Pointwise rotation for homeomorphisms with integrable distortion and controlled compression

TL;DR

The paper analyzes maximal pointwise rotation of planar homeomorphisms with finite distortion, showing that the spiraling rate is governed by the local $L^p$-integrability of distortion and by the inverse's modulus of continuity. Using modulus of path families with distortion weights, it derives sharp bounds on the winding number and shows how these bounds are attained by carefully constructed extremal maps built from annular rotations and radial stretchings. The results include a sharp treatment of the borderline case $p=1$ and establish Hölder regularity aspects for the maps and their inverses, with direct relevance to Yudovich-type flows in planar Euler dynamics. Overall, the work clarifies the trade-off between distortion integrability and rotational distortion, providing precise, optimal spiraling rates for a broad class of maps.

Abstract

We obtain sharp rotation bounds for homeomorphisms $f:\mathbb{C}\to\mathbb{C}$ whose distortion is in $L^p_{loc}$, $p\geq1$, and whose inverse have controlled modulus of continuity. The motivation to study this class of maps comes from so-called Yudovich solutions to planar Euler equations. Furthermore, we present examples proving sharpness in a strong sense, thereby settling the borderline case $p=1$ in \cite[Theorem 3]{CHS}.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• proof
• proof : Proof of Theorem \ref{['submain p']}
• proof : Proof of Theorem \ref{['submain 1']}