zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 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·y ' +2y·x ' ),

where x=(x 1 ,...,x n ), y=(y 1 ,...,y n ) n , and t. 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,qH n ·

The paper deals with the following three basic definitions for quasiconformity. 1. Metric definition. A homeomorphism f:uu ' between domains in the Heisenberg group H n is quasiconformal if

H(p)=limsup r0max 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| =supess pUH(p)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 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:UU ' 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 μ such that |μ| <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+|μ| )/(1-|μ| )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

cap RK ' 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

Kexp(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 μ, |μ| k<1, the f is K-quasiconformal with

K=1+|μ| 1-|μ| ·

So, the equivalence of the metric, analytic, and geometric definitions of quasiconformality is established. It is noteworthy that recently 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:
30C65Quasiconformal mappings in n and other generalizations
31C15Generalizations of potentials and capacities
43A80Analysis on other specific Lie groups