Regularity of SLE in [Formula: see text] and refined GRR estimates

Schramm–Loewner evolution ([Formula: see text] ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by [Formula: see text] times Brownian motion. This yields a (half-plane) valued random field [Formula: see text] . (Hölder) regularity of in [Formula: see text] ), 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 [Formula: see text] . In this paper, we improve their result to joint Hölder continuity up to [Formula: see text] . Moreover, we show that the SLE[Formula: see text] trace [Formula: see text] (as a continuous path) is stochastically continuous in [Formula: see text] at all [Formula: see text] . Our proofs rely on a novel variation of the Garsia–Rodemich–Rumsey inequality, which is of independent interest.