Smoothness of Loewner Slits

TL;DR

This paper investigates how the regularity of the driving function in the chordal Loewner equation influences the smoothness of the generated slit, establishing precise smoothness conditions based on the driving function's regularity.

Contribution

It provides new regularity results linking the smoothness of the driving function to the smoothness of the Loewner slit, including cases with fractional regularity.

Findings

For $1/2 < eta eq 3/2 \,\leq 2$, the slit is $C^{eta + 1/2}$.

When $eta = 3/2$, the slit is weakly $C^{1,1}$.

The results clarify the smoothness transition at fractional regularity levels.

Abstract

In this paper, we show that the chordal Loewner differential equation with $C^{\beta}$ driving function generates a $C^{\beta + 1/2}$ slit for $1/2 < \beta \leq 2$, except when $\beta = 3/2$ the slit is only proved to be weakly $C^{1,1}$.