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A projection theorem and tangential boundary behavior of potentials

Authors: Kohur GowriSankaran and David Singman
Journal: Proc. Amer. Math. Soc. 129 (2001), 397-405
MSC (2000): Primary 31B25
Published electronically: August 29, 2000
MathSciNet review: 1694863
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Let $L_k$ be the Weinstein operator on the half space, $\mathbb{R}^n_+$. Suppose there is a sequence of Borel sets $A_j \subset \mathbb{R}^n_+$ such that a certain tangential projection of $A_j$ onto $\mathbb{R}^{n-1}$ forms a pairwise disjoint subset of the boundary. Let $\nu$ be a finite test measure on the boundary for a specific non-isotropic Hausdorff measure. The measure $\nu$ is carried back to a measure $\lambda$on a subset of $\bigcup A_j$ by the projection. We give an upper bound for the Weinstein potential corresponding to the measure $d\lambda / x_n$ in terms of a universal constant and a Weinstein subharmonic function. We use this upper bound to deduce a result concerning tangential behavior of Weinstein potentials at the boundary with the exception of sets on the boundary of vanishing non-isotropic Hausdorff measure.

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Additional Information

Kohur GowriSankaran
Affiliation: Department of Mathematics, McGill University, Montreal, Quebec, Canada H3A 2K6

David Singman
Affiliation: Department of Mathematics, George Mason University, Fairfax, Virginia 22030

Keywords: Weinstein equation, Littlewood theorem, Weinstein potential, non-isotropic Hausdorff measure, boundary behavior, minimal fine limit
Received by editor(s): August 27, 1998
Received by editor(s) in revised form: April 9, 1999
Published electronically: August 29, 2000
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society