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Foundations for the theory of quasiconformal mappings on the Heisenberg group. (English) Zbl 0876.30019
The paper under review presents the complete framework for a theory of quasiconformal mappings on the Heisenberg group. As a model for the Heisenberg group $H^n$ one can take $\Bbb R^{2n+1}$ as the underlying space and provide it with the group multiplication $$(x,y,t)(x',y',t')=(x+x', y+y', t+t'- 2x\cdot y'+2y\cdot x'),$$ where $x=(x_1,\ldots,x_n)$, $y=(y_1,\ldots,y_n)\in\Bbb R^n$, and $t\in \Bbb R$. This multiplication is non-commutative. The Heisenberg distance derived from the norm $$|(x,y,t)|=((|x|^2+|y|^2)^2+t^2)^{1/4}$$ is given by $$d(p,q)=|p^{-1}q|, p,q\in H^n.$$ The paper deals with the following three basic definitions for quasiconformity. 1. Metric definition. A homeomorphism $f:u\to u'$ between domains in the Heisenberg group $H^n$ is quasiconformal if $$H(p)=\underset{r\to 0}\to {\lim\sup} \frac {\max_{d(p,q)=r}d(f(p),f(q))} {\min_{d(p,q)=r}d(f(p),f(q))}$$ is uniformly bounded. The homeomorphism $f$ is $K$-quasiconformal if, in addition, $$|H |_\infty =\underset {p\in U}\to{\text{supess}}H(p) \le K.$$ According to Mostow it is shown in section 1.3 that quasiconformal mappings in sense of the metric definition are absolutly continuous on a.e. fiber of any smooth fibration determined by a left invariant horizontal vector field (ACL). The notion of differentiability ($P$-differentiability) on the Heisenberg group and the fundamental differentiability results for quasiconformal mappings are due to {\it P. Pansu} [”Quasiisométries des vatiétés de courbure négative”, Thesis, Paris, (1987)]. In particular it was shown that every quasiconformal mapping between domains in the Heisenberg group is a.e. $P$-differentiable. One of the main consequences of this result is the Beltrami system of differential equations which is satisfied by any quasiconformal mappings. This leads to the second definition. 2. Analytic definition for quasiconformality. A homeomorphism $f:U\to U'$ between domains of the Heisenberg group is quasiconformal if it is ACL and a.e. $P$-differentiable and satisfies the Beltrami system with a complex dilation $\mu$ such that $|\mu|_\infty < 1$. Theorems A,B, and C proved in Section 2 show that a homeomorphism which is $K$-quasiconformal according to the metric definition is quasicoonformal according to the analytic definition with $$(1+|\mu|_\infty)/(1-|\mu|_\infty)\le K .$$ The converse will be proved via the geometric definition. Note that there is a natural conformally invariant notion of capacity on the Heisenberg group (conformal invariance means invariance under the action of $SU(n+1,1)$). 3. Geometric definition. A homeomorphism $f$ between domains in the Heisenberg group is quasiconformal if there exists a constant $K'$ such that $$\roman{cap} R\le K'\roman{cap} fR$$ for all rings $R$ contained in the domain of definition. Theorem D in Section 3 shows that a $K$-quasiconformal mapping according to the metric definition is also quasiconformal according to the geometric definition and $K'$ can be taken to be $K^{n+1}$. The following two corollaries are obtained in Section 3. If $f$ is quasiconformal according to the geometrical definition, then $f$ is quasiconformal according to the metric definition. Moreover, the following inequality holds $$K\le \exp (K'c_n C)^{1/(2n+1)},$$ where constants $c_n$ and $C$ are independent of $f$. The best possible estimate in this case is not known. If $f$ is quasiconformal according to the analytic definition with complex dilation $\mu$, $|\mu|_\infty \le k < 1$, the $f$ is $K$-quasiconformal with $$K=\frac{1+|\mu|_\infty}{1-|\mu|_\infty}.$$ So, the equivalence of the metric, analytic, and geometric definitions of quasiconformality is established. It is noteworthy that recently {\it S. K. Vodop’yanov} [Sib. Math. J. 37, No. 6, 1269-1295 (1996; reviewed below)], Theorem 6] has shown that the condition of $P$ -differentiability in the analytic definition of quasiconformality can be omitted.

##### MSC:
 30C65 Quasiconformal mappings in ${\Bbb R}^n$ and other generalizations 31C15 Generalizations of potentials and capacities 43A80 Analysis on other specific Lie groups
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