Oscillating multipliers for some eigenfunction expansions

Abstract

Let P be a non-negative, self-adjoint differential operator of degree d on Rⁿ. Assume that the associated Bochner-Riesz kernel s_R^δ satisfies the estimate, │s_R^δ (x, y)│ = C R^n/d(1+R^1/d|x - y|^-αδ+β for some fixed constants α > 0 and β. We study L^p boundedness of operators of the form m(P), m coming from the symbol class S_p^-α. We prove that m(P) is bounded on L^P 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 m_α (λ) = λ ^-α/2 J_α (t√λ). As a special case we obtain L^p boundedness of solutions to the Wave equation associated to H and L.