Mathematical Sciences: Weighted Norm Inequalities and Differential Equations

Wheeden 9302991这一奖项继续支持对现代偏微分方程和奇异积分算子理论中出现的问题进行数学研究。分析的形式涉及到衡量函数的幂范数与其梯度的幂范数的临界不等式，特别是当这些度量由不同的函数加权时。这样的比较属于索博列夫-庞加莱不平等的范畴。根据这些结果，我们得到了退化抛物型方程散度形式的基本解的上下界，在退化程度是时间相关的情况下，用两个权函数来衡量。当梯度被不同于梯度的矢量场取代时，将完成建立类似的不平等的工作。第二个研究方向是半空间中度量函数(加权)的Carleson型估计，其边值范数相对于不同的权重。对于调和函数，已经得到了很好的，但不完全的结果。将努力确定由非泊松核产生的积分变换在多大程度上产生相同类型的不平等--特别是在非卷积核的情况下。最后，在刻画加权分数次积分与原函数范数相比具有良好界的权重方面还需要做的工作。积分的测量在确定微分方程解的存在性和界方面起着核心作用。这些结果中的许多都涉及仅依赖于基本维度的参数界限。通过对相应导数的积分和范数的仔细分析，可以得到关于解的存在性和正则性的重要信息。***

Wheeden 9302991 This award continues support for mathematical research into problems arising in the modern theory of partial differential equations and singular integral operators. The form of the analysis which takes place involves critical inequalities measuring the power norms of functions with that of their gradients, especially when the measures are weighted by different functions. Such comparisons fall under the heading of Sobolev- Poincare inequalities. From such results one obtains upper and lower bounds for fundamental solutions of divergence form degenerate parabolic equations in case the degeneracy is time- dependent and measured in terms of the two weight functions. Work will be done establishing similar inequalities when the gradient is replaced by vector fields other than the gradient. A second line of investigation concerns Carleson-type estimates for measuring functions (weighted) in a half-space with their boundary value norms taken against different weights. Good, but incomplete, results have been obtained for harmonic functions. Efforts will be made to determine the extent that integral transforms generated by other than the Poisson kernel produce the same type of inequalities - especially in cases of non- convolution kernels. Finally, work is to be done on characterizing weights which give good bounds on weighted fractional integrals when compared with the original function norms. The measurement of integrals plays a central role in determining both existence and bounds on solutions of differential equations. Many of these results involve parameter bounds which are dependent only on the underlying dimension. Through careful analysis of the integrals and norms of the corresponding derivatives one obtains significant information about the existence and regularity of solutions. ***