Sharp Adams inequalities with exact growth conditions on metric measure spaces and applications

Adams inequalities with exact growth conditions are derived for Riesz-like potentials on metric measure spaces. The results extend and improve those obtained recently on ${\mathbb{R}}^n$ by the second author, for Riesz-like convolution operators. As a consequence, we will obtain new sharp Moser-Trudinger inequalities with exact growth conditions on ${\mathbb{R}}^n$, the Heisenberg group, and Hadamard manifolds. On ${\mathbb{R}}^n$ such inequalities will be used to prove the existence of radial ground states solutions for a class of quasilinear elliptic equations, extending results due to Masmoudi and Sani.