Abstract. We find the fundamental solution to the P-Laplace equation in Grushin-type spaces. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized Grushin operator in Euclidean space. We then use this solution to find an infinite harmonic function with specific boundary data and to compute the capacity of annuli centered at the singularity. A solution to the 2-Laplace equation in a wider class of spaces is presented. 1. Grushin-type spaces In this paper, we will find the fundamental solution to the P-Laplace equation for 1 <P < ∞ in a class of Grushin-type spaces with singularities at the sub-Riemannian points. Before presenting the main theorem, we recall the construction of such spaces and their main properties. We begin with Rn, possessing coordinates (x1,x2,...,xn) and vector fields Xi = ρi(x1,x2,...,xi−1) ∂ ∂xi for i =2, 3,...,n where ρi(x1,x2,...,xi−1) is a (possibly constant) polynomial. We decree that ρ1 ≡ 1, so tha
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