Abstract. We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and pharmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage.

In this paper, we prove the Adams inequality in complete metric spaces supporting a Poincaré inequality with a doubling measure. We also prove the trace inequalities for the Riesz potentials.