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

Abstract

Let Fd q be a d-dimensional vector space over a finite field Fq with q elements. We endow the space Fd q with a normalized counting measure dx. Let σ be a normalized surface measure on an algebraic variety V contained in the space (Fd q , dx). We define the restricted averaging operator AV by AV f (x) = f ∗ σ(x) for x ∈ V, where f : (Fd q , dx) → C. In this paper, we initially investigate Lp → Lr estimates of the restricted averaging operator AV . 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 F2 q play a crucial role in proving our results.

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