$L^p$ estimates for maximal averages along one-variable vector fields in ${\mathbf R} ^2$
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- by Michael Bateman PDF
- Proc. Amer. Math. Soc. 137 (2009), 955-963 Request permission
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 \begin{eqnarray} M_{\mathcal R} f (z ) = \sup _{z\in R \in \mathcal R} {1 \over { |R| } } \int _{R} |f|\end{eqnarray} is bounded on $L^p$ for $p>1$ with constants comparable to ${1 \over {\delta } }$.References
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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
- Received by editor(s): January 1, 1800
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 955-963
- MSC (2000): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-08-09583-X
- MathSciNet review: 2457435