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.


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.)
Full Text: DOI EuDML


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