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Optimal Orlicz-Sobolev embeddings. (English) Zbl 1061.46031

The classical Sobolev inequality \(\| u\|_{L^p(G)}\leq C\|\nabla u\|_{L^p(G)}\), where \(G\) is open in \(\mathbb{R}^n\), \(n\geq 2\), \(1\leq p< n\) and \(\nabla u\) is the gradient of \(u\), is transferred to Orlicz-Sobolev spaces in the following manner. Let \(A(r)\) be a Young function such that \(\int_0(r/A(r))^{1/(n-1)}\,dr< \infty\). With \(A\) there is associated another Young function \(B_{A,n}(s)\) (in particular, if \(A(s)= s^p\), \(1\leq p< n\), then \(B_{A,n}(s)\) is equivalent to \(s^p\)). By \(L(n,B_{A,n})\) denote the space of real, measurable functions \(u\) on \(G\) such that \(\| u\|_{L(n,B_{A,n})(G)}=\| s^{-1/n} u^*(s)\|_{L\wedge B(A,n)}< \infty\).
Then, writing \(I= \int^\infty(r/A(r))^{1/(n-1)}\,dr\), we have: (i) if \(I= \infty\), then \(\| u\|_{L(n,B_{A,n})(G)}\leq C_1\|\nabla u\|_{L^A(G)}\) for all \(u\in W^{1,A}_0(G)\), (ii) if \(I<\infty\), then \(\| u\|_{L(n,B_{A,n})(G)}\cap L^\infty(G)\leq C_2\|\nabla u\|_{L^A(G)}\) for all \(u\in W^{1,A}_0(G)\), where the constants \(C_1\), \(C_2\) depend only on \(n\). Also, an anisotropic version of the above theorem is given. The spaces appearing in the above inequalities are proved to be optimal.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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