We revisit regularity of SLE trace, for all κ≠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+κ/8,2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.
