Mathematical Sciences: Weighted Norm Inequalities and Differential Equations

8902952 Wheeden将开展的调查的核心是发展被称为加权范数不等的积分不平等。将给定函数的加权范数与其像在某种变换下的加权范数进行比较(可能使用不同的权重)的不等式是分析微分方程的基本原理。方程的解以积分变换的形式表示，通常涉及奇异核。如果能够建立与变换相关的基本的加权不等式，那么就有可能证明在给定的函数类中可以找到解。分析涉及分数次积分、Sobolev和Poincare不等式、插值不等式、泊松积分和傅立叶变换。将学习退化的椭圆型和抛物型方程，包括梯度场以外的矢量场。此外，我们还将寻找一类半线性退化椭圆型方程正解整体存在的条件。还计划开展工作，以确定经典的不平等可以在多大程度上延续到均匀空间。重点讨论一阶向量场(除普通梯度外)的Soblev和Poincare不等式能否推广到这些空间的问题。

8902952 Wheeden Central to the investigations to be carried out is the development of integral inequalities known as weighted norm inequalities. The inequalities, which compare the weighted norm of a given function with that of its image under some transform (using possibly a different weight) are fundamental in the analysis of differential equations. Solutions of the equations are expressed in terms of integral transforms, often involving singular kernels. If one can establish basic, weighted inequalities relative to the transforms, it is then possible to show that solutions can be found within given classes of functions. The analysis involves fractional integrals, Sobolev and Poincare inequalities, interpolation inequalities, Poisson integrals and Fourier transforms. Degenerate elliptic and parabolic equations, including vector fields other than the gradient fields will be studied. In addition, work will be done seeking conditions for the global existence of positive solutions for a class of semilinear degenerate elliptic equations. Work is also planned in determining the extent to which classical inequalities can be carried over to homogeneous spaces. Emphasis will be placed on the question of whether the Sobolev and Poincare inequalities for first order vector fields (other than the ordinary gradient) can be carried over to these spaces.