Optimal Holder exponent for the SLE path

TL;DR

This paper establishes the exact optimal Hölder exponent for chordal SLE paths, confirming a conjecture by Lind, through sharp derivative moment estimates, and provides a new proof for the lower bound.

Contribution

It determines the precise optimal Hölder exponent for SLE paths and introduces improved estimates for derivative moments, confirming conjectured regularity.

Findings

Confirmed the optimal Hölder exponent for SLE paths.

Provided sharper estimates for derivative moments of inverse maps.

Validated the conjecture by Lind regarding SLE regularity.

Abstract

We prove an upper bound on the optimal H\"older exponent for the chordal SLE path parameterized by capacity and thereby establish the optimal exponent as conjectured by J. Lind. We also give a new proof of the lower bound. Our proofs are based on the sharp estimates of moments of the derivative of the inverse map. In particular, we improve an estimate of the second author.