one-variable vector fields in $ {\mathbf R} ^2$


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
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Abstract: We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let $ v$ be a vector field defined on the unit square such that $ v(x,y) = (1,u(x))$ for some measurable $ u: [0,1] \rightarrow [0,1]$. Let $ \delta$ be a small parameter, and let $ \mathcal R$ be the collection of rectangles $ R$ of a fixed width such that $ \delta$ much of the vector field inside $ R$ is pointed in (approximately) the same direction as $ R$. We show that the operator defined by


$\displaystyle M_{\mathcal R} f (z ) = \sup _{z\in R \in \mathcal R} {1 \over { \vert R\vert } } \int _{R} \vert f\vert$     (1)

is bounded on $ L^p$ for $ p>1$ with constants comparable to $ {1 \over {\delta} }$.


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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

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