Mathematical Sciences: Applications of Harmonic and Parabolic Measure to Problems in Function Theory and PartialDifferential Equations

这个项目涉及到将测度论的某些进展与偏微分方程和经典函数论相结合的数学研究。后一领域的问题有几个根源。一种是Littlewood猜想，它预测多项式和有理函数的球面导数的平均值。这个猜想已被证明对有理函数是尖锐的，但对多项式不是。我们将在有理情况下求出精确的常数，在另一种情况下求出多项式次数的正确幂。这些结果有望从对调和度量的更好估计中得出。这些都应该是最近关于支持调和测量的集合的维度的工作的结果。关于椭圆型偏微分方程解的增长的工作将涉及到根据集合的Riesz容量来度量双调和方程的解的增长，其中梯度为零。对于较低的维度，已经得到了很好的结果。在抛物型方程和热方程领域，将讨论抛物型测度相对于勒贝格测度的绝对连续性问题。对于二维域来说，情况并不总是如此，这一点已经得到证实。在这方面，在边界上建立了绝对连续的充要条件。目前的工作着眼于将该理论扩展到更高的维度。接下来是抛物型方程的解的表示理论的应用。

This project involves mathematical research combining certain advances in measure theory with partial differential equations and classical function theory. Problems in the latter area stem from several sources. One is the Littlewood conjecture which predicts the mean value of the spherical derivature of polynomials and rational functions. The conjecture has been shown to be sharp for rational functions but not for polynomials. Work will be done in obtaining sharp constants in the rational case and working out the correct power of the polynomial's degree in the other. These results are expected to derive from better estimates for harmonic measure. These should follow from recent work on the dimension of sets which can support harmonic measure. Work on elliptic partial differential equations will be concerned with measuring the growth of solutions to the biharmonic equation in terms of the Riesz capacity of set where the gradient vanishes. Sharp results for lower dimensions have been obtained. In the area of parabolic and heat equations, the question of absolute continuity of the parabolic measure with respect to Lebesgue measure will be taken up. That this is not always the case has already been established for two-dimensional domains. In this context, good necessary and sufficient conditions on the boundary for absolute continuity have been established. The present work looks to extend the theory to higher dimensions. Applications to the representation theory for solutions of parabolic equations will follow.