Higher-order Sobolev-type embeddings on Carnot-Carathéodory spaces. (English) Zbl 1377.46019
The author provides sufficient conditions for higher-order Sobolev-type embeddings for the class of rearrangement-invariant function spaces on bounded domains of Carnot-Carathéodory spaces. The most general condition takes the form of the inequality
\[
\left\|\int\limits_t^1\frac{f(s)}{I(s)}\left(\int\limits_t^s\frac{dr}{I(r)} \right)^{m-1}\, ds \right\|_{\mathbb Y(0,1)} \leq C\|f\|_{\mathbb X(0,1)},
\]
where \(\mathbb X\) and \(\mathbb Y\) are given rearrangement-invariant spaces, \(I(s)\) is a lower bound for the isoperimetric function \(I_{X,\Omega}(s)\) of a domain \(\Omega\) with respect to a system of vector fields \(X\). Under this condition, the embedding \(V^m_X\mathbb X(\Omega) \to\mathbb Y(\Omega)\) holds. The proof partly follows the approach of D.E.Edmunds et al. [J. Funct. Anal. 170, No. 2, 307–355 (2000; Zbl 0955.46019)]. Particular results are provided for regular domains (\(X\)-PS domains), and in the case of Lebesgue spaces classical-like results are obtained. In the case of the Heisenberg group and \(\mathbb H\)-John domains, the condition is shown to be necessary as well.
Reviewer: Nikita Evseev (Novosibirsk)
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 53C17 | Sub-Riemannian geometry |
Keywords:
higher-order Sobolev embeddings; Carnot-Carathéodory spaces; rearrangement-invariant spacesCitations:
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