The optimal uniform $L^p$ resolvent region conjecture on the torus

Let $n \geq 3$ and let $\Delta_{\mathbb{T}^n}$ be the Laplacian on the flat torus $\mathbb{T}^n:= \mathbb{R}^n / \mathbb{Z}^n$. Define

$$\mathcal{R}_{\mathrm{opt}}:= \big\{ z = (\lambda + i \mu)^2 \in \mathbb{C}: \lambda, \mu \in \mathbb{R}, \lambda \geq 1, \, |\mu| \geq \lambda^{-1 + \varepsilon} \big\}.$$

Optimal resolvent-region conjecture. For every $\varepsilon > 0$, there is a constant $C_{\varepsilon}$ such that, whenever $z \in \mathcal{R}_{\mathrm{opt}}$,

$$\\|u\\|_{L^{\frac{2n}{n-2}}(\mathbb{T}^n)} \leq C_{\varepsilon} \\| (\Delta_{\mathbb{T}^n} + z)u\\|_{L^{\frac{2n}{n+2}}(\mathbb{T}^n)}.$$

The conjecture would identify the optimal uniform region for the critical torus resolvent estimate; the paper notes that the proven region is not sharp and proposes this as the natural endpoint improvement.