We establish the optimal L p , p = 2(d + 3)/(d + 1), eigenfunction bound for the Hermite operator H = −∆ + |x| 2 on R d . Let Π λ denote the projection operator to the vector space spanned by the eigenfunctions of H with eigenvalue λ. The optimal L 2 -L p bounds on Π λ , 2 ≤ p ≤ ∞, have been known by the works of Karadzhov and Koch-Tataru except p = 2(d + 3)/(d + 1). For d ≥ 3, we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere2010 Mathematics Subject Classification. 42B99 (primary), 42C10 (secondary). Key words and phrases. Hermite functions, spectral projection. * This is what was proved in [11], where a care with the notation ℓ ∞ λ L p seems necessary.

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We consider the Bochner-Riesz means for the Hermite and special Hermite expansions and study their L p boundedness with the sharp summability index in a local setting. In two dimensions we establish the boundedness on the optimal range of p and extend the previously known range in higher dimensions. Furthermore, we prove a new lower bound on the L p summability index for the Hermite Bochner-Riesz means in R d , d ≥ 2. This invalidates the conventional conjecture which was expected to be true. 2010 Mathematics Subject Classification. 42B99 (primary); 42C10 (secondary).

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