Sharp spherical $L^p\to L^4$ extension estimates over finite fields in odd dimensions

Abstract

Let $q$ be an odd prime power, let $d\ge3$ be odd, and let $S_j=\{\xi\in \mathbb F_q^d:\xi_1^2+\cdots+\xi_d^2=j\}$, where $j\ne 0$. We prove the sharp extension estimate ${L^{4(d+1)/(3d+1)}(S_j, d\sigma)\to L^4(\mathbb F_q^d,dc)}$ whenever $\eta_q((-1)^{(d-1)/2}j)=-1$, where $\eta_q$ is the quadratic character. This confirms the odd-dimensional spherical extension conjecture of Koh, Pham, and Vinh over every finite field of odd characteristic. Together with the known estimate in the complementary sign class, this determines the sharp ${L^p\to{L^4}}$ extension range for all nonzero-radius spheres in odd dimensions. We also apply the endpoint estimate to obtain sufficient conditions for the unique recovery of sparse signals from incomplete Fourier data.