Sharp inequalities for derivatives and potentials in the critical cases of the Sobolev embedding theorem

许多真实的世界现象都是用偏微分方程来建模的，这些方程涉及函数及其导数。为了实际的目的，理解这些方程的解的性质是极其重要的。尽管通常不可能明确地计算这些解，但可以通过对它们大小的定量估计（通常是通过不等式）获得大量信息。本研究的目的是获得许多新的优化不等式相关的基本方程在数学，物理学和几何：从方程描述曲面的曲率，湍流的涡流模型中产生的平均场方程，等等。要获得的估计是尖锐的（即，它们不能被改进），并且它们包含关于底层几何和物理模型的深层信息。在这个项目中，主要研究人员试图找到最好的常数在几个亚当斯，Moser-Trudinger，和Onofri不等式在各种设置，尖锐的渐近微分和伪微分算子的基本解，以及最佳嵌入重排不变空间的Sobolev空间。 在Cauchy-Riemann球面上，对于相当一般的谱定义的伪微分算子，得到了尖锐的不等式，其特征在于最佳常数显式地依赖于特征值。该方法包括一个新的渐近分析的基本解决方案，这样的运营商。在欧几里得空间上，我们将得到无穷测度域上的尖锐不等式，填补了自亚当斯开创性工作以来25年的空白。这些方法是新的，它们足够强大，可以扩展到其他非紧的设置，如海森堡群和赋予双曲度量的空间。在几乎不可积函数的约化Sobolev空间上得到了Sharp Brezis-Merle型不等式和最优嵌入.该方法是基于一般测度空间上的亚当斯-和Orlicz-型不等式的新理论，对于具有缓变核的积分算子。

Many real world phenomena are modeled using partial differential equations, equations that involve functions and their derivatives. For practical purposes it is extremely important to understand the behavior of the solutions of such equations. Even though it is often impossible to compute these solutions explicitly, a great deal of information can be garnered from quantitative estimates on their size, usually by means of inequalities. The purpose of this research is to obtain many new optimal inequalities related to fundamental equations in mathematics, physics, and geometry: from equations describing the curvature of a surface, to mean-field equations arising in vortex models for turbulent flows, and more. The estimates to be obtained are sharp (i.e., they cannot be improved), and they incorporate deep information about the underlying geometric and physical models. In this project, the principal investigator seeks to find best constants in several Adams, Moser-Trudinger, and Onofri inequalities in various settings, sharp asymptotics for fundamental solutions of differential and pseudodifferential operators, and optimal embeddings of Sobolev spaces in rearrangement invariant spaces. On the Cauchy-Riemann sphere, sharp inequalities are to be obtained for rather general spectrally defined pseudodifferential operators, featuring best constants that depend explicitly on the eigenvalues. The method includes a new asymptotic analysis of the fundamental solutions of such operators. On Euclidean spaces, sharp inequalities will be obtained on domains of infinite measure, filling a twenty-five-year-old gap that was left since Adams's seminal work. The methods are new, and they are powerful enough to allow extensions to other noncompact settings such as the Heisenberg group and spaces endowed with hyperbolic metrics. Sharp Brezis-Merle-type inequalities and optimal embeddings are to be obtained on reduced Sobolev spaces of barely integrable functions. The method is based on a new theory of Adams- and Orlicz-type inequalities on general measure spaces, for integral operators with slowly varying kernels.