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Register a new userExtension Theorems for Paraboloids in the Finite Field Setting
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In this paper we study the $L^p-L^r$ boundedness of the extension operators associated with paraboloids in vector spaces over finite fields.In higher even dimensions, we estimate the number of additive quadruples in the subset $E$ of the paraboloids, that is the number of quadruples $(x,y,z,w) in E^4$ with $x+y=z+w.$ As a result, in higher even dimensions, we improve upon the standard Tomas-Stein exponents which Mockenhaupt and Tao obtained for the boundedness of extension operators for paraboloids by estimating the decay of the Fourier transform of measures on paraboloids. In particular, we obtain the sharp $L^p-L^4$ bound up to endpoints in higher even dimensions. Moreover, we also study the $L^2-L^r$ estimates.In the case when -1 is not a square number in the underlying finite field, we also study the $L^p-L^r$ bound in higher odd dimensions.The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.
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In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, our previous results. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.
We study the finite field extension estimates for Hamming varieties $H_j, jin mathbb F_q^*,$ defined by $H_j={xin mathbb F_q^d: prod_{k=1}^d x_k=j},$ where $mathbb F_q^d$ denotes the $d$-dimensional vector space over a finite field $mathbb F_q$ with $q$ elements. We show that although the maximal Fourier decay bound on $H_j$ away from the origin is not good, the Stein-Tomas $L^2to L^r$ extension estimate for $H_j$ holds.
The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important.
We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas-Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimates for Gauss and Kloosterman sums and their variants play an important role.
In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezouts theorem, we obtain the sufficient and necessary conditions on general curves where sharp $L^p-L^r$ extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao and Iosevich and Koh. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.
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