A Hörmander type theorem in finite fields

Abstract Let p be a prime and given a kernel K : F p × F p → C , define a discrete integral operator as follows: T ( f ) ( x ) = ∑ y ∈ F p f ( y ) K ( x , y ) , where f is any complex-valued function defined on F p . We proved that if the kernel K satisfies certain natural size condition and cancellation conditions, the l 2 → l 2 -operator norm of T is bounded by p − γ for some positive number γ. This result can be viewed as a discrete analogue of Hormander theorem. As an application, we recovered a power-saving estimate of certain bilinear average operator in finite fields by X. Li, Sawin and the author.