含有p-Laplace算子的散度型次椭圆方程在数学与物理领域有着深刻的背景与广泛的应用，在Heisenberg群上研究其弱解的正则性具有重要的探索意义和理论价值。本项目拟考虑Heisenberg群上具有不连续系数的p-Poisson型方程，研究弱解的非线性Calderón-Zygmund正则性与点态位势估计。本项目拟借助Sharp极大算子与函数空间的刻画等调和分析的工具，采用统一的方法得到弱解水平梯度在Lebesgue、Lorentz、Orlicz、BMO、Hölder、Campanato、Besov与Triebel-Lizorkin等空间中的正则性；借助于非线性Havin-Maz’ya-Wolff位势，拟建立弱解与其水平梯度的点态位势估计。本项目旨在丰富和推广Heisenberg群上的非线性Calderón-Zygmund正则性与点态位势理论的研究，促进调和分析与偏微分方程的发展。

The nonlinear sub-elliptic partial differential equations involving the p-Laplacian in divergence form have a profound background and extensive applications in the fields of Mathematics and Physics. The study on regularity estimates for weak solutions to the relative equations on the Heisenberg groups is of great theoretical value and realistic significance. This project will focus on some applications of Harmonic Analysis in the field of Partial Differential Equations. We will deal with the nonlinear Calderón-Zygmund regularities and pointwise potential estimates for weak solutions with their horizontal gradients to the p-Poisson type equations with discontinuous coefficients on the Heisenberg groups. By using the Fefferman-Stein sharp maximal operator and characterization of function spaces in Harmonic Analysis, we are expected to adopt a unified approach to establish global regularities in various spaces, such as Lebesgue space, Lorentz space, Orlicz space, BMO space, Hölder space, Campanato space, Besov space and Triebel-Lizorkin space. By using the Havin-Maz’ya-Wolff potential, we will establish pointwise potential estimates for weak solutions as well as their horizontal gradients. It will be expected to enrich and extend the nonlinear Calderón-Zygmund regularity theory and pointwise potential theory for the p-Poisson type equations on the Heisenberg groups, promote the developments of Harmonic Analysis and Partial Differential Equations on the Heisenberg groups, and gain some meaningful achievements.