Regularity of SLE in $(t,\kappa)$ and refined GRR estimates

Schramm-Loewner evolution (SLE$_\kappa$) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $\sqrt{\kappa}$ times Brownian motion. This yields a (half-plane) valued random field $\gamma = \gamma (t, \kappa; \omega)$. (H\"older) regularity of in $\gamma(\cdot,\kappa;\omega$), a.k.a. SLE trace, has been considered by many authors, starting with Rohde-Schramm (2005). Subsequently, Johansson Viklund, Rohde, and Wong (2014) showed a.s. H\"older continuity of this random field for $\kappa < 8(2-\sqrt{3})$. In this paper, we improve their result to joint H\"older continuity up to $\kappa < 8/3$. Moreover, we show that the SLE$_\kappa$ trace $\gamma(\cdot,\kappa)$ (as a continuous path) is stochastically continuous in $\kappa$ at all $\kappa \neq 8$. Our proofs rely on a novel variation of the Garsia-Rodemich-Rumsey (GRR) inequality, which is of independent interest.