On the equivalence between trace and capacitary inequalities for the abstract contractive space of Bessel potentials

Abstract

Let $F_{r,p}=V_{r,p}(L^p(X,m))$ be the abstract space of Bessel potentials and $\mu$ a positive smooth Radon measure on $X$. For $2\leq p\leq q < \infty$, we give necessary and sufficient criteria for the boundedness of $V_{r,p}$ from $L^p(X,m)$ into $L^p(X,\mu)$, provided $F_{r,p}$ is contractive. Among others, we shall prove that the boundedness is equivalent to a capacitary type inequality. Further we give necessary and sufficient conditions for $F_{r,p}$ to be compactly embedded in $L^q(\mu)$. Our method relies essentially on establishing a \textit{capacitary strong type inequality}.