is fulfilled (see Lemma \[lem:kernel\_continuity\]), we need a couple of auxiliary results.

\[prop:RCK\_bounded\] Assume that for all $f\in  L_{r,m}(G) $, $f$ and $K$ are convolvable (in the sense that $f \cdot \lambda(x)\widecheck{K} \in L_1(G)$ for almost all $x\in G$) and $f\ast K\in  L_{r,m}(G) $, then the right convolution operator $$RC_K:  L_{r,m}(G) \to  L_{r,m}(G) , \qquad RC_Kf= f\ast K,$$ is bounded.

For $r=2$ the result is stated in [@Halmos Proposition 3.10], whose proof holds true for any $p$. Indeed, by the closed graph theorem, it is enough to show that $RC_K$ is a closed operator. Take a sequence $(f_n)_{n\in\mathbb{N}}$ converging to $f\in   L_{r,m}(G) $ such that $(RC_Kf_n)_{n\in\mathbb{N}}$ converges to $g\in L_{r,m}(G) $. By a sharp version of the Riesz-Fischer theorem, see [@Hitchhiker Theorem 13.6], there exists a positive function $g\in L_{r,m}(G) $ such that, possibly passing twice to a subsequence, there exist two null sets $E,F$ such that for all $y\in G \setminus E$ and $x\in G \setminus F$ $$\begin{aligned}
{1}
 & |f_n(y)|\leq g(y),   \\
 & \lim_{n\to\infty} f_n(y) = f(y), \\
 & \lim_{n\to\infty}RC_K f_n(x) = g(x).\end{aligned}$$ Furthermore, by definition of convolution and possibly re-defining the null set $F$, we get that for all $x\in G \setminus F$ and all $n\in\mathbb{N}$ the 