Cesàro means of Fourier transforms and multipliers on 𝐿¹(𝑅)

Abstract

We prove that the Cesàro mean σ \sigma of a multiplier λ \lambda on L 1 ( R ) {L^1}({\mathbf {R}}) is also a multiplier on L 1 ( R ) {L^1}({\mathbf {R}}) . In the particular cases when (i) λ \lambda is odd, we prove that σ \sigma is the Fourier transform of an odd function in the Hardy space H 1 ( R ) {H^1}({\mathbf {R}}) , and (ii) λ \lambda is even, we give a necessary and sufficient condition in order that σ \sigma be a Fourier transform of an even function in L 1 ( R ) {L^1}({\mathbf {R}}) . As a corollary, we obtain a nontrivial condition for λ \lambda in order to be a multiplier on L 1 ( R ) {L^1}({\mathbf {R}}) ; namely, \[ ∫ 0 ∞ | 1 t ∫ 0 t { λ ( ξ ) − λ ( − ξ ) } d ξ | d t t > ∞ . \int _0^\infty {\left | {\frac {1}{t}\int _0^t {\{ \lambda (\xi ) - \lambda ( - \xi )\} \,d\xi } } \right |} \frac {{dt}}{t} > \infty . \] We also prove Hardy type inequalities for multipliers and Hilbert transforms.