Mathematical Sciences: Convolution Estimates and Sobolev Inequalities

这个项目涉及的基本数学问题 谐波分析在主要的研究方向将 是对多维卷积的研究， 支持曲线。目的是在E.斯坦的 观察到奇异测度与 Lebesgue空间中的函数可能会导致相关的函数 勒贝格空间。本研究将侧重于这些措施 专注于平滑曲线。卷积的整个问题 沿着曲线有其根源的理论分化的 积分，这反过来又是积分解的公式化 偏微分方程 第二条调查线将集中在获得夏普 非椭圆型常系数Sobolev不等式 微分算子，反映了某些 正交群 Sobolev型不等式在 得到了偏微分算子的存在性结果。 它们形成了一类不等式，其中函数的范数 是由其规范（可能是不同的规范）主导的 梯度离心 在这项工作中将处理两个特殊情况， 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.