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A compact embedding theorem for a class of degenerate Sobolev spaces. (English) Zbl 0789.46026

Summary: Let \(\Omega\) be an open bounded set in \(\mathbb{R}\) and let \(p\in [1,\infty)\). If \({\mathcal A}(x,\xi)=\langle A(x)\xi,\xi\rangle=\Sigma_ j\langle X_ j(x),\xi\rangle^ 2\) is a positive semidefinite quadratic form on all points of \(\Omega\), denote by \(\mathring W^ p_ A(\Omega)\) the subspace of \(L^ p(\Omega)\) obtained by completing \(C^ \infty_ 0(\Omega)\) with respect to the norm \[ \| u\|_{A,p}=\Bigl(\| u;L^ p(\Omega)\|^ p+\int_{\Omega}\langle A(x)\nabla u(x)\rangle^{p/2}dx\Bigr)^{1/p}. \] In this note we present a compact imbedding theorem of \(\mathring W^ p_ A(\Omega)\) into \(L^ p(\Omega)\) as a consequence of a geometric property of the family of \(A\)- sub-unitary curves and apply this result if
i) \(A\) defines a weak sub-Riemannian structure,
ii) the quadratic form associated to \(A\) is the symbol of a second order operator satisfying a Hölder condition.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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