Estimates for capacities and traces of potentials (Q2266789)

Let S be a normed function space containing \(C_ 0^{\infty}(R^ n)\). The norm on S is denoted \(\| \cdot;S\|\). For any compact subset e of \(R^ n\), let be \(cap(e;S)=\inf \{\| u;S\|^ p:\) \(u\in C_ 0^{\infty}(R^ n)\); \(u\geq 1\) on \(e\}\). (Here \(p>1\), and in applications p is a parameter appearing in the definition of S.) This definition can be extended in the usual way to a wider class of sets, including the Borel sets. Let \(\Phi\) : [0,\(\infty)\to [0,\infty)\) be an increasing function such that \(t\Phi (t^{-1})\) decreases and tends to 0 as \(t\to \infty\). In certain cases where S is either a space of Riesz potentials or a space of Bessel potentials, it is proved that if \(\mu\) is a measure on \(R^ n\) such that \(\mu (B)\leq \Phi (cap(B;S))\) for each open ball B, then \(\mu (E)\leq C\Phi (C cap(E;S))\) for each Borel set E. As corollaries, necessary and sufficient conditions for the continuity of some imbeddings of Riesz and Bessel potentials are obtained.