Abstract
The operator T, defined by convolution with the affine arc length measure on the moment curve parametrized by \(h(t)=(t,t^{2},\ldots ,t^{d})\) is a bounded operator from \(L^{p}\) to \(L^{q}\) if \((\frac{1}{p}, \frac{1}{q})\) lies on a line segment. In this article we prove that at non-end points there exist functions which extremize the associated inequality and any extremizing sequence is pre compact modulo the action of the symmetry of T. We also establish a relation between extremizers for T at the end points and the extremizers of an X-ray transform restricted to directions along the moment curve. Our proof is based on the ideas of Michael Christ on convolution with the surface measure on the paraboloid.
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Acknowledgements
I would like to thank my Ph.D. adviser, Betsy Stovall, for suggesting this problem and for many insightful comments and numerous discussions during the course of this work. This work would have been impossible without her guidance. I also like to thank Almut Burchard and Michael Christ for their many valuable remarks, in particular Christ’s suggestion to look at the X-ray transform for the end point case. I would like to thank Andreas Seeger for pointing out few references and many suggestions that improved the exposition of this article. This work was supported in part by NSF grants DMS-1266336 and DMS-1600458. Part of this work was carried out when the author visited the Mathematical Sciences Research Institute as a program associate in the Harmonic Analysis program in Spring 2017, supported by DMS-1440140. A version of this article formed part of the author’s Ph.D. thesis. I would like to thank the referees for their comments and suggestions which significantly improved the exposition of this article.
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Communicated by Hans G. Feichtinger.
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This work is supported in part by NSF Grants DMS-1266336 and DMS-1600458.
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Biswas, C. Existence of Extremizers for a Model Convolution Operator. J Fourier Anal Appl 25, 2653–2689 (2019). https://doi.org/10.1007/s00041-019-09677-x
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DOI: https://doi.org/10.1007/s00041-019-09677-x