一类单位逼近卷积函数的边界渐近问题

在经典的分析中，调和函数的边界奇异性质因无合适的工具去研究而被忽略在所谓的“例外集”中，而这类被忽略的部分在理论及应用中却最容易出现，利用分形几何的技巧作为工具，就能解决一类这样的问题，弥补经典理论的不足。本项目利用分形几何、重分形分析、位势理论等方法技巧研究调和函数、热核函数、以及一类单位逼近函数的局部和整体边界性质。申请人的博士论文中已经对一类单位逼近函数的局部和整体性质给出了部分结果，但还有很多问题尚未解决。本项目首先对Gauss- Weierstrass核所定义的热核函数的整体边界性质进行研究，包括不同情况下的u(.,y)边界增长性，u(x,y)的L^p边界可积性。其次对一类单位逼近所定义卷积函数的整体边界性质进行研究，进一步完善已有结果。其中一类单位逼近所定义卷积函数的整体边界性质是本项目重点解决的问题，困难是由于单位逼近形式更为抽象，无法沿用Poisson积分的处理方法。

In the classical analysis，there is no suitable tool to resolve the singular boundary properties of some harmonic functions. Some problems of the harmonic functions are ignored as the so-called "exceptional set", which is common in the theory and application. However, we can resolve some problems and cover the insufficient of the classical theory by using the methods and techniques in fractal geometry.By using some methods and techniques in fractal geometry, multifractals, and potentials， this project intends to study some boundary properties of the harmonic function, parabolic function and the convolution transform of measures by an approximate. In the doctoral paper, some local and global properties are given, but there are many problems need to study. In this project, we are first intend to research the global boundary properties of the convolution transform of measures by the Gauss- Weierstrass kernel ,including the growth properties of the u(.,y) near the boundary and the integrability of u(x,y). Furthermore, the global boundary properties of the convolution transform of measures by an approximate will be studied also. Because of the abstract of the approximate, the difficult point is that we can not use the method of the Poisson integral to resolve this problem.