Discrete Fourier restriction associated with Schrödinger equations

Abstract

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain's level set result on Strichartz estimates associated with Schrödinger equations on torus. Some sharp estimates on L 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 (Z). It was conjectured by Bourgain inThe understanding of this conjecture is still incomplete. For instance, the desired upper bounds for A 5,1,N , A 3,2,N or A 2(d+2) d ,d,N for d ≥ 3 are not yet obtained. The most crucial estimate established by Bougain in [1] is certain (sharp) level set estimate. In this paper we provide a different proof of the level set estimate.These problems arise from the study of periodic nonlinear Schrödinger equations:(1.4) ∆ x u + i∂ t u + u|u| p−2 = 0 u(x, 0) = u 0 (x) .

Restricted weak-type endpoint estimates for k-spherical maximal functions

“…The method here is common in number theory and proceeds by using Plancherel's theorem to rewrite the L p -norm as the number of solutions to a system of equations. We note that Hu and Li [11][12][13] recently used this method to study related discrete restriction problems. Since the proof of the following lemma is a standard technique in the circle method, we refer the reader to Chapter 5, Section 5.1 of [27], in particular inequality (5.4), or Chapter 4, Section 2 of [19] for proofs.…”

Discrete Fourier restriction via Efficient Congruencing

“…Beginning with work of Stein (see [17], and [2] and [18] for more recent broader context), there is by now an extensive body of research concerning the norms of operators restricting such Fourier series to integral points n lying on manifolds of various dimensions. Thus, for example, in work concerning the non-linear Schrödinger and KdV equations, Bourgain [4,5] has considered the situation with k = 2 and the restriction n = (n, n l ) to the Fourier series Rg = n∈Zĝ (n, n l )e(nα 1 + n l α 2 ) (l = 2, 3), as well as higher dimensional analogues (see [10,11] for recent work on these problems). Such results have also found recent application in additive combinatorics in work concerning the solutions of translation invariant equations with variables restricted to dense subsets of the integers (see [8,14,15]).…”

The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability

“…His proof relies on Weyl's sum estimates, the Hardy-Littlewood circle method, and the Tomas-Stein restriction theorem. Partial improvements were obtained in [26,47].…”

“…The case is much more difficult when generalizing to any given p and d. Hu and Li in [47] presented a variant of the proof of Bourgain's result (3.4), which makes use of the Hardy-Littlewood circle method and estimates on level sets just as Bourgain's does. Their proof of Bourgain's level set estimates is, however, somewhat simpler.…”

“…When q is large, the desired estimate (3.4) follows immediately from the following result (cf. [6,47]). , there exists a constant C, independent of N , such that…”