On the regularity of SLE trace

We revisit regularity of SLE trace, for all $\kappa \neq 8$, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia--Rodemich--Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index $\min (1 + \kappa / 8, 2) $, improving on previous works of Werness, and also (optimal) H\"older regularity \`a la Johansson Viklund and Lawler.