Singular Integrals and Complex Analysis in One and Several Variables

AbstractLanzani这个项目涉及的几个问题的积分表示的全纯函数的一个和几个复杂的变量，相关的（奇异）积分边界算子，椭圆边值问题。主要的基本主题包括估计柯西奇异积分算子的情况下，重点粗糙边界的域。这个项目的关键是一个基本的观察， 为了确定某个函数，比如说，f，在比如说一个球内的值，通过计算涉及基准f和一个非线性函数K（“核”）的（表面）积分，只测量f在球表面上的值。 将点P处的f值视为该点的温度可能会有所帮助：通过（容易地）测量球表面的温度（认为球是由固体材料制成的，例如金属），然后可以找出球内部的温度值，而不必到达内部，简单地通过将表面温度数据绘制在积分公式中，然后计算积分。由于物质的微观性质，在一个“粗糙”的物体上进行这些测量和计算要比在一个“光滑”的球上进行这些测量和计算自然得多，比如一个立方体（有角和边）。在数学模型中，这种新的情况转化为对核K的额外约束，这使得在积分计算中有效地使用K变得更加困难。 在这个项目中，我们研究某些（“复杂”）的类似物，这些数学模型的背景下，“粗糙”域。

AbstractLanzaniThis project concerns several questions about integral representations for holomorphic functions of one and several complex variables, related (singular) integral boundary operators, and elliptic boundary value problems. The main underlying theme consists of estimates of Cauchy singular integral operators with focus on the case of domains with rough boundaries. The crux of this project is the basic observation that it is often possible to determine the values a certain function, say, f, takes inside, say, a ball, by only measuring the values f takes on the surface of the ball, via the computation of a (surface) integral involving the datum f and an auxiliaryfunction K ( the "kernel"). It may help to think of the value f at a pointP as the temperature at this point: by (easily) measuring temperature on the surface of a ball (think of the ball as been made of a solid material, such as metal), one can then find out the value of the temperature inside the ball without having to reach the inside, simply by plotting the surface temperature data in an integral formula, and then computing the integral. Because of the microscopic nature of matter, it is much more natural to make these measurements and calculations on a "rough" body, say a cube (which has corners and edges), than a "smooth" ball. In the mathematical model, this new situation translates into additional constraints on the kernel K which make it much harder to effectively use K in the computation of the integrals. In this project we study certain ("complex")analogues of these mathematical models in the context of "rough" domains.