In this paper, we establish a structural inequality of the ∞-subLaplacian ${▵}_{0,\infty }$ in a class of the semi-simple Lie group endowed with the horizontal vector fields $X_1,\dots ,X_{2n}$. When $1<p\le 4$ with $n=1$ and $1<p<3+\frac{1}{n-1}$ with $n\ge 2$, we apply the structural inequality to obtain the local horizontal $W^{2,2}$-regularity of weak solutions to p-Laplacian equation in the semi-simple Lie group. Compared to Euclidean spaces ${\mathbb{R}}^{2n}$ with $n\ge 2$, the range of this p obtained is already optimal.

In the Heisenberg group ${\mathbb{H}}^n$, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form ${\partial }_{t}u-{\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i\left(Xu\right)=K(x,t,u,Xu)$, where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For $2\le p\le 4$, we establish the local Lipschitz regularity ($u\in C_{\mathrm{loc}}^{0,1}$), with the horizontal gradient satisfying $Xu\in L_{\mathrm{loc}}^{\infty }$; (ii) For $2\le p<3$, we establish the local second-order horizontal Sobolev regularity ($u\in H{W}_{\mathrm{loc}}^{2,2}$), with the second-order horizontal derivative satisfying $XXu\in L_{\mathrm{loc}}^2$. These results solve an open problem proposed by Capogna et al.