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Endpoint eigenfunction bounds for the Hermite operator

Eunhee Jeong, Sanghyuk Lee, Jaehyeon Ryu

Abstract

We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the Hermite operator $\mathcal H=-Δ+|x|^2$ on $\mathbb R^d$. Let $Π_λ$ denote the projection operator to the vector space spanned by the eigenfunctions of $\mathcal H$ with eigenvalue $λ$. The optimal $L^2$--$L^p$ bounds on $Π_λ$, $2\le p\le \infty$, have been known by the works of Karadzhov and Koch-Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$, 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 sphere $\sqrtλ\mathbb S^{d-1}$.

Endpoint eigenfunction bounds for the Hermite operator

Abstract

We establish the optimal , eigenfunction bound for the Hermite operator on . Let denote the projection operator to the vector space spanned by the eigenfunctions of with eigenvalue . The optimal -- bounds on , , have been known by the works of Karadzhov and Koch-Tataru except . For , 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 sphere .
Paper Structure (20 sections, 30 theorems, 236 equations)

This paper contains 20 sections, 30 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. The projection operator $\Pi_\lambda$
  3. The kernel of $\Pi_\lambda$
  4. The phase function $\mathcal{P}$
  5. Preliminary decomposition
  6. Sectorial decomposition of annuli
  7. The kernel of $\mathfrak P_\lambda[\psi_j]$
  8. An $L^2$-estimate for $\mathfrak P_\lambda[ \eta_j ]$
  9. Proof of Corollary \ref{['cor:weighted']}
  10. Asymmetric improvement: Proof of Proposition \ref{['improvedbound']}
  11. The sum over $\nu\in\mathcal{N}_{\bf e}$
  12. The sum over $\nu\in \mathcal{N}_{\bf c}$
  13. 2nd-order derivative of $\mathcal{P}$
  14. Reduction to $L^2$ estimates
  15. $L^2$ estimate: Proof of Lemma \ref{['propp1']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $d\ge 3$. Then,

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof : Proof of Lemma \ref{['large-sep']}
  • Lemma 2.6
  • ...and 42 more