2022/03/26 by Michael Cowling, Cowling, Michael G. · 1 citation
Mathematics · #22E45 #22E46 #46L05 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #FOS: Mathematics #Representation Theory (math.RT) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2203.13989
openalex publication_date 2022/03/26 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
Let G be a connected semisimple Lie group with finite centre and K be a maximal compact subgroup thereof. Given a function u on G, we define A u to be the root mean square average over K, acting both on the left and the right, of u. We show that for all unitary representations π of G, there exists a unique minimal positive-real-valued spherical function ϕλ on G such that A ⟨ π(⋅) ξ, η⟩ ≤ \Vert ξ\VertHπ \Vert η\VertHπ ϕλ. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates \vert ⟨ π(⋅) ξ, η⟩ \vert ≤ C(ξ, η) ϕλ for K-finite or smooth vectors ξ and η, and it exhibits different decay rates in different directions at infinity in G. Further, if we assume the latter inequality with arbitrary C( ξ, η), we can prove the former inequality and then return to the latter inequality with explicit knowledge of C( ξ, η). On the other hand, it holds everywhere in G, in contrast to asymptotic estimates which are not global. We also provide some applications.