Bounds on the Hermite spectral projection operator

Abstract

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