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Applications of geometric measure theory to the study of Gauss- Weierstrass and Poisson integrals. (English) Zbl 0793.31001
Let \(u(x,t)\) denote the Poisson integral in \(\mathbb{R}^ n\times (0,\infty)\) of a signed measure \(\mu\) on \(\mathbb{R}^ n\), let \(m_ q\) denote \(q\)- dimensional Hausdorff measure, and let \(Z\) be a Borel subset of \(\mathbb{R}^ n\) which is \(\sigma\)-finite with respect to \(m_ q\).
It is shown that \(Z\) is positive for \(\mu\) if and only if \(\liminf_{t\to 0} t^{n-q} u(x,t)\geq 0\) for \(\mu\)-almost all \(x\) in \(Z\). Another result states that the restriction of \(\mu\) to \(Z\) is absolutely continuous with respect to \(m_ q\) if and only if \(\limsup_{t\to 0} t^{n-q} | u(x,t)|<\infty\) for \(\mu\)-almost all \(x\) in \(Z\). A further result characterizes “rectifiable” subsets of \(\mathbb{R}^ n\), again in terms of the limiting behaviour of \(t^{n-q} u(x,t)\).
Corresponding theorems for the Gauss-Weierstrass integral \(v(x,t)\) of \(\mu\) are also given. Here one considers the limiting behaviour of \(t^{(n-q)/2} v(x,t)\).

MSC:
31B25 Boundary behavior of harmonic functions in higher dimensions
28A78 Hausdorff and packing measures
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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