Abstract
The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide new L2→Lr extension estimates for paraboloids in certain odd dimensions with −1 non-square, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using the first association scheme graph to analyze energy sets, and as a consequence, we obtain new Lp→L4 extension theorems for spheres of primitive radii in odd dimensions, which break the Stein-Tomas result toward Lp→L4 which has stood for more than ten years. Most significantly, it follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the Lp→L4 estimates for spheres with primitive radii are much stronger than those for paraboloids. The second purpose is to show that there is a connection between the restriction conjecture associated to paraboloids and the Erdős-Falconer distance conjecture over finite fields. The last is to prove that the Erdős-Falconer distance conjecture holds in odd dimensional spaces when we study distances between two sets: one set lies on a variety (a paraboloid or a sphere), and the other set is arbitrary in vector spaces over finite fields.
Highlights
The Fourier extension problem is one of the most important open problems in Euclidean harmonic analysis and it has several applications to other fields
We will present new extension theorems for spheres Sj in odd dimensional spaces. It is well-known that in the Euclidean space, the extension theorems for paraboloids and spheres are the same, but in the setting of finite fields, the problems are completely different
Compared to the case of paraboloids or cones, it has been believed that the spherical extension problem is much harder to understand, since the Fourier transform of the sphere is closely related to the Kloosterman sum whose explicit form is not known
Summary
The Fourier extension problem is one of the most important open problems in Euclidean harmonic analysis and it has several applications to other fields. Holds for all functions f on V , where the constant C is independent of q, the size of the underlying finite field Fq. Here, and throughout the paper, dc denotes the counting measure supported on Fqd , and the inverse Fourier transform (f dσ)∨ takes the following form:. In this paper we will address new extension estimates for the case when the algebraic variety V is a paraboloid or a sphere. We will prove the conjectured exponents in the specific case when one set lies on varieties such as spheres and paraboloids, and the other set is arbitrary in Fqd. The key new ingredient is to make a use of the explicit Gauss sum value, which enables us to remove certain tricky terms appearing in estimating a lower bound of the distance set
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