基于不同函数空间的Calderón-Zygmund理论研究

随着函数空间理论的不断发展和大量非线性问题的不断涌现，经典Calderón-Zygmund理论的局限性日趋显现。本项目将借助于空间理论、调和分析、非线性分析和偏微分方程等工具，通过方程结构及其解在不同函数空间的合理定义，尝试探索非线性Calderón-Zygmund理论的新方法，进而应用于双（多）相变偏微分方程的定性理论研究。具体研究内容为：（1）建立散度型三相变问题在Campanato空间的Calderón-Zygmund估计；（2）建立测度数据双相变问题在分数阶Sobolev-Orlicz空间的Calderón-Zygmund估计和点态位势估计；（3）证明散度型双相变抛物型问题在Lebesgue空间的Calderón-Zygmund估计和Harnack不等式。本项目的研究不仅有利于函数空间理论的丰富和发展，而且为Calderón-Zygmund理论在其他非线性问题中的应用提供新的思路。

The limitations of the classical Calderón-Zygmund theory become increasingly apparent with the development of function space theory and the emergence of a large number of nonlinear problems. By virtue of the tools of space theory, harmonic analysis, nonlinear analysis and partial differential equations, in this project we attempt to explore the new methods of nonlinear Calderón-Zygmund theory through the structures of equations and the reasonable definition of solutions in different function spaces, then apply them to the study of qualitative theory for the partial differential equations with double (multiple) phases. Our research focuses on the following: (1) establishing the Calderón-Zygmund estimate for three-phase problem of divergence type in Campanato space; (2) developing the Calderón-Zygmund estimate and pointwise potential estimate for double phase problem with measure data in fractional Sobolev-Orlicz space; (3) proving the Calderón-Zygmund estimate and Harnack inequality for double phase parabolic problem of divergence type in Lebesgue space. The research of this project is not only beneficial to the enrichment and development of function space theory, but also provides a new way for the application of Calderón-Zygmund theory in other nonlinear problems.