Inner Functions in Lipschitz, Besov, and Sobolev Spaces

Abstract

We study the membership of inner functions in Besov, Lipschitz, and Hardy-Sobolev spaces, finding conditions that enable an inner function to be in one of these spaces. Several results in this direction are given that complement or extend previous works on the subject from different authors. In particular, we prove that the only inner functions in either any of the Hardy-Sobolev spaces $H_{\alpha }^p$ with $1/p\le \alpha <\mathrm{\infty }$ or any of the Besov spaces $B_{\alpha }^{p,\mathrm{q}}$ with $0<p,q\le \mathrm{\infty }$ and $\alpha \ge 1/p$, except when $p=\mathrm{\infty }$, $\alpha =0$, and $2<q\le \mathrm{\infty }$ or when $0<p<\mathrm{\infty }$, $q=\mathrm{\infty }$, and $\alpha =1/p$ are finite Blaschke products. Our assertion for the spaces $B_0^{\mathrm{\infty },q}$, $0<q\le 2$, follows from the fact that they are included in the space $\text{i}tVMOA$. We prove also that for $2<q<\mathrm{\infty }$, $\text{i}tVMOA$ is not contained in $B_0^{\mathrm{\infty },q}$ and that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values of $\alpha $ relating the membership of an inner function $I$ in the spaces under consideration with the distribution of the sequences of preimages $\left\{I^{-1}\left(a\right)\right\}$ , $\left|a\right|<1$. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.