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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Integral geometric properties of capacities

Author: Pertti Mattila
Journal: Trans. Amer. Math. Soc. 266 (1981), 539-554
MSC: Primary 31B15; Secondary 28A75, 31C15
MathSciNet review: 617550
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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\vert {x - y} \right\vert^{n - m}} \geqslant 1$ for $ \left\vert {x - y} \right\vert \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$.

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Article copyright: © Copyright 1981 American Mathematical Society