Improved remainder estimate under strong convexity

Let $A^0$ be a scalar operator and let

$$\Sigma^0_\lambda=\{\xi\colon A^0(\xi)=\lambda\}$$

be a strongly convex surface, meaning that

$$\pm\sum_{j,k}A^0_{\xi_j\xi_k}(\xi)\eta_j\eta_k\geq\epsilon|\eta|^2$$

for all $\xi\in\Sigma^0_\lambda$ and all $\eta$ satisfying

$$\sum_jA^0_{\xi_j}(\xi)\eta_j=0,$$

where the sign depends on the connected component of $\Sigma_\lambda$ containing $\xi$. Improved remainder estimate. Under this additional assumption, the last term in the right-hand side of the estimate in Theorem 2.8 should be replaceable by

$$C_s h^s(|x|+1)^{k-d}.$$

The proposed improvement would make the remainder decay in $x$ rather than grow as $(|x|+1)^k$; the source presents it as a hoped-for consequence of strong convexity, and no resolution is supplied.