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

许多现实世界的现象都是用偏微分方程式来模拟的，偏微分方程式涉及到函数及其导数。出于实际目的，了解这类方程的解的性态是极其重要的。尽管通常不可能显式地计算这些解，但从对其大小的定量估计中可以获得大量信息，通常是通过不等式的方式。这项研究的目的是获得许多与数学、物理和几何中的基本方程有关的新的最优不等式：从描述曲面曲率的方程，到湍流涡流模型中产生的平均场方程，等等。将获得的估计是尖锐的(即，它们不能被改进)，并且它们包含了关于基本几何和物理模型的深入信息。在这个项目中，主要的研究人员试图在不同的环境下找到几个Adams，Moser-Trudinger和Onofri不等式中的最佳常数，微分和伪微分算子基本解的尖锐渐近性，以及Sobolev空间在重排不变空间中的最优嵌入。在Cauchy-Riemann球面上，得到了相当一般谱定义的拟微分算子的尖锐的不等式，其特征是最佳常数显式地依赖于本征值。该方法包括对这类算子基本解的一种新的渐近分析。在欧几里得空间上，将在无穷测度域上得到尖锐的不等式，填补了自亚当斯开创性工作以来留下的一个长达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致环境，如海森堡群和赋予双曲度量的空间。在几乎可积函数的约化Soblev空间上，得到了尖锐的Brezis-Merle型不等式和最优嵌入。该方法基于一般度量空间上的Adams和Orlicz型不等式的一个新理论，适用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.