Examples of the domains on some metric spaces
by
Alexandre Greshnov
Sobolev Institute of Mathematics (Novosibirsk)

Connected simply connected nilpotent Lie group G is called Carnot group [1] if its algebra Lie V satysfies following conditions: V=\oplus_i=1^m V_i, V_i=[V₁;V_i-1], [V₁;V_m]=0. Let d(x, y) be the Carnot-Carathéodory distance associated to G and \delta_s:G --> G be thehomogeneous group of dilatation on G.

DEFINITION 1. Let \Cal D subset G and z in \partial\Cal D. We say that \Cal D has interior homogeneous cone condition (IHCC) at z if there is a ball B(x₀(z), r(z)) such that \gamma_z={z\delta_s(x) | s in (0, s₀(z)), x in B(x₀(z), r(z))} subset \Cal D fosr some s₀(z) > 0. Exterior homogeneous cone condition (EHCC)is defined for G\closure\Cal D likewise.

DEFINITION 2. Domain \Cal D subset G satysfies IHCC if \Cal D has IHCC at every z in \partial\Cal D and there is a constant K > 0 such that s₀(z), r(z) > K.

The notion about homogeneous cone condition was introdused by L.Capogna and N.Garofalo [2]. This notion turned out very usefull in construction of uniform and NTA-domains with smooth boundaries on 2-step Carnot groups [2]. Also the examples of the domains with IHCC are very impottant in the theory of the embedding of the functional spaces in subriemannian geometry [3]. The domains with IHCC and EHCC are close with uniform, NTA and John domains on Carnot groups. We established the following properties.

PROPOSITION 1. On Carnot groups there exist John domains which are not satysfying the IHCC and EHCC together.

PROPOSITION 2. On Carnot groups there exist the domains which are satysfying IHCC and EHCC together and are not uniform.

The work partially supported by RFBR and the State Russian Programm [[Univercities of Russia]].

[1] Pansu P. Métriques de Carnot-Carathéodory et quasiisométries de espaces symétrique de rang un// Ann. of Math.-1989-V.119-P.1-60.

[2]Capogna L., Garofalo N. Boundary behavior of non-negative solutions of subelliptic equations in NTA-domains for Carnot-Carathéodory metrics// Fourier Anal. Appl.-1998-V.4-N.4-P.403-432.

[3]Romanovskii N. N. Integral representatoins and embedding theorems of functions defined on the Heisenberg groups Hⁿ// (to appear)

Date received: August 13, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caia-08.