Uniform two-thirds decay conjecture for the discrete Klein–Gordon oscillatory integral

Let $d$ be the spatial dimension, let $\tau\in\mathbb{R}$, let $v\in\mathbb{R}^d$, and let $\zeta\in\mathcal{C}_c^{\infty}(\mathbb{R}^d)$. Define

$$J_{\Phi_v,\zeta}=\int_{\mathbb{R}^d}e^{i\tau\Phi_v(\xi)}\zeta(\xi)\,\mathrm{d}\xi,$$

where $\Phi_v(\xi)=v\cdot\xi-\gamma_h(\xi)$ and

$$\gamma_h(\xi)=\sqrt{h^2+2\sum_{j=1}^d\cos(\xi_j)}.$$

Uniform two-thirds decay conjecture. For any $\zeta\in\mathcal{C}_c^{\infty}(\mathbb{R}^d)$,

$$\sup_{v\in\mathbb{R}^d}|J_{\Phi_v,\zeta}|\leq C(\zeta)(1+|\tau|)^{-\frac{2}{3}}.$$

This conjecture proposes a uniform oscillatory-integral decay rate suggested by the continuum limit $h\to0$, where the discrete Klein–Gordon dispersion relation approaches that of the discrete wave equation. The preceding discussion notes a $\tau^{-3/4}$ decay result in the two-dimensional case with $h=1$ and contrasts it with the slower $\tau^{-2/3}$ rate suggested by the wave-equation analysis; the conjectured estimate is presented as an unresolved question.