Bennett, Carbery and Tao (2006) considered the
-linear restriction estimate
in
and established
the near optimal
estimate under transversality assumptions only. In 2017, we showed that
the trilinear restriction estimate improves its range of exponents under
some curvature assumptions. In this paper we establish almost sharp
multilinear estimates for a class of hypersurfaces with curvature for
. Together
with previous results in the literature, this shows that curvature improves the range of
exponents in the multilinear restriction estimate at all levels of lower multilinearity, that
is, when
.
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