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

根据A.P. Calderon的计划（在1978年赫尔辛基ICM的演讲中简述），物理和工程中的许多基本边值问题可以通过以势算子的形式开始求解，然后将问题简化为分析边界上伪微分算子系统的泛函数解析性质来解决。当存在非光滑结构时出现的新现象是，这些边界算符属于一个更大的类，由奇异积分算符组成。在这种情况下，缺乏符号演算使得对这些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弗雷德霍姆性和可逆性这样微妙的问题时。虽然在这期间这些想法有了戏剧性的发展，但仍有许多工作要做。当前提案的主要目标是根据调和分析和偏微分方程领域的最新进展，研究与数学物理中基本问题（如麦克斯韦方程、矢量泊松问题、霍奇分解和斯托克斯流体动力学系统）相关的奇异积分算子的具体性质。为了获得最大的适用性，重要的是要在一般Sobolev-Besov空间的背景下考虑这些问题，当底层结构是最低限度光滑的。

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.