characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Question

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$?

In the case of the torus $\mathbb{T}$, it is easy to show that a Fourier multiplier $T \colon L^\infty(\mathbb{T}) \to L^\infty(\mathbb{T}), \sum_n a_ne^{in\theta}\mapsto \sum_n b_na_ne^{in\theta}$ is unital if and only if $b_0=1$.

Answer 1

The bounded Fourier multiplier operators on $L^\infty$ correspond to convolutions with finite Borel measures. The Fourier transform of $1$ is a Dirac located at the origin. Therefore the unital operators correspond to convolutions with measures $\mu$ satisfying

\begin{equation*} 1 = \widehat{\mu}(0) \end{equation*}