Weak version of restriction estimates for spheres and paraboloids in finite fields (1401.6997v1)
Abstract: We study Lp-Lr restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension $d$ is even, then it is conjectured that the L{(2d+2)/(d+3)}-L2 Stein-Tomas restriction result can be improved to the L{(2d+4)/(d+4)}-L2 estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured Lp-L2 restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in $d$ dimensions and those for homogeneous varieties in (d+1)dimensions.