Wiener Tauberian theorems for \(SL_ 2(\mathbb{R} )\)
The author proves a Wiener Tauberian theorem for \(L^p (SL_2 (\mathbb{R}))\), \(1\leq p<2\). Let \(G\) be the group \(SL_2 (\mathbb{R})\) and \(K\) its maximal compact subgroup \(SO(2, \mathbb{R})\). Let \(M\) be \(\{\pm I\}\). We show that if the Fourier transforms of a set of functions in \(L^p(G)\) do not vanish simultaneously on any irreducible \(L^{p- \varepsilon}\)-tempered representation for some \(\varepsilon >0\), where they are assumed to be defined, and if for each \(M\)-type at least one of the matrix coefficients of any of those Fourier transforms does not `decay too rapidly at \(\infty\)' in a certain sense, then this set of functions generate \(L^p (G)\) as an \(L^1 (G)\)-bimodule. As a key step towards this main theorem we prove a W-T theorem for \(L^p\)-sections of certain line bundles over \(G/K\). W-T theorems on \(SL_2 (\mathbb{R})\) have been proved so far, for biinvariant \(L^1\) functions and for \(L^1\) functions on the symmetric space \(SL_2 (\mathbb{R})/ SO(2, \mathbb{R})\), where the generator is left \(K\)-finite. Our results are on the space of all \(L^p\) functions (resp. sections), \(p\in [1, 2)\) of \(SL_2 (\mathbb{R})\) (resp. of line bundles over \(SL_2 (\mathbb{R})/ SO (2, \mathbb{R})\)), without any restriction of \(K\)-finiteness on the generators.
scientific article; zbMATH DE number 950222