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Trace inequalities for Carnot-Carathéodory spaces and applications. (English) Zbl 0938.46036
The authors are interested in traces of Sobolev spaces defined on domain in \(\mathbb{R}^n\) by the family of vector fields \(\{X_1,\dots, X_m\}\) with real-valued locally Lipschitz coefficients. The trace inequalities are formulated in terms of nonnegative Borel measures \(\mu\) on \(\mathbb{R}^n\). The classical example is a Borel measure given by the volume of a compact \(C^1\)-submanifold. The inequalities of the following types are investigated: \[ \begin{aligned} \Biggl(\int_{B(x,r)}|f- f_{B,\mu}|^q d\mu\Biggr)^{1/q} & \leq C\Biggl(\int_{B(x,\beta r)}|XF|^p dx\Biggr)^{1/p},\\ \Biggl(\int_{B(x,r)}|f|^p d\mu\Biggr)^{1/q} & \leq C\Biggl(\int_{B(x,\beta r)}|Xf|^p dx\Biggr)^{1/p},\end{aligned} \] where \(1\leq p\leq q<\infty\), \(\beta>1\), \(f_{B,\mu}= {1\over\mu(B)} \int_{B(x,r)}f d\mu\) and \(Xf= (X_1f,\dots, X_mf)\) is a gradient associated to the vector fields. It is assumed that the vector fields satisfy the following conditions:
(H.1) the metric Carnot-Carathéodory topology is equivalent to the Euclidean one,
(H.2) the Carnot-Carathéodory balls satisfy the doubling condition,
(H.3) the weak-\(L^1\) Poincaré type inequality holds for the gradient \(X\).
The following types of vector fields satisfy the assumptions (H.1)–(H.3): the Hörmander finite rank vector fields, the Baouendi-Grushkin vector fields, Lipschitz vector fields associated to the subelliptic operators.
Different assumptions concerning the domains and the Borel measures are regarded. In particular, bounded domains in Heisenberg groups with \(C^2\)-boundaries are considered. Several applications are mentioned.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65H10 Numerical computation of solutions to systems of equations
58J55 Bifurcation theory for PDEs on manifolds
46N20 Applications of functional analysis to differential and integral equations
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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