Refined regularity of SLE

We prove refined (variation and H"older-type) regularity statements for the SLE trace (under capacity parametrisation). More precisely, we show that the trace has finite $\psi$-variation for $\psi(x) = x^d(\log 1/x)^{-d-\varepsilon}$andH\ddot{o}lder-typemodulus$\varphi(t) = t^\alpha(\log 1/t)^{\beta}$where$d$and$\alpha$aretheoptimal$p$-variation and H"older exponents of SLE${}_{\kappa }$ which have been previously identified by Viklund, Lawler (2011) and Friz, Tran (2017). For SLE${}_8$, we simplify a step in the proof by Kavvadias, Miller, and Schoug (2021), and get the modulus $\varphi(t) = (\log 1/t)^{-1/4}(\log\log 1/t)^{2+\varepsilon}$. Finally, for $\kappa \ge 8$, we prove regularity estimates for the uniformising maps that hold uniformly in time, namely $\sup_t |\hat f_t’(u+iv)| \lesssim v^{2\alpha-1}(\log 1/v)^\beta$incase$\kappa>8$and$v^{-1}(\log 1/v)^{-1/4}(\log\log 1/v)^{1+\varepsilon}$incase$\kappa=8$. Our results are obtained from analysing the forward Loewner differential equation (in contrast to the other mentioned works which analyse the backward equation).