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A Universal Stein-Tomas Restriction Estimate for Measures in Three Dimensions

  • Alex IosevichEmail author
  • Svetlana Roudenko
Chapter

Summary

We study restriction estimates in \({\mathbb{R}}^{3}\) for surfaces given as graphs of low regularity functions. We obtain a “universal” mixed-norm estimate for the extension operator \(f \rightarrow \widehat{ f\mu }\) in \({\mathbb{R}}^{3}\). We also prove that this estimate holds for any Frostman measure supported on a compact set of Hausdorff dimension greater than two. The approach is geometric and is influenced by a connection with the Falconer distance problem.

Keywords

Restriction estimates Measures 

Notes

Acknowledgements

A. I. thanks Michael Loss of Georgia Institute of Technology for a helpful suggestion regarding the regularity assumptions in the main result. He also thanks Andreas Seeger for a very helpful conversation about Haar measures. A.I. was partially supported by the NSF grant DMS-0456306. S. R. was partially supported by the NSF grant DMS-0531337. Part of this work was done while both authors participated in the MTBI Program at the Arizona State University and are grateful for their support.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University of MissouriColumbiaUSA
  2. 2.Arizona State UniversityTempeUSA

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