Cesàro means of Fourier transforms and multipliers on $L^ 1(\textbf {R})$

Abstract:

We prove that the Cesàro mean $\sigma$ of a multiplier $\lambda$ on ${L^1}({\mathbf {R}})$ is also a multiplier on ${L^1}({\mathbf {R}})$. In the particular cases when (i) $\lambda$ is odd, we prove that $\sigma$ is the Fourier transform of an odd function in the Hardy space ${H^1}({\mathbf {R}})$, and (ii) $\lambda$ is even, we give a necessary and sufficient condition in order that $\sigma$ be a Fourier transform of an even function in ${L^1}({\mathbf {R}})$. As a corollary, we obtain a nontrivial condition for $\lambda$ in order to be a multiplier on ${L^1}({\mathbf {R}})$; namely, \[ \int _0^\infty {\left | {\frac {1}{t}\int _0^t {\{ \lambda (\xi ) - \lambda ( - \xi )\} d\xi } } \right |} \frac {{dt}}{t} < \infty .\] We also prove Hardy type inequalities for multipliers and Hilbert transforms.