Joint BCAM-UPV/EHU Analysis and PDE seminar: Endpoint bounds on the Hermite spectral projection

In this talk we are concerned with L2​-Lq bounds on the Hermite spectral projection operator Πλ in ℝd​​. For d≥2,  the optimal L2​-Lq bound on Πλhas been known except for the endpoint case q=2(d+3)/(d+1).

However, the endpoint L2​-L2(d+3)/(d+1) bound  has been left unsettled for a long time. We prove this missing endpoint case for every d≥3. Our result is based on a new phenomenon: improvement of bounds due to asymmetric localization near the sphere.