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The Birnbaum--Orlicz spaces generalize the Lebesgue spaces (see for a precise definition). The space $L_\Phi(\Omega)$ is defined for convex functions $\Phi:(0,\infty)\rightarrow(0,\infty)$ with $\Phi(0)=0$ and $\Phi(\infty)=\infty$. The norm in $L_\Phi$ is denoted $\|\cdot\|_\Phi$. When $\Phi(t)=t^p$, then $L_\Phi=L^p$ and $\|\cdot\|_\Phi=\|\cdot\|_p$.

Define the Sobolev space $W^{1,\Phi}_0(\Omega)$ to be the closure of ${\mathcal D}(\Omega)$ under the norm $$f\mapsto\|\nabla f\|_\Phi.$$ Let me recall some of the Sobolev embeddings, when $\Omega$ is bounded. If $1<p<n$, we have $\dot W^{1,p}(\Omega)\subset L^q(\Omega)$, with $$\frac1q+\frac1n=\frac1p.$$

Actually, if $p=n$, $W^{1,n}_0(\Omega)$ is included in $L_\Phi(\Omega)$ where $\Phi(t)=\exp(t^{n/(n-1)})-1$.

Question: is there a theory of embedding for spaces $W^{1,\Phi}(\Omega)$. I suspect that one can find an other convex function $\Psi$ such that $W^{1,\Phi}\subset L_\Psi$.

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Yes, you are right. A good place to start are some surveys written by Andrea Cianchi. For example,

Cianchi, Andrea(I-FRNZ-AMA) On some aspects of the theory of Orlicz-Sobolev spaces. (English summary) Around the research of Vladimir Maz'ya. I, 81–104, Int. Math. Ser. (N. Y.), 11, Springer, New York, 2010.


Cianchi, Andrea(I-FRNZ-AMA) Optimal Orlicz-Sobolev embeddings. (English summary) Rev. Mat. Iberoamericana 20 (2004), no. 2, 427–474.

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