Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the sharpness of Strichartz estimates and spectrum of compact Lie groups

Duván Cardona, Brian Grajales, Michael Ruzhansky

Abstract

We prove Strichartz estimates on any compact connected simple Lie group. In the diagonal case of Bourgain's exponents $p=q,$ we provide a new regularity order $s_{0}^{R}(p)$ in the sense that our (reverse) Strichartz estimates are valid when $s> s_{0}^{R}(p)$ and when $p\rightarrow 2^{+}.$ As expected our Sobolev index satisfies the estimate $ s_{0}^{R}(p)>s_{0}(d)=\frac{d}{2}-\frac{d+2}{p}.$ 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 Schrödinger equations, we link the problem of finding Strichartz estimates on compact Lie groups with the problem of counting the number of representations $r_{s,2}(R)$ of an integer number $R$ as a sum of $s$ squares, and then, we implicitly use the very well known bounds for $r_{s,2}(R)$ from the Hardy-Littlewood-Ramanujan circle method. In our analysis, we explicitly compute the parametrisation of the spectrum of the Laplacian (modulo a factor depending on the geometry of the group) in terms of sums of squares. As a byproduct, our approach also yields explicit expressions for the spectrum of the Laplacian on all compact connected simple Lie groups, providing also a number of results for Strichartz estimates in the borderline case $p=2.$

On the sharpness of Strichartz estimates and spectrum of compact Lie groups

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 Schrödinger 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 analysis, we explicitly compute the parametrisation of the spectrum of the Laplacian (modulo a factor depending on the geometry of the group) in terms of sums of squares. As a byproduct, our approach also yields explicit expressions for the spectrum of the Laplacian on all compact connected simple Lie groups, providing also a number of results for Strichartz estimates in the borderline case
Paper Structure (8 sections, 17 theorems, 165 equations, 1 table)

This paper contains 8 sections, 17 theorems, 165 equations, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Main result
  4. Tools from Lie theory and from analytic number theory
  5. Spectrum of the Laplacian on compact simple Lie groups
  6. Tools from analytic number theory
  7. Reverse Strichartz estimates
  8. Data statement

Key Result

Theorem 1.1

Let $2\leq p<\infty.$ Let $(G,g)$ be a compact connected simple Lie group of rank $\ell\geq 2$ endowed with a Riemmanian bi-invariant metric $g$ and let $\tilde{G}$ be its universal cover. Let Let $T=2\pi b \gamma,$ where $b$ is a non-negative integer depending on $G$ and $\gamma$ is a geometric constant defined by the metric $g.$ Then where if $m\geq 5,$$s> s_{0}^R(p,q)$ if $m=2,3,4$ and

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1: Theorem of the highest weight
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • ...and 24 more