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

根据A. P.卡尔德龙的计划（在1978年的赫尔辛基ICM演讲中作了概述），物理学和工程学中的许多基本边值问题都可以从势算子形式的解的表示开始，然后把问题归结为分析边界上伪微分算子系统的泛函分析性质。这种新现象，当存在非光滑结构时，这些边界算子属于一个更大的类，由奇异积分算子组成。在这种情况下，缺乏符号演算使得对这类算子的研究从根本上变得更加困难，特别是当人们必须解决像Fredholmness和可逆性这样的微妙问题时。虽然在这期间这些思想经历了一个令人激动的发展，但还有很多工作要做。本文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，研究数学物理中的基本问题，如Maxwell方程、矢量Poisson问题、Hodgedecompositions和Stokes流体力学系统中的奇异积分算子的特殊性质。为了获得最大的适用性，在一般Sobolev-Besov空间的上下文中考虑这些问题以及当底层结构是最小光滑时是很重要的。

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.