Hard Problems in Hard Analysis

我们研究了谐波分析中或与之相关的四个主要未解决的问题。它们是Kakeya问题，Lipschitz微分问题，限制问题，以及thenavierstokes方程的全局可解性。对于Kakeya问题，我们继续与Tao合作改进和差方法中的指数。对于Lipschitz向量场的微分问题，我们尝试将我们的工作应用于任意方向上的极大函数，以了解反例的限制是什么，并希望不存在任何限制。对于限制问题，我们尝试将新的结果应用于Kakeya，并更好地理解Bourgain将Kakeya结果转换为关于限制的结果的机器。对于Navier Stokes，我们首先用一种广义小波系数的形式离散一切。在pavlovic的工作中，这已经产生了对超耗散情况的Caffarelli-Kohn-Nirenberg定理的推广。我们希望从这个角度从非线性项中发现级联效应的一种局部色散性质。然后，我们希望将Clay问题与一个给定色散的二进模型联系起来。可以想象，我们不可能解决所有这些问题。分析是通过检查系统各部分的贡献来证明对系统的估计。一个这样的系统是Navier Stokes方程，它控制着不可压缩粘性流体的行为。一个重要的开放问题是，这个方程从光滑初始数据开始，是否可以在没有强迫项的情况下发展出奇点。这类似于一个人的浴缸里自发地开始了一场旋风。这似乎不大可能，但分析工具还不够强大，无法排除这种可能性。我们的方法是将问题离散化，即尝试用一个关于有限数量对象的近似问题，并通过组合学研究这些对象之间可能的相互作用。大多数物理数学都可以用这种方式来看待，因为物质不是连续的，而是由粒子组成的。我们将用同样的观点来研究上述问题和其他一些重要问题。

We investigate four major unsolved problems in or bordering on harmonicanalysis. These are the Kakeya problem, the Lipschitz differentiationproblem, the restriction problem, and global solvability for theNavier Stokes equation. For the Kakeya problem, we continue ourwork with Tao on improving exponents in the sums-differences approach.For the problem of differentiation by Lipschitz vector fields, we tryto apply our work on maximal functions in arbitrary directions tounderstand what are the limitations on a counterexample and hopefullythat none can exist. For the restriction problem, we try to applythe new results on Kakeya and to better understand Bourgain'smachine for converting Kakeya results to ones about restriction.For Navier Stokes, we first discretize everything in theform of a kind of generalized wavelet coefficients. In work withPavlovic, this has already produced a generalization ofthe Caffarelli-Kohn-Nirenberg theorem to the case of hyperdissipation.We hope from this point of view to discover a sort of local dispersionproperty for the cascading effect from the nonlinear term. Then wehope to tie in the Clay problem with a dyadic model in which thisdispersion is a given. As might be imagined, we are unlikely to solveall these problems.Analysis concerns the proof of estimates on interesting systems byexaming the contributions of all their parts. One such system isthe Navier Stokes equation which governs the behaviour of incompressibleviscous fluids. An important open problem is whether this equationstarting with smooth initial data can develop singularities withouta forcing term. This would akin to a cyclone beginning spontaneouslyin one's bathtub. It seems rather unlikely but the tools of analysisare not yet strong enough to rule it out. Our approach is todiscretize the problem, that is to try to approximate the problemby one about a finite number of objects and investigate possibleinteractions of those objects by means of combinatorics. Mostphysically arising mathematics can be looked at this way because matteris not continuous but rather composed of particles. We will workon the above problem and some other important problems which may beapproached with the same point of view.