On the sharpness of Strichartz estimates and spectrum of compact Lie groups
Duv\'an Cardona, Brian Grajales, Michael Ruzhansky

TL;DR
This paper establishes Strichartz estimates on compact simple Lie groups, linking spectral analysis with number theory, and provides explicit spectral formulas and regularity conditions for these estimates.
Contribution
It introduces a new regularity order for Strichartz estimates on compact Lie groups and connects the problem to counting representations as sums of squares using number theory methods.
Findings
Proves Strichartz estimates for all compact connected simple Lie groups.
Provides explicit spectral parametrization in terms of sums of squares.
Derives new regularity conditions for estimates when p approaches 2.
Abstract
We prove Strichartz estimates on any compact connected simple Lie group. In the diagonal case of Bourgain's exponents we provide a new regularity order in the sense that our (reverse) Strichartz estimates are valid when and when As expected our Sobolev index satisfies the estimate Motivated by the recent progress in the field, in the spirit of the analytical number theory methods developed by Bourgain in the analysis of periodic Schr\"odinger equations, we link the problem of finding Strichartz estimates on compact Lie groups with the problem of counting the number of representations of an integer number as a sum of squares, and then, we implicitly use the very well known bounds for from the Hardy-Littlewood-Ramanujan circle method. In our…
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Taxonomy
TopicsAdvanced Mathematical Physics Problems · Mathematical Analysis and Transform Methods · Advanced Harmonic Analysis Research
