Integral representations and embedding theorems of functions defined on the Heisenberg groups
by
N. N. Romanjvskii
Sobolev institute of mathematics (Novosibirsk, Russia)

Let X₁, ..., X_2n+1 be the standard frame of Lie algebra of Heisenberg group Hⁿ. Let Q be a linear differential operator defined on smooth vector-functions such that Qf is vector-functions with components \sum_i=1^m\sum_|a|=kC_{i, j, a}X^af_i, where C_{i, j, a} are constants, a is (2n+1)-tuple of integers, |a|=a₁+...+a_2n+2a_2n+1, X^a=X₁^a₁... X_2n+1^a_2n+1. We suppose that dim(ker Q) < \infty.

We define W_{Q, p}(U) be the completion of a space of smooth on U m-vector-functions with norm |f|_{Q, p}=||f||_Lp(U)+||Qf||_Lp(U).

Theorem 1. Let U subset Hⁿ be a bounded domain with C² boundary, 1 < p < \infty. Then there exists a linear bounded operator E_Q : W_{Q, p}(U) --> W₀^{k, p}(U) such that E_Qf(x)=f(x) for a. e. x in U.

Theorem 2. Let U subset Hⁿ be a bounded domain with C² boundary, v=2n+2. Then W_{Q, p}(U) subset W_{vp/(v-(k-l)p)}^l(U), l <= k, 1 < p < v/(k-l); W_{Q, p}(U) subset C^k-v/p(U), v/k < p < \infty.

Theorem 3(Coercive inequalities). Let U be a bounded domain with C² boundary. Then norms of spaces W_{Q, p}(U) and W_p^k(U) are equivalent.

The work partially supported by RFBR and the State Russian program Üniversities of Russia".

Date received: August 14, 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-11.