Complex geodesics, their boundary regularity, and a Hardy--Littlewood-type lemma

TL;DR

This paper explores the boundary regularity of complex geodesics in convex domains, showing how boundary curvature affects their extension, and introduces a Hardy--Littlewood-type lemma relevant to this context.

Contribution

It provides a sufficient condition for boundary regularity of complex geodesics in convex domains and establishes a new Hardy--Littlewood-type lemma.

Findings

Existence of convex domains with non-continuous boundary geodesics

Boundary curvature influences geodesic boundary extension

Introduction of a Hardy--Littlewood-type lemma

Abstract

We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to $\partial\mathbb{D}$. This example suggests that continuity at the boundary of the complex geodesics of a convex domain $\Omega\Subset \mathbb{C}^n$, $n\geq 2$, is affected by the extent to which $\partial\Omega$ curves or bends at each boundary point. We provide a sufficient condition to this effect (on $\mathcal{C}^1$-smoothly bounded convex domains), which admits domains having boundary points at which the boundary is infinitely flat. Along the way, we establish a Hardy--Littlewood-type lemma that might be of independent interest.