On Schrödinger groups of fractional powers of Hermite operators

Let H = −Δ + |x|² be the Hermite operator on Rⁿ with n ≥ 2. In this paper, we prove the boundedness of Schrödinger groups of fractional powers of H on Lebesgue and Hardy spaces. More precisely, we prove that (a) for p ∈ (1, ∞), γ > 0 and β/γ ≥ n|1/p − 1/2|, ||H^−β/2e^itHγ/² f||_Lp(Rn) ≤ C(1 + |t|)ⁿ|1/p^−1/2|||f||_Lp_(Rn₎, ∀t ∈ R, and (b) for p ∈ (0, 1], γ > 0 and β/γ ≥ n(1/p − 1/2), ||H^−β/2e^itHγ /² f||_HHp _(Rn) ≤ C(1 + |t|)ⁿ(1/p^−1/2)||f||_HHp _(Rn), ∀t ∈ R, where H_H^p (Rⁿ) is the Hardy space associated with the operator H. These estimates improve related result of Thangavelu [22] and have some interesting applications.