作为数学的核心学科之一，调和分析对偏微分方程理论的应用尤为突出。特别是最近十几年，利用调和分析的方法来研究一类用微分形式表示的偏微分方程正逐渐受到关注。然而，对其相应的算子理论的研究才刚刚开始。因此，从调和分析的角度建立微分形式的算子理论无论是对微分形式还是对偏微分方程的发展都具有重要的意义。本项目主要致力于研究：1. 探讨将调和分析中一些经典算子推广到微分形式的可行性及在微分形式Lp空间的有界性；2. 利用二进Haar移位算子，Bellman函数技术和双权范数不等式，计算或估计同伦算子、Green算子、位势算子和Beurling-Ahlfors算子等的Lp-范数，建立更严格的范数不等式；3. 结合退化椭圆方程的非线性位势理论，明确微分形式A-调和方程弱解的性质，为进一步研究微分形式A-调和方程弱解的正则性和可积性及完善微分形式算子理论提供理论基础。

As one of the core subjects of mathematics, harmonic analysis plays an important role in the theory of Partial Differential Equations. Especially in the last ten years, the use of harmonic analysis to study a class of partial differential equations for differential forms has drawn more and more attention from mathematicians. However, the research on the corresponding operator theory is just getting started. Therefore, whether for differential forms or for the development of PDEs, it is from the perspective of harmonic analysis to establish the operator theory for differential forms of great significance. The project mainly aims to the following problems: 1. Discuss the feasibility that some classical operators in the field of harmonic analysis are extended to the differential forms and the boundedness of these operators in Lp spaces for differential forms; 2. Computer or at least obtain good estimates for the p-norms of the homotopy operator, Green’s operator, the potential operator and Beurling-Ahlfors operator by the dyadic Haar shift operators, Bellman function techniques, two-weight norm inequalities and establish sharp type norm inequalities; 3. Combine nonlinear potential theory of degenerate elliptic equations, to further study the properties of solutions to the A-harmonic equation for differential forms. These results will provide a theoretical basis for studying the regularity and integrability of weak solutions to A-harmonic equations and improving the operator theory of differential forms.