Let $G$ be a locally compact group and $f \in L^1(G).$ Define a linear map $L_f : L^2(G) \longrightarrow L^2(G)$ by $g \longmapsto f \ast g,\ $ $g \in L^2(G).$ Show that $L_f$ is bounded.
Here we need to show that $\|f \ast g\|_2 \leq C \|g\|_2$ for some $C \gt 0.$ It is not so inconsequent to expect that this constant has something to do with $\|f\|_2.$ But inspite of pondering over it for a long time I couldn't able to come up with the inequality I desired to. Could anyone give some suggestion in this regard which would enable me to lead to some progress? Any such cooperation will be highly appreciated.
Thanks a bunch.
EDIT $:$ What I got is the following $:$
$$\|f \ast g\|_2 \leq \|g\|_2 \left (\int_G \int_G |f(t - s)|^2\ ds\ dt \right )^{\frac {1} {2}}.$$
Now what to do?