A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on \(\mathrm{SL}(2, \mathbb{R} )\)
The paper provides a genuine analogue of the Wiener Tauberian theorem for certain Lorentz spaces \(L^{1,p}(G)\), \(1 \leq p < 2\), where \(G\) is the group \(\mathrm{SL}(2,\mathbb{R})\); that is, an ideal generated by \(\Lambda\subseteq L^{1,p}(G)\) shall be dense in \(L^{1,p}(G)\) if the Fourier transforms \(\{\hat{f}: f \in \Lambda\}\) admit a certain non-vanishing condition and satisfy a so-called ``not decay too rapidly'' requirement.\par The proof of the theorem consists in first showing the theorem on suitably well-chosen subsets of \(L^{1,p}(G)\), where a dense set can be constructed as the span of certain eigenfunctions of the Casimir operator. The theorem on these subsets is proven by combining this construction with ideas on the resolvent transform. The theorem in general is then shown by techniques inspired by \textit{R.~Sarkar} [Pac. J. Math. 177, 291--304 (1997; Zbl 0868.43010)].
scientific article; zbMATH DE number 7318800
scientific article