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This paper proves Harnack's inequality for solutions to a class of quasilinear subelliptic differential equations. The proof relies on various embedding theorems into nonisotropic Lipschitz and BMO spaces associated with the vector fields $X_{1},\ldots, X_{m}$ satisfying Hormander's condition. The nonlinear subelliptic equations under study include the important p-sub-Laplacian equation, e.g., $$\sum_{j=1}^{m}X_{j}^{*}\left(|Xu|^{p-2}X_{j}u\right) =A|Xu|^{p}+B|Xu|^{p-1}+C|u|^{p-1}+D,\\ 1<p<\infty $$where $|Xu|=\sum_{j=1}^{m}\left(|X_{j}u|^{2}\right)^{\frac{1}{2}}$ and $A$ is a constant; $B$, $C$ and $D$ can be in appropriate function spaces. We note that $A$ can be nonzero.

Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations