The celebrated Wiener Tauberian theorem asserts that for $ f \in L^1(\mathbb{R})$, the closed ideal generated by the function $f$ is equal to the whole of $ L^1(\mathbb{R})$ if and only if its Fourier transform $\hat f $ is nowhere vanishing on $\mathbb{R}$. The analogous result holds for locally compact abelian groups.

However in 1955, L. Ehrenpreis and F. I. Mautner observed that the corresponding result is not true for the commutative Banach algebra $L^1(G//K)$ of $K$-biinvariant functions on $G$ and proved Wiener Tauberian theorem with additional conditions, for $G= \mathrm{SL(2,\mathbb{R})}$ and $ K=\mathrm{SO}(2) $. Their result is ameliorated by Y. Ben Natan et al. In their paper, the authors studied the analog of the Wiener Tauberian theorem for the Banach algebra $ L^1( \mathrm{SL(2,\mathbb{R})} //\mathrm{SO}(2))$.

In this talk, we will discuss an analog of the Wiener Tauberian theorem for the Lorentz spaces $L^{p,1}(\mathrm {SL}(2, \mathbb{R}))$, $1\leq p<2$.

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Last updated: 06 Jun 2023