Smoothness of the boundary values of functions bounded and holomorphic in the disk

Abstract:

The non-Euclidean counterparts of Hardy-Littlewood’s theorems on Lipschitz and mean Lipschitz functions are considered. Let $1\le p < \infty$ and $0 < \alpha \le 1$. For $f$ holomorphic and bounded, $|f|< 1$, in $|z|< 1$, the condition that is necessary and sufficient for $f$ to be continuous on $|z|\le 1$ with the boundary function $f({e^{it}}) \in \sigma {\Lambda _\alpha }$, the hyperbolic Lipschitz class. Furthermore, the condition that the $p$th mean of $f^{\ast }$ on the circle $|z|=r < 1$ is $O({(1 - r)^{\alpha - 1}})$ is necessary and sufficient for $f$ to be of the hyperbolic Hardy class $H_\sigma ^{p}$ and for the radial limits to be of the hyperbolic mean Lipschitz class $\sigma \Lambda _\alpha ^{p}$.