Danielli, D. A compact embedding theorem for a class of degenerate Sobolev spaces. (English) Zbl 0789.46026 Rend. Semin. Mat., Torino 49, No. 3, 399-420 (1991). 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 ifi) \(A\) 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. Cited in 6 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:semidefinite quadratic form; compact imbedding theorem; \(A\)-sub-unitary curves; weak sub-Riemannian structure; symbol of a second order operator; Hölder condition PDFBibTeX XMLCite \textit{D. Danielli}, Rend. Semin. Mat., Torino 49, No. 3, 399--420 (1991; Zbl 0789.46026)