## 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:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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