Restriction of Averaging Operators to Algebraic Varieties over Finite Fields

Abstract

We study L p → L r estimates for restricted averaging operators related to algebraic varieties V of d-dimensional vector spaces over finite fields Fq with q elements. We observe properties of both the Fourier restriction operator and the averaging operator over V ⊂ F d q . As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions d ≥ 2, and cones in odd dimensions d ≥ 3. In addition, when the variety V is a cone lying in an even dimensional vector space over Fq and −1 is a square number in Fq, we also obtain sharp estimates except for two endpoints.

Restricted averaging operators to cones over finite fields

“…In addition, to prove the optimal results in the case when the cone C contains a d/2-dimensional subspace, it will be enough to obtain the critical points P 1 and P 2 . In fact, when d ≥ 3 is odd, the critical point P 0 was obtained in [15], which gives the complete answer to the restricted averaging problem for cones in odd dimensions. On the other hand, when d ≥ 4 is even, it is in general impossible to obtain the point P 0 , because the cone C may contain a d/2-dimensional subspace.…”

“…Such a sharp result was obtained by a direct application of the complete solution to the Fourier restriction problem for curves in two dimensions. The authors in [15] observed from the Fourier decay estimate that the optimal restricted averaging inequalities can be obtained if the variety V ⊂ (F d q , dx) satisfies the following two conditions: Here, and throughout this paper, we write E(x) for the characteristic function χ E on the set E ⊂ F d q . Also recall that X Y is used to denote that there exists C > 0 independent of q = |F q | such that X ≤ CY.…”

“…Therefore, it would be interesting to prove sharp restricted averaging inequalities for non-regular varieties. The cone C ⊂ (F d q , dx) defined as in (1.2) has unusual structures in that it is not a regular variety in even dimensions d ≥ 4, but it is a regular variety in odd dimensions d ≥ 3 (see Corollary 4.3 in [15]). For this reason, we are interested in establishing the sharp restricted averaging problem on cones C ⊂ (F d q , dx) in even dimensions d ≥ 4.…”

“…The necessary conditions for the boundedness of A C (p → r) were given in [15]. For example, the lemma below follows immediately from Lemma 2.1 in [15]. Lemma 1.2.…”

“…In [15], it was proved that the inequality (1.5) holds if d ≥ 4 is even and the test functions f are characteristic functions on (F d q , dx). It was also proved in [15] that the dual estimate of the inequality (1.6) holds for all characteristic test functions on the cone C in even dimensions d ≥ 4. Hence, Theorem 1.3 provides the improved endpoint estimates in even dimensions d ≥ 6.…”