# Estimates for matrix coefficients of representations

## TL;DR

This paper introduces a new estimate for matrix coefficients of certain representations of SL(2,R) and the metaplectic representation, unifying previous $L^p$ and pointwise estimates, with applications to dispersive PDEs.

## Contribution

It develops a novel estimate that combines the strengths of existing $L^p$ and pointwise bounds for specific unitary representations, advancing understanding in representation theory and PDEs.

## Key findings

- New estimate for SL(2,R) and metaplectic representations
- Unifies $L^p$ and pointwise matrix coefficient estimates
- Provides a dispersive $L^2$ estimate for Schrödinger equation

## Abstract

Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, $L^p$ estimates, which are a dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of $\mathrm{SL}(2,\mathbb{R})$ and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new $L^2$ estimate of dispersive type for the free Schr\"odinger equation in $\mathbb{R}^n$.