Mathematical Sciences: Weighted Norm Inequalities and Differential Equations

这个奖项继续支持对现代偏微分方程理论和奇异积分算子中出现的问题的数学研究。分析的形式涉及到测量函数的幂范数与其梯度的幂范数的关键不等式，特别是当测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。从这些结果可以得到散度形式的退化抛物方程的基本解的上界和下界，这种情况下的退化是时变的，并且是用两个权函数来测量的。当梯度被向量场代替而不是梯度时，我们将建立类似的不等式。第二项研究涉及半空间中测量函数（加权）的carleson型估计，其边界值规范针对不同的权重。对于调和函数已经得到了好的但不完整的结果。将努力确定由泊松核以外的其他核生成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，要做的工作是描述权值，与原始函数范数相比，这些权值给出了加权分数阶积分的良好界。积分的测量在确定微分方程解的存在性和界上起着中心作用。这些结果中的许多都涉及仅依赖于底层维度的参数边界。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. ***