We give a new non-capacitary characterization of positive Borel measures $\mu$ on ${\Bbb R}^n$ such that the potential space $I_\alpha*L^p$ is imbedded in $L^q(d\mu)$ for $1< q < p<+\infty$, that is, the trace inequality $\|I_\alpha f\|_{L^q(d\mu)} \le C \, \|f\|_{L^p(d x)}$ holds, for Riesz potentials $I_\alpha = (- \Delta)^{-\alpha/2}$. A weak-type trace inequality is also characterized. A non-isotropic version on the unit sphere ${\Bbb S}^n$ is studied, as well as the holomorphic case for Hardy--Sobolev spaces $H_\alpha^p$ in the ball. 1991 Mathematics Subject Classification: primary 31C15, 42B20; secondary 32A35.