# Strichartz estimates for the metaplectic representation

## TL;DR

This paper establishes Strichartz estimates for the metaplectic representation, extending dispersive analysis to non-compact Lie groups using modulation and Lebesgue spaces, with implications for harmonic analysis and representation theory.

## Contribution

It provides the first uniform weak-type sharp estimates and Strichartz estimates for the metaplectic representation, linking dispersive phenomena to time-frequency analysis tools.

## Key findings

- Proves sharp weak-type estimates for matrix coefficients.
- Establishes Strichartz estimates for the metaplectic group.
- Identifies modulation spaces as suitable function spaces for analysis.

## Abstract

Strichartz estimates are a manifestation of a dispersion phenomenon, exhibited by certain partial differential equations, which is detected by suitable Lebesgue space norms. In most cases the evolution propagator $U(t)$ is a one parameter group of unitary operators. Motivated by the importance of decay estimates in group representation theory and ergodic theory, Strichartz-type estimates seem worth investigating when $U(t)$ is replaced by a unitary representation of a non-compact Lie group, the group element playing the role of time. Since the Schr\"odinger group is a subgroup of the metaplectc group, the case of the metaplectic or oscillatory representation is of special interest in this connection. We prove uniform weak-type sharp estimates for matrix coefficients and Strichartz estimates for that representation. The crucial point is the choice of function spaces able to detect such a dispersive effect, which in general will depend on the given group action. The relevant function spaces here turn out to be the so-called modulation spaces from Time-frequency Analysis in Euclidean space, and Lebesgue spaces with respect to Haar measure on the metaplectic group. The proofs make use in an essential way of the covariance of the Wigner distribution with respect to the metaplectic representation.