On distortion of the moduli of rings under locally quasiconformal mappings in $\mathbb R^{n}$

S. Yu. Graf

Problemy analiza, Tome 4 (2015) no. 2, pp. 32-44

Résumé

Some of the earlier results of author concerning distortion of the moduli of ring domains under planar locally quasiconformal mappings are generalized on the case of locally quasiconformal mappings in $\mathbb R^n$, $n\ge 2$. The main result of the article represents the sharp double-sided estimation of modulus $M(D)$ of the image $D$ of the concentric spherical ring $K(r,R)=\{x\in\in\mathbb R^n:\,r|x|$ under locally quasiconformal homeomorphism $f$: $$ \int^{R}_{r}P^{1/(1-n)}_f(t)\frac{dt}{t}\le Mod(D)\le \int^{R}_{r}P^{1/(n-1)}_f(t)\frac{dt}{t}. $$ Here the function $P_f$ is the majorant of dilatation of mapping $f$ and $P_f$ is well defined as $$ P_f(t)=\lim_{\varepsilon\to 0+}\operatorname{essup}\{p_f(x):\;t-\varepsilon\le|x|\le t+\varepsilon\}. $$ As the consequence of the main inequalities the sharp estimations of the derivative $f'(0)$ of the normalized locally quasiconformal automorphisms $f$ of the unit ball in the terms of majorant of the dilatation of function $f$ are proved. The sharpness of the results is demonstrated by examples of non-trivial locally quasiconformal mappings with unbounded dilatation that provide the equalities in estimations. The main theorems were obtained by means of method of moduli of families of curves and hypersurfaces in $\mathbb R^n$.

Détail

Keywords: locally quasiconformal mapping, modulus of ring domain.