The $(n-1)$-linear restriction estimate in $\mathbb{R}^n$ with degenerate curvature

We revisit the $(n-1)$-linear restriction estimate in $\mathbb{R}^n$ proved by the first author in \cite{Be-6} and refine the result in the case when the curvature assumptions contain small parameters. The standard $L^p$ estimate is replaced by a more complex norm/quasi-norm of type $\ell^p L^\frac{