On boundedness of oscillating multipliers on stratified Lie groups

In this paper, we study the oscillating spectral multipliers associated with the sub-Laplacian L on an arbitrary stratified Lie group G. We prove the boundedness of the operators m_α,β,t(L) = ψ(L)L-^β/2e^itLα/2 on Hardy spaces H^p(G) for all p ∈ (0 , ∞) and β/α ≥ Q| 1/p - 1/2 | , where ψ is a smooth function on [0 , ∞) vanishing on [0, a] and equal to 1 on [b, ∞) for some 0 < a < b < ∞, and Q is the homogeneous dimension of G. This extends the existing results and can be applied to obtain L^p estimates for Riesz means of the Schrödinger operators associated with the fractional powers of L.