Local smoothing for the Hermite wave equation

TL;DR

This work develops a local smoothing theory for the Hermite wave equation by constructing a parametrix as a Fourier integral operator and linearizing it to reduce to Klein–Gordon propagation. It leverages \(\ell^2\) decoupling, plus novel decoupling results for radially degenerate elliptic and uniformly degenerate surfaces, to obtain essentially sharp derivative losses in higher dimensions and endpoint-sharp results in 1D. The analysis connects local smoothing to Hermite Bochner–Riesz means and clarifies how phase-space localization and mass parameters shape dispersion and smoothing across regimes. The results advance understanding of smoothing phenomena for Schrödinger- and wave-like equations with harmonic potentials and have implications for spectral multiplier theory in the Hermite setting.

Abstract

We show local smoothing estimates in $L^p$-spaces for solutions to the Hermite wave equation. For this purpose, we obtain a parametrix given by a Fourier Integral Operator, which we linearize. This leads us to analyze local smoothing estimates for solutions to Klein-Gordon equations. We show $\ell^2$-decoupling estimates adapted to the mass parameter to obtain local smoothing with essentially sharp derivative loss. In one dimension as consequence of square function estimates, we obtain estimates sharp up to endpoints. Finally, we elaborate on the implications of local smoothing estimates for Hermite Bochner--Riesz means.

Theorems & Definitions (49)

• Conjecture 1.1: Local smoothing conjecture for the Hermite wave equation
• Theorem 1.2: Local smoothing in one dimension
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Proposition 2.2: Existence of solutions to the eikonal equation
• proof
• Lemma 2.3
• Lemma 2.4
• proof