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The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case of $f,g\in L^1(\mathbb R^n)$ with compact support.

Now, consider a Lie group $G$ endowed with its Haar measure, and $p\geq1$. We may formulate a property

(T1) If $f,g\in L^p(G)$ are nonzero and compactly supported, the convolution $f*g$ is nonzero.

This is true for $G=\mathbb R^n$, but fails for abelian groups of the form $G=\mathbb R^n \times \mathbb T^m$, where $\mathbb T^m$ is the $m$-dimensional torus, and $m>0$. [Counterexample: let $f$ be the characteristic function of $K\times \mathbb T^m$, where $K\subseteq \mathbb R^n$ is compact, and $g(x,y)=f(x,y)h(y)$, where $h\in L^p(\mathbb T^m)$ and $\int_{\mathbb T^m}h =0$].

However, we may strengthen the assumptions in (T1), obtaining

(T2) There exists an open set $U\subseteq G$ with compact closure, such that if $f,g\in L^p(G)$ are nonzero and supported in left translates of $U$, the convolution $f*g$ is nonzero.

This is actually true for any abelian Lie group $G$, since in this case $G$ is locally isomorphic to $\mathbb R^k$.

Of course, if (T1) or (T2) hold for some $p$, then they also hold for all $q>p$. The most interesting cases are $p=1$ and $p=2$.

So, my question is:

Which Lie groups are known to satisfy property (T1) or (T2) with $p=1$ or $p=2$?

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This nice question is of course related to the zero-divisor conjecture for discrete groups (considered as zero-dimensional Lie groups if you want). Asking this for connected topological groups (or Lie groups), a first more specific question would be: Does the existence of torsion in $G$ lead to zero-divisors in $L^1(G)$? –  Andreas Thom Aug 13 '10 at 12:18
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