Gabriel's problem for harmonic Hardy spaces

TL;DR

This paper investigates inequalities for harmonic functions in the unit disk, extending Gabriel's classical results for analytic functions to harmonic functions, especially focusing on the cases where p is less than or equal to 1, and establishing optimal bounds.

Contribution

It extends Gabriel's inequalities from analytic to harmonic functions, introduces new bounds for 0<p<1, and proves a refined inequality for p=1 with potential applications.

Findings

Inequalities hold for p>1 but generally fail for 0<p≤1.

A new inequality is established for 0<p<1, shown to be optimal.

Refined inequality for circles holds at p=1, with a maximal theorem included.

Abstract

We obtain inequalities of the form $$\int_C |f(z)|^p |dz| \leq A(p) \int_{\mathbb{T}} |f(z)|^p |dz|, \quad (p>1)$$ where $f$ is harmonic in the unit disk $\mathbb{D}$, $\mathbb{T}$ is the unit circle, and $C$ is any convex curve in $\mathbb{D}$. Such inequalities were originally studied for analytic functions by R. M. Gabriel [Proc. London Math. Soc. 28(2), 1928]. We show that these results, unlike in the case of analytic functions, cannot be true in general for $0< p \le 1$. Therefore, we produce an inequality of a slightly different type, which deals with the case $0<p<1$. An example is given to show that this result is "best possible", in the sense that an extension to $p=1$ fails. Then we consider the special case when $C$ is a circle, and prove a refined result which surprisingly holds for $p=1$ as well. We conclude with a maximal theorem which has potential applications.