I would like to try Stein's next proposition, singular integrals and differentiability properties.
Proposition 1. Let T be a bounded linear transformation mapping $L^1(\mathbb{R}^n)$ to itself. Then a necessary and sufficient condition that $T$ commutes with traslations is that exists a measure $\mu$ in $\mathcal{B}(\mathbb{R}^n)$ so that $T(f)=f\ast \mu$ for all $f\in L^1(\mathbb{R}^n)$. One has then $\left\|T\right\|=\left\|\mu\right\|.$
How can I proves this?
