A method of rotations for Lévy multipliers

Abstract

ABSTRACT. We use a method of rotations to study the L p boundedness, 1 < p < ∞, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric α-stable processes, 0 < α < 2. Our proof does not use the fact that 0 < α < 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; see conjecture 1 below.