Regularity of Loewner Curves

Abstract

The Loewner equation encrypts a growing simple curve in the plane into a real-valued driving function. We show that if the driving function $\lambda$ is in $C^{\beta }$ with $\beta >2$ (or real analytic) then the Loewner curve is in $C^{\beta +\frac{1}{2}}$ (respectively analytic). This is a converse to a result by Earle and Epstein and extends a result of Wong.

arXiv comment32 pages, 5 figures