Geometric harmonic analysis for diffusive heat equations

谐波分析与偏微分方程的相互作用由来已久，最早可以追溯到拉普拉斯方程的狄利克雷问题。目前，对PDE各个方面的研究是相互作用的丰硕成果。本研究计划将特别强调几何容量测量理论的方法，通过线性-非线性热方程的柯西问题，围绕上述两个核心区域之间的边界进行研究。要研究的特定主题包括：*由非齐次热方程的解产生的抛物型Riesz势和容量的时空估计。非线性热方程的适定性，如半线性耗散、耗散准地转、广义Euler和Navier-Stokes方程。* Carleson型测度与Riesz型能力通过对齐次热和薛定谔方程初始数据的解的先验估计。*与热半群算子相关的各种函数空间的泛函论和/或算子论刻画。与势的凹凸性和一些耗散方程有关的尖锐的几何和/或解析不等式（例如，容性Brunn-Minkowski和对数Sobolev估计）。主要的思想和技术将涉及发展函数空间及其算子的调和分析的一些几何特征，建立在许多人在过去几十年里在复调和分析、算子势理论和凸几何方面取得的重大进展的基础上。期望本研究能为演化方程和凸几何分析领域提供有用的工具。

The interaction between harmonic analysis and partial differential equations (PDE) has a long history going back to the Dirichlet problem for the Laplace equation. Nowadays, a fruitful ground for the interplay is the study of various aspects of PDE. With special emphasis on the geometric capacity-measure theory approach, this proposed research program will be centered about the boundary between the above two core areas via the Cauchy problem for the linear-nonlinear heat equations. Particular topics to be investigated include:* Spacetime estimates for the parabolic Riesz type potentials and capacities generated by solutions to the inhomogeneous heat equations.* Wellposedness of the nonlinear heat equations such as the semilinear dissipative, dissipative quasi-geostrophic, and generalized Euler and Navier-Stokes equations.* Carleson type measures versus Riesz type capacities through a priori estimations of solutions to the homogeneous heat and Schroedinger equations in terms of the initial data.* Function-theoretic and/or operator-theoretic characterizations of various function spaces associated to the heat semigroup operators.* Sharp geometric and/or analytic inequalities (e.g. the capacitary Brunn-Minkowski and logarithmic Sobolev estimates) related to the convexity of potentials and some dissipative equations.The principle ideas and techniques will involve developing some geometric features of harmonic analysis over function spaces and their operators, building on many significant advances in complex-harmonic analysis, operator-potential theory, and convex geometry made in the last decades by a number of people. It is expected that this research will provide useful tools for the field of evolution equations and convex geometric analysis.