fa.functional analysis - For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?

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

Now, consider a Lie group $G$ endowed with its Haar measure, and $pgeq1$. We may formulate a property

(T1) If $f,gin L^p(G)$ 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 $Ktimes mathbb T^m$, where $Ksubseteq mathbb R^n$ is compact, and $g(x,y)=f(x,y)h(y)$, where $hin L^p(mathbb T^m)$ and $int_{mathbb T^m}h =0$].

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

(T2) There exists an open set $Usubseteq G$ with compact closure, such that if $f,gin L^p(G)$ 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 $p=2$?