Bounds on the Hermite spectral projection operator

We study L^p–L^q bounds on the spectral projection operator Π_λ associated to the Hermite operator H=|x|²−Δ in R^d. We are mainly concerned with a localized operator χ_EΠ_λχ_E for a subset E⊂R^d and undertake the task of characterizing the sharp L^p–L^q bounds. We obtain sharp bounds in extended ranges of p,q. First, we provide a complete characterization of the sharp L^p–L^q bounds when E is away from λS^d−1. Secondly, we obtain the sharp bounds as the set E gets close to λS^d−1. Thirdly, we extend the range of p,q for which the operator Π_λ is uniformly bounded from L^p(R^d) to L^q(R^d).