A criterion for straightening a Lipschitz surface in the Lizorkin–Triebel sense. III

We obtain two new equivalent quasinorms for unweighted isotropic Besov and Lizorkin–Triebel spaces in the epigraph of a Lipschitz function. The question on the straightening is studied, i. e., the question on the existence of a diffeomorphism taking the epigraph into a halfspace which preserves the Lizorkin–Triebel spaces of the same indices. A criterion for the straightening is established in terms of dyadic weighted inequality where oscillations of a given function on stretched dyadic cubes are involved.