Suppose that $m:\mathbb R \to \mathbb C$ satisfies: there exists $C > 0$ such that
$$\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}.$$
That is, $m$ is an $L^{p}$-multiplier. Let $M(L^{p})$ denote the space of $L^p$-multipliers.
Put $m_{\delta}(x)= m(\delta x), (\delta >0),$ $T_{k}m(x)=m(x-k), (k\in \mathbb R).$
My questions:
(1) If $m\in M(L^p) $, then can we say that $m_\delta, T_{k}m \in M(L^{p})$?
(2) Let $m\in M(L^p)$ and $\phi : \mathbb R \to \mathbb R$ be such that $m\circ \phi \in M(L^{p})$. What can we say about $\phi$?
If this is known, references would be helpful.