Nonlinear Energy Forms and Lipschitz Spaces on the Koch Curve

We consider the nonlinear convex energy forms

on the Koch curve

K K

and we prove that the corresponding domains coincide with the spaces

\mathit{Lip}_{\alpha, D_f} (p, \infty, K) L i p α , D f ( p , ∞ , K )

. Then we give a precise interpretation of the smoothness index

\alpha α

in terms of the structural constants of the fractal.