 Axial Poisson extension of a circular boundary patch

Take boundary data one on a disk of radius $a>0$ and zero outside. The Poisson kernel for the upper half-space gives on the axisThe axis value tends to one at the boundary and behaves as $a^2/(2z^2)$ far away. The boundary data at the circular rim are understood almost everywhere: changing a value on the rim, a set of planar measure zero, leaves the integral unchanged. This does not assert a continuous boundary extension across the jump.