Functions which are restrictions of $L^{p}$-multipliers

Abstract:

Raouf Doss has given a sufficient condition for a measurable function $\phi$ on a measurable subset $\Lambda$ of an LCA group $\Gamma$ to be the restriction (l.a.e.) to $\Lambda$ of the Fourier transform of a bounded measure, i.e., a Fourier multiplier of type (1, 1). We generalise Doss’ theorem, and prove that, if the measurable function $\phi$ on $\Lambda$ is approximable on finite subsets of $\Lambda$ by trigonometric polynomials which are Fourier multipliers of type (p, p) on $\Gamma$ of norms no greater than C, then $\phi$ is equal locally almost everywhere to the restriction to $\Lambda$ of a Fourier multiplier of type (p, p) and norm no greater than C.