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Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 167–183 | Cite as

Lipschitz Linearization of the Maximal Hyperbolic Cross Multiplier

  • Olli SaariEmail author
  • Christoph Thiele
Article

Abstract

We study the linearized maximal operator associated with dilates of the hyperbolic cross multiplier in dimension two. Assuming a Lipschitz condition and a lower bound on the linearizing function, we obtain \(L^{p}(\mathbb {R}^{2}) \rightarrow L^{p}(\mathbb {R}^{2})\) bounds for all \(1<p <\infty \). We discuss various related results.

Keywords

Hyperbolic cross Multipliers Maximal functions Square functions 

Mathematics Subject Classification

Primary 42B15 42B25 

Notes

Acknowledgements

The authors would like to thank Andreas Seeger for bringing this problem to their attention, and Joris Roos for suggesting to look at general \(\beta \). Part of the research was done during the first author’s stay at the Mathematical Institute of the University of Bonn, which he wishes to thank for its hospitality. The first author acknowledges support from the Väisälä Foundation, the Academy of Finland, and the NSF Grant No. DMS-1440140 (the paper was finished while in residence at MSRI, Berkeley, California, during the Spring 2017 semester). The second author acknowledges support from the Hausdorff center of Mathematics and DFG Grant CRC 1060.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Systems AnalysisAalto University School of ScienceAaltoFinland
  2. 2.Institute of MathematicsUniversity of BonnBonnGermany

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