Calderon-Zygmund Operators on Sobolev-Besov Spaces and Boundary Problems with Minimal Smoothness Assumptions

按照A.P.Calderon的程序(1978年在他的赫尔辛基ICM演讲中概述)，物理和工程中的许多基本边值问题可以从位势算子形式的解的表示开始，然后将问题归结为分析边界上拟微分算子组的泛函解析性质。当存在非光滑结构时，新的现象是这些边界算子属于由奇异积分算子组成的更大类。在这种背景下，缺乏符号演算使得对这类运算符的研究从根本上变得更加困难，特别是在人们必须解决弗雷德霍尔姆和可逆性等微妙问题的时候。尽管在其间的几年里，这些想法经历了广泛的发展，但仍有许多工作要做。本建议的主要目的是根据调和分析和偏微分方程领域的最新进展，研究与数学物理中的基本问题有关的奇异积分算子的特殊性质，如麦克斯韦方程、矢量泊松问题、霍奇德分解和斯托克斯流体动力学系统。为了获得最大的适用性，重要的是要在一般的Sobolev-Besov空间的背景下以及当底层结构是极小光滑的情况下考虑这些问题。

Following A.P. Calderon's program (sketched in his Helsinki ICM address in 1978), many basic boundary value problems in physics and engineering can be attacked by starting with a representation of the solution in the form of a potential operator, and then reducing matters to analyzing the functional analytic propertiesof a system of pseudodifferential operators on the boundary.The new phenomenon, occurring when non-smooth structures arepresent, is that these boundary operators belong to a largerclass, consisting of singular integral operators. The lack ofa symbolic calculus in this context makes the study of suchoperators fundamentally harder, particularly when one has toaddress such delicate issues like Fredholmness and invertibility.While in the intervening years these ideas have undergone adramatic development, much remains to be done. The main goal of the current proposal is to investigate thespecific nature of the singular integral operators arising inconnection with basic problems in mathematical physics, such asMaxwell's equations, vector Poisson problems, Hodgedecompositions, and Stokes's system of hydrodynamics, in lightof recent advances in the fields of harmonic analysis andpartial differential equations. For maximum applicability, itis important to consider these problems in the context ofgeneral Sobolev-Besov spaces and when the underlying structures are minimally smooth.