Minimal fine limits for a class of potentials
There is rich literature regarding the boundary behavior of the potentials \(p=G \mu \), where \(G\) is the harmonic Green's function. In particular, Carleson in his thesis proved, in the case of the half-space \(\mathbb{R}^n _+\), that if \(\int y^{\omega } _n (1+ |y|)^{-n} d \mu (y) < \infty\), \( 0< \omega \leq 1\), then \(p\) has a normal boundary limit of zero everywhere at \(\partial \mathbb{R}^n _+\) but a set of vanishing \((n-2+ \omega)\)-dimensional Hausdorff measure. In this very thoroughly written article the authors are concerned with the potentials \(G_k \mu \), where \(G_k\) is Green's function of the Weinstein equation \(\Delta f + \frac{k}{x_n} \frac{\partial f}{\partial x_n} =0\), \(k=\text{const}\), in the half-space. They prove that if \(\mu \) is a positive Radon measure, then (a) If \(\max (k, 2-n) < \omega \leq 1\) and \(\int y^{\omega } _n (1+ |y|)^{k-n} d \mu (y) < \infty \), then the minimal fine limit of \(G_k \mu \) is zero at each boundary point of \(\mathbb{R}^{n-1}\) except for a set \(E \subset \mathbb{R}^{n-1}\) having zero \((n-2+ \omega)\)-dimensional Hausdorff measure. (b) If \(\max (2-k, 2-n)< \omega \leq 1\) and \(\int y^{k+ \omega -1} _n (1+ |y|)^{2-k-n} d \mu (y) < \infty \), then the minimal fine limit of \(x_n ^{k-1}G_k \mu \) is zero at each boundary point of \(\mathbb{R}^{n-1}\) except for a set \(E \subset \mathbb{R}^{n-1}\) having zero \((n-2+ \omega)\)-dimensional Hausdorff measure. Moreover, it is shown that the size of the exceptional sets in the statement is best possible.
scientific article; zbMATH DE number 1546776