Suppose $m(x)$ is a $L^p(R^n)$ multiplier, $f\in L^1(R^n)$,prove that the convolution $m\star f$ is also a $L^p(R^n)$ multiplier, and satisfy $$\|T_{m\star f\quad}\|_{L^p\rightarrow L^p}\leq\|f\|_1\|T_m\|_{L^p\rightarrow L^p}$$.
I can only prove the case $p=2$ by proving that $f\star m$ is bounded. And I try to use interpolation but failed.
Any idea will be helpful. Thanks.