Related papers: Extension Theorems for Paraboloids in the Finite F…

We study the extension estimates for paraboloids in d-dimensional vector spaces over finite fields F_q with q elements. We use the connection between L^2 based restriction estimates and L^p\to L^r extension estimates for paraboloids. As a…

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…

We prove that the finite field Fourier extension operator for the paraboloid is bounded from $L^2\to L^r$ for $r\geq \frac{2d+4}{d}$ in even dimensions $d\ge 8$, which is the optimal $L^2$ estimate. For $d=6$ we obtain the optimal range $r>…

Let $P$ denote the $3$-dimensional paraboloid over a finite field of odd characteristic in which $-1$ is not a square. We show that the Fourier extension operator associated with $P$ maps $L^2$ to $L^{r}$ for $r > \frac{32}{9} \approx…

In this paper we study $L^p-L^r$ estimates of both extension operators and averaging operators associated with the algebraic variety $S=\{x\in {\mathbb F}_q^d: Q(x)=0\}$ where $Q(x)$ is a nondegenerate quadratic form over the finite field…

Sharp restriction theory and the finite field extension problem have both received a great deal of attention in the last two decades, but so far they have not intersected. In this paper, we initiate the study of sharp restriction theory on…

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…

We propose a new approach to the Fourier restriction conjectures. It is based on a discretization of the Fourier extension operators in terms of quadratically modulated wave packets. Using this new point of view, and by combining natural…

Let $\mathbb{H}$ be a $(d-1)$-dimensonal hyperbolic paraboloid in $\mathbb{R}^d$ and let $Ef$ be the Fourier extension operator associated to $\mathbb{H},$ with $f$ supported in $B^{d-1}(0,2)$. We prove that $\|Ef\|_{L^p (B(0,R))} \leq…

We apply geometric incidence estimates in positive characteristic to prove the optimal $L^2 \to L^3$ Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when…

We provide $L^p \to L^q$ refinements on some Fourier restriction estimates obtained using polynomial partitioning. Let $S\subset \mathbb{R}^3$ be a compact $C^\infty$ surface with strictly positive second fundamental form. We derive sharp…

We establish some weighted $L^2$ estimates for the Fourier extension operator in $\mathbb{R}^2$ and discuss several applications to $L^p$ problems. These include estimates for the maximal Schr\"odinger operator and the maximal extension…

We study L^p-L^r restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension $d$ is even, then it is conjectured that the L^{(2d+2)/(d+3)}-L^2 Stein-Tomas…

We obtain some sharp $L^p$ weighted Fourier restriction estimates of the form $\|Ef\|_{L^p(B^{n+1}(0,R),Hdx)} \lessapprox R^{\beta}\|f\|_2$, where $E$ is the Fourier extension operator over the truncated paraboloid, and $H$ is a weight…

We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's…

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…

We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$, allowing for exponents $p$ below the Stein-Tomas critical exponent $\frac{2(N+1)}{N-1}$.…

In this article, we develop a linear profile decomposition for the $L^p \to L^q$ adjoint Fourier restriction operator associated to the sphere, valid for exponent pairs $p<q$ for which this operator is bounded. Such theorems are new when $p…

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf…

Using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques, L^p boundedness theorems for p > 2 are obtained for maximal operators over a wide range of…