Mappings of finite distortion: Formation of cusps III

Abstract

Abstract. We give sharp integrability conditions on the distortion of a planar homeomorphism that maps a standard cusp onto the unit disk.

Joint rotational and stretching multifractal spectra of mappings with integrable distortion

“…Let f : C → C be a mapping of finite distortion, for which K f (z) ∈ L p loc (C). Then it is well known that its pointwise stretching is bounded by (1.1) |f (z) − f (x)| ≥ e −c (1/|z−x|) 2/p , where c = c f,p,z and |z − x| > 0 is small, see, for example, [10] and [3]. Moreover, recently in [9] the pointwise rotation of such mappings was bounded by , where c = c f,p,z and |h| > 0 is small.…”

A note to “Mappings of finite distortion: formation of cusps II”

“…Theorem A is analogous to the result of [4] which states that the corresponding bound for λ is 1/s in the case when f is, in addition, assumed or required to be quasiconformal on B.…”

“…In order to gain insight on the relationship between the geometry of the images of the disk under homeomorphism of integrable distortion and the degree of integrability we have [4,9,5] considered the model domains…”

Mappings of Lp-integrable distortion: regularity of the inverse

“…This optimal modulus of continuity was established by Clop and Herron [5] and earlier in dimension two by Koskela and Takkinen [13]. Regardless the optimality, it is natural to examine radially symmetric mappings f : B → R n ,…”