Weighted norm inequalities for singular integrals

申请人建议继续研究奇异、分数和最大积分的加权范数不等式，以及退化椭圆型偏微分方程解的存在性和正则性。Calderon-Zygmund奇异积分和分数次积分出现在研究几乎所有偏微分方程的最关键的情况下，从薛定谔算子到量子力学，再到流体中的Navier-Stokes方程。对这类算子的两个加权范数不等式的研究不仅扩大了它们的应用范围，而且揭示了所考虑的单个算子的核的重要性质，这些性质在没有这样的研究的情况下往往是隐藏的。 候选人最近的研究产生了许多新的工具，例如具有更高消失矩的加权Alpert小波和测试限于特殊函数的加倍立方体，长期目标是将这些想法和其他想法构建到现有理论的结构中，以提供对奇异积分的两个加权范数不等式的全面处理，以及它们在几何测度论和偏微分方程组中的应用。 在短期内，我们将致力于(1)将刻画希尔伯特变换有界性的NTV猜想的解从一个L2空间推广到另一个空间，推广到更一般情况下的高维Calderon-Zygmund算子，以及(2)与学生合作将我们最近关于希尔伯特变换的局部TB定理推广到更高维，并寻求潜在的应用。 对于退化偏微分方程，我们提出了一个推广我们最近关于无穷退化算子的局部有界性、连续性和最大值原理的工作的程序，这也有助于为研究经典的光滑情形指明方向。这些研究的影响应该有助于推动椭圆型方程的经典正则性结果推广到无限退化区域。

The applicant proposes to continue investigations into weighted norm inequalities for singular, fractional and maximal integrals, and into existence and regularity for degenerate elliptic partial differential equations. Calderon-Zygmund singular integrals and fractional integrals arise in the most critical cases of the study of virtually all partial differential equations, from Schrodinger operators to quantum mechanics to Navier-Stokes equations in fluid flow. The study of two weight norm inequalities for these operators not only extends the scope of application, but reveals important properties of the kernels of the individual operators under consideration, that often remain hidden without such investigation. A number of new tools have arisen from the candidate's recent research, such as weighted Alpert wavelets with higher vanishing moments and testing restricted to special functions for doubling cubes, and a long term goal is to build these ideas and others into the fabric of existing theory to provide a comprehensive treatment of two weight norm inequalities for singular integrals, and their application to geometric measure theory and partial differential equations. In the short term, we will work more specifically to (1) extend our solution to the NTV conjecture characterizing boundedness of the Hilbert transform from one L2 space to another one, to higher dimensional Calderon-Zygmund operators in more general settings, and (2) to extend our recent local Tb theorem for the Hilbert transform to higher dimensions in collaboration with students, and pursue potential applications. In regard to degenerate partial differential equations, we propose a broad program of extending our recent work on local boundedness, continuity and maximum principles for infinitely degenerate operators, which could help point the way to investigating the classical smooth case as well. The impact of these investigations should help motivate the extension of the classical regularity results on elliptic equations to the infinitely degenerate regime.