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arXiv · 1906.02060

Estimates for matrix coefficients of representations

Tommaso Bruno·Michael G. Cowling·Fabio Nicola·Anita Tabacco
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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 Schrödinger equation in $\mathbb{R}^n$.

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Tommaso Bruno, Michael G. Cowling, Fabio Nicola, Anita Tabacco. 2019-06-05. Estimates for matrix coefficients of representations. https://arxiv.org/abs/1906.02060

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