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Register a new userDiscrete Fourier restriction associated with Schrodinger equations
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In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgains level set result on Strichartz estimates associated with Schrodinger equations on torus. Some sharp estimates on $L^{frac{2(d+2)}{d}}$ norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on $L^2(mathbb Z)$.
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In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term $ F(u)p_x u$ provided $Fin C^5$ and the initial data $phiin H^s$ with $s>1/2$.
A recently proposed discrete version of the Schrodinger spectral problem is considered. The whole hierarchy of differential-difference nonlinear evolution equations associated to this spectral problem is derived. It is shown that a discrete version of the KdV, sine-Gordon and Liouville equations are included and that the so called `inverse class in the hierarchy is local. The whole class of related Darboux and Backlund transformations is also exhibited.
We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an S-integral solution. These differential operators are naturally associated with Teichmueller curves in genus 2. They are counterexamples to conjectures by Chudnovsky--Chudnovsky and Dwork. We also determine the field of moduli of primitive Teichmueller curves in genus 2, and an explicit equation in some cases.
We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.
Given $ -infty< lambda < Lambda < infty $, $ E subset mathbb{R}^n $ finite, and $ f : E to [lambda,Lambda] $, how can we extend $ f $ to a $ C^m(mathbb{R}^n) $ function $ F $ such that $ lambdaleq F leq Lambda $ and $ ||F||_{C^m(mathbb{R}^n)} $ is within a constant multiple of the least possible, with the constant depending only on $ m $ and $ n $? In this paper, we provide the solution to the problem for the case $ m = 2 $. Specifically, we construct a (parameter-dependent, nonlinear) $ C^2(mathbb{R}^n) $ extension operator that preserves the range $[lambda,Lambda]$, and we provide an efficient algorithm to compute such an extension using $ O(Nlog N) $ operations, where $ N = #(E) $.
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