Mathematical Sciences: Convolution Estimates and Sobolev Inequalities

本项目涉及调和分析中的基本数学问题。主要的研究方向之一将是研究具有曲线上支撑的度量的多维卷积。本文的目的是扩展E.Stein的观察，即奇异测度与勒贝格空间中的函数的卷积可能导致相关的勒贝格空间中的函数。本研究将重点研究那些集中在光滑曲线上的措施。曲线卷积的整个问题都植根于积分微分理论，而积分微分理论又是偏微分方程组的积分解的公式。第二行的研究将集中于获得反映某些正交群的对称性的非椭圆常系数微分算子的尖锐的Soblev不等式。Sobolev型不等式是获得偏微分算子存在性结果的关键。它们形成了一类不等式，其中函数的范数由其梯度的范数(可能是不同的范数)所支配。本文将讨论两种特殊情况，经典波算子和Klein-Gordon算子。

This project concerns fundamental mathematical questions in harmonic analysis. Among the principal research directions will be the study of multidimensional convolutions with measures supported on curves. The object is to expand on E. Stein's observation that the convolution of singular measures with functions in Lebesgue spaces may result in functions in related Lebesgue spaces. This research will focus on those measures concentrated on smooth curves. The entire question of convolution along cuves has its roots in the theory of differentiation of integrals, which in turn is a formulation of integral solutions of partial differential equations. A second line of investigation will focus on obtaining sharp Sobolev inequalities for non-elliptic, constant coefficient differential operators which reflect the symmetry of certain orthogonal groups. Sobolev-type inequalities are crucial in obtaining existence results for partial differential operators. They form a class of inequalities in which the norm of a function is dominated by the norm (possibly a different norm) of its gradient. Two particular cases will be addressed in this work, the classical wave operator and the Klein-Gordon operator.