Note on potential theory for functions in Hardy classes

TL;DR

This paper extends set functions related to Hardy class functions in the unit disk, providing uniform estimates for a supremum quantity involving functions in H^p with boundary constraints.

Contribution

It demonstrates that the set functions from previous work can be extended to all subsets of the unit disk, yielding nearly-optimal uniform estimates for a key supremum.

Findings

Extended set functions to all subsets of the unit disk.

Derived nearly-optimal estimates for D_p(E, ε, R).

Provided bounds for functions in Hardy spaces with boundary conditions.

Abstract

The purpose of this note is to show that the set functions defined in \cite{trong-tuyen} can be suitably extended to all subsets $E$ of the unit disk $\mathbb{D}$. In particular we obtain uniform nearly-optimal estimates for the following quantity D_p(E,\epsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, ||g||_{H^p}\leq 1, (1-|\zeta |)|g(\zeta)| \leq \epsilon \forall \zeta\in E\}.