A Restriction Estimate with a Log-Concavity Assumption

Abstract

The purpose of this paper is to prove an optimal restriction estimate for a class of flat curves in ${\mathbb {R}} ^d$, $d\ge 3$. Namely, we consider the problem of determining all the pairs (p, q) for which the $L^p-L^q$ estimate holds (or a suitable Lorentz norm substitute at the endpoint, where the $L^p-L^q$ estimate fails) for the extension operator associated to $\gamma (t) = (t, {\frac{t^2}{2!}}, \ldots , {\frac{t^{d-1}}{(d-1)!}}, \phi (t))$, $0\le t\le 1$, with respect to the affine arclength measure. In particular, we are interested in the flat case, i.e. when $\phi (t)$ satisfies $\phi ^{(d)}(0) = 0$ for all integers $d\ge 1$. A prototypical example is given by $\phi (t) = e^{-1/t}$. The paper (Bak et al., J. Aust. Math. Soc. 85:1–28, 2008) addressed precisely this problem. The examples in Bak et al. (2008) are defined recursively in terms of an integral, and they represent progressively flatter curves. Although these include arbitrarily flat curves, it is not clear if they cover, for instance, the prototypical case $\phi (t) = e^{-1/t}$. We will show that the desired estimate does hold for that example and indeed for a class of examples satisfying some hypotheses involving a log-concavity condition.