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On Fourier restriction type problems on compact Lie groups. (English) Zbl 07786040

Summary: In this article, we obtain new results for Fourier restriction-type problems on compact Lie groups. We first provide a sharp form of \(L^p\) estimates of irreducible characters in terms of their Laplace-Beltrami eigenvalue, and as a consequence provide some sharp \(L^p\) estimates of joint eigenfunctions for the ring of conjugate-invariant differential operators. Then, we improve upon the previous range of exponent for scale-invariant Strichartz estimates for the Schrödinger equation, and provide new \(L^p\) bounds of Laplace-Beltrami eigenfunctions in terms of their eigenvalue similar to known bounds on tori. A key ingredient in our proof of these results is a barycentric-semiclassical subdivision of the Weyl alcove in a maximal torus. On each component of this subdivision we carry out the analysis of characters and exponential sums, and the circle method of Hardy-Littlewood and Kloosterman.

MSC:

43A80 Analysis on other specific Lie groups
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
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