SHARP Lp → Lr ESTIMATES OF RESTRICTED AVERAGING OPERATORS OVER CURVES ON PLANES IN FINITE FIELDS

Let Fdq be a d-dimensional vector space over a finite field Fq with q elements. We endow the space Fdq with a normalized counting measure dx. Let σ be a normalized surface measure on an algebraic variety V contained in the space (Fdq, dx). We define the restricted averaging operator A V by A V f (x) = f * σ(x) for x ∈ V, where f : (Fdq, dx) → C. In this paper, we initially investigate L p → L r estimates of the restricted averaging operator A V . As a main result, we obtain the optimal results on this problem in the case when the varieties V are any nondegenerate algebraic curves in two dimensional vector spaces over finite fields. The Fourier restriction estimates for curves on F play a crucial role in proving our results.