2023/10/17 by Tapendu Rana, Michael Ruzhansky, Rana, Tapendu +1
Mathematics · #33C80 #43A90 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Primary: 22E30 #Secondary: 22E46
paper · pdf · doi:10.48550/arxiv.2310.11371
openalex publication_date 2023/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we explore the boundedness properties of pseudo-differential operators on radial sections of line bundles over the Poincaré upper half plane, even when dealing with symbols of limited regularity. We first prove the boundedness of these operators when the symbol is smooth. To achieve this, we establish a connection between the operator norm of the local part of our pseudo-differential operators and the corresponding Euclidean pseudo-differential operators. Additionally, we introduce a class of rough symbols that lack any regularity conditions in the space variable and investigate the boundedness properties of the associated pseudo-differential operators. As a crucial part of our proof, we provide asymptotic estimates and functional identities for certain matrix coefficients of the principal and discrete series representations of the group SL(2,ℝ).