The Hermite spectral projection estimate on the full exponent square

Let $d$ be the dimension, let $(1/p,1/q)$ lie in the exponent square ${\mathlarger{\square}}$, and let $\chi_{\lambda,\mu}$ and $\Pi_\lambda$ be the spectral localization and Hermite spectral projection operators appearing in the estimate

$$\|\chi_{\lambda,\mu}\Pi_\lambda\chi_{\lambda,\mu}\|_{p\to q}\leq C\lambda^{\beta(p,q)}\mu^{\gamma(p,q)}.$$

Hermite spectral projection estimate. For $(1/p,1/q)\in {\mathlarger{\square}}$, the estimate

$$\|\chi_{\lambda,\mu}\Pi_\lambda\chi_{\lambda,\mu}\|_{p\to q}\leq C\lambda^{\beta(p,q)}\mu^{\gamma(p,q)}$$

holds. The estimate extends the previously known case $p=q'$ to other exponent pairs, and the stated power of $\mu$ is sharp up to a constant; whether the estimate holds throughout the full exponent square is left as a plausible expectation in the source.