The Bochner-Riesz conjecture for Hermite expansions

Let $S_\lambda^\delta(\mathcal H)$ denote the Hermite Bochner–Riesz means, and for $1\leq p\leq\infty$ write

$$\|S_\lambda^\delta(\mathcal H)\|_{p}:=\sup_{\|f\|_p\leq 1}\|S_\lambda^\delta(\mathcal H)f\|_p.$$

For $d$ dimensions and $p\in[1,\infty]\setminus\{2\}$, set

$$\delta(d,p):=\max\left\{d\left|\frac{1}{p}-\frac{1}{2}\right|-\frac{1}{2},0\right\}.$$

The Bochner-Riesz conjecture for Hermite expansions. The uniform estimate

$$\|S_\lambda^\delta(\mathcal H)\|_{p}\leq C$$

holds if and only if

$$\delta>\delta(d,p).$$

The conjecture identifies the necessary condition inherited from the classical Bochner–Riesz problem with the optimal condition for Hermite Bochner–Riesz means. The paper notes that only partial results are known in higher dimensions, while in one dimension the problem is nearly settled apart from endpoint cases.