  In this paper we study the boundedness of extension operators associated with
spheres in vector spaces over finite fields.In even dimensions, we estimate the
number of incidences between spheres and points in the translated set from a
subset of spheres. As a result, we improve the Tomas-Stein exponents, our
previous results. The analytic approach and the explicit formula for Fourier
transform of the characteristic function on spheres play an important role to
get good bounds for exponential sums.
