$L^p$-multipliers: a new proof of an old theorem

Abstract:

New proofs are given for the following results of Hirschman and Wainger: Let $\psi \in {C^\infty }({\mathbb {R}^n})$ vanish in a neighborhood of the origin; $\psi (\xi ) = 1$ for large $\xi$. Then \[ |\xi {|^{ - \beta }}\psi (\xi )\exp (i|\xi {|^\alpha })\] is a multiplier in ${L^p}({\mathbb {R}^n})$ for $|1/p - 1/2| < \beta /n\alpha$; is not a multiplier in ${L^p}\left ( {{\mathbb {R}^n}} \right )$ for $|1/p - 1/2| > \beta /n\alpha$.