Is there an equivalent to $\|f \ast g\|_p \leq \|f\|_1 \|g\|_p$ on groups? I'm new to Fourier Analysis on LCA groups and I'm wondering if there is an equivalent to the inequality
$$\|f \ast g\|_p \leq \|f\|_1 \|g\|_p,~~~~~\forall f \in L^1(\mathbb{R}), \forall g \in L^p(\mathbb{R}).$$
I couldn't find anything of the kind in Rudin's Fourier Analysis on Groups. 

There is an equivalent version of Minokowski's inequality for locally compact groups, namely the following:

Let $G$ be a locally compact group, and let $1\leq p \leq \infty$. Suppose $f \in L^1 (G)$ and $g \in L^p (G)$. Then $f \ast g$ exists $\mu-a.e.$ and
  $$ \|f \ast g \|_{L^p} \leq \| f \|_{L^1} \| g \|_{L^p}. $$

The proof of this result is primarily the same as for $G = \mathbb{R}$. The result is stated and proved in Grafakos' Classical Fourier Analysis as Theorem 1.2.10.
