Fundamental solution for the Q-Laplacian and sharp Moser–Trudinger inequality in Carnot groups

For a general Carnot group G with homogeneous dimension Q we prove the existence of a fundamental solution of the Q-Laplacian uQ and a constant aQ>0 such that exp(−aQuQ) is a homogeneous norm on G. This implies a representation formula for smooth functions on G which is used to prove the sharp Carnot group version of the celebrated Moser–Trudinger inequality.