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Taking the Fourier transform and using $L^2$ orthogonality you are equivalently trying to estimate $$\sum_{i + j +k = 0} \hat{u}_i \hat{u}_j \hat{u}_{k} |i+j|^{n+1}|k|^{n}$$ Now from the equality $i + j +k = 0$ we have that either $|k| >\max(|i|,|j|)$ in which case $i$ and $j$ have the same sign, $|i| > \max(|j|,|k|)$ in which case $j$ and $k$ ...
[EDITED] $\def\conv{\mathop{\rm conv}\nolimits}$The answer is yes if $x_1,\dots,x_n\in(0,\pi]$ are pairwise distinct (are these `reasonable restrictions'?). Consider a set $C=\{x(t)=(\cos tx_1,\dots,\cos tx_n)\colon t\in\mathbb Z\}\subset \mathbb R^n$. Let $D$ be the closure of the convex hull of $C$. I claim that $0$ is the interior point of $D$. This ...