A majorant problem for the periodic Schrodinger group (Harmonic Analysis and Nonlinear Partial Differential Equations)
A majorant problem for the periodic Schrodinger group (Harmonic Analysis and Nonlinear Partial Differential Equations)
复制标题
周期性薛定谔群的一个主要问题（调和分析和非线性偏微分方程）
DOI：
--
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
N. Bez
中科院分区：
文献类型：
--
作者：
Jonathan Bennett;N. Bez
对于（x，t）∈ T.这里我们写e（x）：= e for x ∈ T，其中T：= [0，1]。当然，u（x，t）= Ea（x，t）形式上满足自由薛定谔方程<$xu（x，t）= 2πi <$tu（x，t），其中周期性初始数据u（0）等于第n个傅立叶系数等于an的函数。此外，E的伴随是将傅里叶变换限制在离散抛物线{（n，n）：n ∈ Z}上的映射。当序列a被序列A优选时，我们将写一个A，|一个|对于每个n ≤ An。这里我们关心的是算子E在多大程度上满足L（T）上的优数性质;也就是说，给定一个A，在什么意义上A ∈ A ∈ Lp（T2）优于A ∈ A ∈ Lp（T2）？这个问题以前似乎没有明确提出过，在本说明中，我们提供了一些初步结果。这个问题当然很容易让人想起经典的Hardy-Littlewood优数问题，它起源于[8]，其中证明了对于每个p ∈ [2，∞），存在一个有限常数Bp，使得Lp（T）≤ Bp Lp（T）
for (x, t) ∈ T. Here we are writing e(x) := e for x ∈ T, where T := [0, 1]. Of course, u(x, t) = Ea(x, t) formally satisfies the free Schrodinger equation ∂ xu(x, t) = 2πi∂tu(x, t) with periodic initial data u(0) equal to the function whose nth Fourier coefficient is equal to an. Moreover, the adjoint of E is the mapping which restricts the Fourier transform to the discrete parabola {(n, n) : n ∈ Z}. We shall write a ≼ A when the sequence a is majorised by the sequence A in the sense that |an| ≤ An for each n. Our concern here is to what extent the operator E satisfies a majorant property on L(T); that is, given a ≼ A, in what sense is ∥Ea∥Lp(T2) majorised by ∥EA∥Lp(T2)? It appears that this question has not been explicitly posed before and in this note we offer some preliminary results. The question is of course very reminiscent of the classical Hardy–Littlewood majorant problem for Fourier series, originating in [8], where it is conjectured that for each p ∈ [2,∞) there exists a finite constant Bp such that ∥∥∥∥∑ n∈Z ane(nx) ∥∥∥∥ Lp(T) ≤ Bp ∥∥∥∥∑ n∈Z Ane(nx) ∥∥∥∥ Lp(T)