Extension and averaging operators for finite fields

In this paper we study L − L estimates of both extension operators and averaging operators associated with the algebraic variety S = {x ∈ Fdq : Q(x) = 0} where Q(x) is a nondegenerate quadratic form over the finite field Fq. In the case when d ≥ 3 is odd and the surface S contains a (d − 1)/2-dimensional subspace, we obtain the exponent r where the L − L extension estimate is sharp. In particular, we give the complete solution to the extension problems related to specific surfaces S in three dimension. In even dimensions d ≥ 2, we also investigates the sharp L − L extension estimate. Such results are of the generalized version and extension to higher dimensions for the conical extension problems which Mochenhaupt and Tao ([10]) studied in three dimensions. The boundedness of averaging operators over the surface S is also studied. In odd dimensions d ≥ 3 we completely solve the problems for L − L estimates of averaging operators related to the surface S. On the other hand, in the case when d ≥ 2 is even and S contains a d/2-dimensional subspace, using our optimal L −L results for extension theorems we, except for endpoints, have the sharp L − L estimates of the averaging operator over the surface S in even dimensions.