# Imbedding theorems for Orlicz-Sobolev spaces

Establishing imbedding theorems goes back to T.K. Donaldson and N.S. Trudinger, and R.A. Adams (see [a4], [a10], [a2]). For the sake of simplicity, consider the space $W ^ { 1 } L _ { \Phi } ( \Omega )$ with an $N$-function $\Phi$ (cf. Orlicz–Sobolev space), where $\partial \Omega$ is sufficiently smooth (a Lipschitz boundary, for instance; see [a1]). Define $g _ { \Phi } ( t ) = \Phi ^ { - 1 } ( t ) t ^ { - 1 - 1 / n }$, $t > 0$. The case $\int _ { 1 } ^ { \infty } g _ { \Phi } ( t ) d t = \infty$ corresponds to the sublimiting case for usual Sobolev spaces (cf. also Sobolev space) and in this case $W ^ { 1 } L _ { \Phi } ( \Omega )$ is imbedded into $L _ { \Phi ^ * } ( \Omega )$, where $\Phi ^ { * } ( t ) = \int _ { 0 } ^ {| t | } g _ { \Phi } ( s ) d s$, $t \in {\bf R} ^ { 1 }$. The case $\int _ { 1 } ^ { \infty } g _ { \Phi } ( t ) d t < \infty$ corresponds to $k p > n$ for $W _ { p } ^ { k } ( \Omega )$. If one defines the generalized Hölder spaces $C ^ { 0 , \sigma ( t ) } ( \Omega )$, where $\sigma$ is a increasing continuous function on $[ 0 , \infty )$ such that $\sigma ( 0 ) = 0$, as the space of continuous functions on $\overline{\Omega}$, for which $\operatorname { sup } _ { x \neq y \in \Omega } | u ( x ) - u ( y ) | ( \sigma | x - y | ) ^ { - 1 } < \infty$, then the target space for the imbeddings is a Hölder space of this type with $\sigma ( t ) = \int _ { t ^ { - n }} g_ {\Phi }^ { \infty } ( s ) d s$.
A (partial) ordering of such functions $\sigma$ (and of $N$-functions) can be introduced in the following way: If $\Phi _ { 1 }$ and $\Phi _ { 2 }$ are $N$-functions, then $\Phi _ { 1 } \prec \Phi _ { 2 }$ if $\operatorname { lim } _ { t \rightarrow \infty } \Phi _ { 1 } ( t ) / \Phi _ { 2 } ( s t ) = 0$ for every $_ { S } \in {\bf R} ^ { 1 }$. Further, $\sigma _ { 1 } \prec \sigma _ { 2 }$ if $\sigma _ { 2 } \sigma _ { 1 } ^ { - 1 }$ is a function of the same type. If now $W ^ { 1 } L _ { \Phi } ( \Omega )$ is imbedded into $L _ { \Phi _ { 2 } } ( \Omega )$ and $\Phi _ { 1 } \prec \Phi _ { 2 }$, then $W ^ { 1 } L _ { \Phi } ( \Omega )$ is compactly imbedded into $L _ { \Phi _ { 1 } } ( \Omega )$, and if $W ^ { 1 } L _ { \Phi } ( \Omega )$ is imbedded into $C ^ { 0 , \sigma _ { 2 } ( t )} ( \Omega )$ and $\sigma _ { 1 } \prec \sigma _ { 2 }$, then $W ^ { 1 } L _ { \Phi } ( \Omega )$ is compactly imbedded into $C ^ { 0 , \sigma _ { 1 } ( t ) } ( \Omega )$.
The sublimiting case was handled also by the method of Fourier analysis (cf. e.g. [a9], [a8]), by considering potential Orlicz–Sobolev spaces (nevertheless, in this case non-reflexive spaces are excluded). The problem of the best target space in the scale of Orlicz spaces has been dealt with in, e.g., [a3]). Recently (1998), logarithmic Sobolev spaces, which are nothing but Orlicz–Sobolev spaces with generating function of the type $t ^ { p } ( \operatorname { log } ( 1 + t ) ) ^ { \alpha }$, tuning the scale of Sobolev spaces, have been used in connection with limiting imbeddings into exponential Orlicz spaces and/or logarithmic Lipschitz spaces (see, e.g., [a7], [a5], [a6]).