Oscillating multipliers for some eigenfunction expansions

Let P be a non-negative, self-adjoint differential operator of degree d on Rn. Assume that the associated Bochner-Riesz kernel sRδ satisfies the estimate, │sRδ (x, y)│ = C Rn/d(1+R1/d|x - y|-αδ+β for some fixed constants α > 0 and β. We study Lp boundedness of operators of the form m(P), m coming from the symbol class Sp-α. We prove that m(P) is bounded on LP if α > n(1-p)/α │1/p - 1/2│. We also study multipliers associated to the Hermite operator H on Rn and the special Hermite operator L on Cn given by the symbols mα (λ) = λ -α/2 Jα (t√λ). As a special case we obtain Lp boundedness of solutions to the Wave equation associated to H and L