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

We study L p − L r 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 ( 2 d + 2 ) / ( d + 3 ) − L 2 Stein–Tomas restriction result can be improved to the L ( 2 d + 4 ) / ( d + 4 ) − L 2 estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured L p − L 2 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.