Let $f\in L^1(\mathbb R^n)$. Define operator $T_f(g)=|f|\ast g$ for functions $g$ on $\mathbb R^n$. The set of measurable functions $f$ on $\mathbb R^n$, such that $T_f$ is bounded from $L^p(\mathbb R^n)$ to $L^p(\mathbb R^n)$ is $L^1(\mathbb R^n)$, which is a Banach algebra.

There are locally compact topological (unimodular) groups where the set of such functions can be bigger than $L^1$. (That is a stronger Young's inequality is true there.) The group $G = \mathrm {SL}(2, \mathbb R)$ is an example.

Is this true that for $G = \mathrm{SL}(2, \mathbb {R})$, the set of measurable functions $f$ such that $T_f$ is bounded from $L^p(G)$ to $L^p(G)$ is a Banach algebra?