A Method of Rotations for L\'evy Multipliers

Abstract

We use a method of rotations to study the $L^p$ boundedness, $1<p<\infty$, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric $\alpha$-stable processes, $0<\alpha<2$. Our proof does not use the fact that $0<\alpha<2$, and therefore allows us to obtain a larger class of multipliers which are bounded on $L^p$. As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the $L^p$ boundedness of the Beurling-Ahlfors transform.