Sharp local $L^p$ estimates for the Hermite eigenfunctions

We investigate the concentration of eigenfunctions for the Hermite operator $H=-\Delta+|x|^2$ in $\mathbb{R}^n$ by establishing local $L^p$ bounds over the compact sets with arbitrary dilations and translations. These new results extend the local estimates by Thangavelu and improve those derived from Koch-Tataru, and explain the special phenomenon that the global $L^p$ bounds decrease in $p$ when $2\le p\le \frac{2n+6}{n+1}$. The key $L^2$-estimates show that the local probabilities decrease away from the boundary $\{|x|=\lambda\}$, and then they satisfy Bohr's correspondence principle in any dimension. The proof uses the Hermite spectral projection operator represented by Mehler's formula for the Hermite-Schr\"odinger propagator $e^{-it H}$, and the strategy developed by Thangavelu and Jeong-Lee-Ryu. We also exploit an explicit version of the stationary phase lemma and H\"ormander's $L^2$ oscillatory integral theorem. Using Koch-Tataru's strategy, we construct appropriate examples to illustrate the possible concentrations and show the optimality of our local estimates.