Regularity of SLE in and refined GRR estimates

Schramm-Loewner evolution ( $$ ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $$ times Brownian motion. This yields a (half-plane) valued random field $$ . (Hölder) regularity of in $$ ), a.k.a. SLE trace, has been considered by many authors, starting with Rohde and Schramm (Ann Math (2) 161(2):883-924, 2005). Subsequently, Johansson Viklund et al. (Probab Theory Relat Fields 159(3-4):413-433, 2014) showed a.s. Hölder continuity of this random field for $$ . In this paper, we improve their result to joint Hölder continuity up to $$ . Moreover, we show that the SLE $$ trace $$ (as a continuous path) is stochastically continuous in $$ at all $$ . Our proofs rely on a novel variation of the Garsia-Rodemich-Rumsey inequality, which is of independent interest.