Mathematical Sciences: Weighted Norm Inequalities and Differential Equations

要进行的研究的中心是发展被称为加权范数不等式的积分不等式。将给定函数的加权范数与其在某种变换（可能使用不同的权值）下的像的加权范数进行比较的不等式是微分方程分析的基础。方程的解用积分变换来表示，通常包含奇异核。如果一个人可以建立基本的，与变换相关的加权不等式，那么就有可能证明在给定的函数类中可以找到解。分析涉及分数积分，Sobolev和Poincare不等式，插值不等式，泊松积分和傅里叶变换。退化椭圆和抛物方程，包括向量场以外的梯度场将被研究。此外，还研究了一类半线性退化椭圆型方程整体正解存在的条件。还计划在确定经典不平等可以转移到齐次空间的程度上进行工作。重点将放在一阶向量场（除了普通梯度）的Sobolev和Poincare不等式是否可以转移到这些空间的问题上。

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.