Quasiregular curves: H\"older continuity and higher integrability

TL;DR

This paper proves that quasiregular curves exhibit local H"older continuity with a specific exponent and also possess higher integrability properties, advancing understanding of their regularity and integrability behavior.

Contribution

It establishes the H"older continuity and higher integrability of quasiregular curves with respect to a covector, providing new regularity results for these mappings.

Findings

Quasiregular $ ext{ extomega}$-curves are locally H"older continuous.

They exhibit higher integrability properties.

The H"older exponent depends on the quasiregularity constant and the covector norm.

Abstract

We show that a $K$-quasiregular $\omega$-curve from a Euclidean domain to a Euclidean space with respect to a covector $\omega$ is locally $(1/K)(\lVert \omega\rVert/|\omega|_{\ell_1})$-H\"older continuous. We also show that quasiregular curves enjoy higher integrability.