Mathematical Sciences: Weighted Norm Inequalities and Differential Equations

Wheeden 9302991 该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的数学研究。 所进行的分析形式涉及到衡量函数的幂范数及其梯度的临界不等式，特别是当这些度量由不同的函数加权时。 这种比较属于索博列夫-庞加莱不等式的范畴。 如果简并性是时间相关的并且根据两个权重函数来测量，则从这些结果可以获得简并抛物线方程的发散基本解的上限和下限。 当梯度被梯度以外的向量场取代时，将完成建立类似不等式的工作。 第二条研究涉及卡尔森型估计，用于测量半空间中的函数（加权），其边界值范数针对不同的权重采取。 对于调和函数，已经获得了良好但不完整的结果。 我们将努力确定泊松核以外的积分变换产生相同类型的不等式的程度 - 特别是在非卷积核的情况下。 最后，要完成表征权重的工作，与原始函数范数相比，这些权重可以为加权分数积分提供良好的界限。 积分的测量在确定微分方程解的存在性和界限方面起着核心作用。其中许多结果涉及仅依赖于基础维度的参数界限。 通过仔细分析相应导数的积分和范数，可以获得有关解的存在性和规律性的重要信息。 ***

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. ***