Weak version of restriction estimates for spheres and paraboloids in finite fields (Q2252299)

Let \({\mathbb F}_q^d\), \(d\geq2\), be the \(d\)-dimensional vector space over a finite field \({\mathbb F}_q\). Let \(V\) be an algebraic variety in the dual space \(({\mathbb F}_q^d,dx)\) of \(({\mathbb F}_q^d,dm)\) equipped with the normalized surface measure \(d\sigma\). \textit{G. Mockenhaupt} and \textit{T. Tao} [Duke Math. J. 121, No. 1, 35--74 (2004; Zbl 1072.42007)] investigated the Fourier restriction \(L^p\)-\(L^r\) problem for \(V\), that is, to determine \(1\leq p,\ r\leq\infty\) such that the following inequality holds: \[ \|\hat f\|_{L^r(V,d\sigma)}\leq C\|f\|_{L^p({\mathbb F}_q^d.dm)} \] for all functions \(f\) on \({\mathbb F}_q^d\) and they showed the \(L^{\frac{4}{3}}\)-\(L^2\) restriction estimate for the parabola lying in \({\mathbb F}_q^2\). We know that, if \(d\geq3\) is odd and \(-1\) is a square number, then the Stein-Tomas exponent \(p_0=\frac{2d+2}{d+3}\) gives the sharp exponent for the \(L^p\) \(-\) \(L^2\) restriction estimate for the spheres or the paraboloids. However, when \(d\) is even, we don't know the sharp exponent except \(d=2\). In this paper, H. Kang and D. Koh conjecture that the sharp exponent is \(\frac{2d+4}{d+4}\) in the even case and they show a weak version of the restriction problem, that is, it is true when the test functions \(g\) under consideration are restricted to \(d\)-coordinate functions satisfying for \((m', m_d)\in {\mathbb F}_q^{d-1}\times{\mathbb F}_q\) and \(s\in{\mathbb F}_q\backslash\{0\}\) \[ g(m', m_d)=g(m' sm_d) \] or homogeneous functions of degree zero satisfying for \(s\in{\mathbb F}_q\backslash\{0\}\) \[ g(sm)=g(m). \] To obtain their main result they use a connection between the restriction estimates for homogeneous varieties in dimension \((d+1)\) and those for the spheres and paraboloids in \(d\)-dimension.