Integral geometric properties of capacities

Abstract:

Let $m$ and $n$ be positive integers, $0 < m < n$, and ${C_K}$ and ${C_H}$ the usual potential-theoretic capacities on ${R^n}$ corresponding to lower semicontinuous kernels $K$ and $H$ on ${R^n} \times {R^n}$ with $H(x,y) = K(x,y){\left | {x - y} \right |^{n - m}} \geqslant 1$ for $\left | {x - y} \right | \leqslant 1$. We consider relations between the capacities ${C_K}(E)$ and ${C_H}(E \cap A)$ when $E \subset {R^n}$ and $A$ varies over the $m$-dimensional affine subspaces of ${R^n}$. For example, we prove that if $E$ is compact, ${C_K}(E) \leqslant c\smallint {C_H}(E \cap A)d{\lambda _{n,m}}A$ where ${\lambda _{n,m}}$ is a rigidly invariant measure and $c$ is a positive constant depending only on $n$ and $m$.