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

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.