Bounded Harmonic Functions on the Disk

For any bounded measurable function $f\in L^\infty(S^1)$, the Poisson integral $P[f]$ gives a bounded harmonic functions in $B_1$. This function is continuous (equivalently, uniformly continuous) if and only if $f$ itself is continuous.

Here you can find what the Poisson kernel is and how it's used to build harmonic functions in the ball.

Said otherwise, $f\mapsto P[f]$ is a linear isometry of $L^p(S^1)$ onto $h^p(B_1)$. We used the case $p=\infty$, where $h^\infty(B_1)$ stands for the the bounded harmonic functions in the ball. You can read more about it here.