A projection theorem and tangential boundary behavior of potentials

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.