Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields

In this paper we pose the $\infty$-Laplace Equation as a Dirichlet Problem in a class of Grushin-type spaces whose vector fields are of the form \begin{equation*} X_k(p):=\sigma_k(p)\frac{\partial}{\partial x_k} \end{equation*} and $\sigma_k$ is not a polynomial for indices $m+1 \leq k \leq n$. Solutions to the $\infty$-Laplacian in the viscosity sense have been shown to exist and be unique in [3], when $\sigma_k$ is a polynomial; we extend these results by exploiting the relationship between Grushin-type and Euclidean second-order jets and utilizing estimates on the viscosity derivatives of sub- and supersolutions in order to produce a comparison principle for semicontinuous functions.