Boundary behavior of analytic functions on certain Banach spaces
Hector N. Salas
TL;DR
This work addresses boundary behavior of analytic functions in Banach spaces where polynomials are dense and point evaluations are continuous. Using Baire category, it shows that if a space contains a function with $\limsup_{r\to1}|f(re^{i\theta})|=\infty$ on a full-measure subset of $\mathbb{T}$, then almost every function in the space exhibits the same extremal boundary growth on a full-measure set, i.e., the property is residual. The results apply to classical examples such as $S_\nu$ for $\nu<0$ and $\mathcal{D}_{p-1}^p$ for $2<p$, with corollaries like $H^2=S_0$ being of first category inside $\hat{S}_0=\cap_{\nu<0}S_\nu$, and $H^p$ being first category in $\mathcal{D}_{p-1}^p$ for $2<p$. The paper also discusses variants with $L^1$-average continuous point evaluations and poses several open questions about continuity, embeddings, and extensions of the framework to related spaces.
Abstract
For Banach spaces of analytic functions on the disc for which the polynomials are dense and their pointt evaluations continuous, we prove the following: If they contain a function such that the limit superior of its modulus is infinite almost everywhere on the unit circle, then the same is true for a residual set of functions.