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]).

There are, however, still many open problems in the theory. Apart from difficulties of rather technical nature, the whole scale of these spaces presumably cannot be handled by known methods of interpolation and/or extrapolation of Sobolev spaces or even more general Besov or Triebel–Lizorkin spaces.

References

[a1]	R.A. Adams, "Sobolev spaces" , Acad. Press (1975)
[a2]	R.A. Adams, "General logarithmic Sobolev inequalities and Orlicz imbeddings" J. Funct. Anal. , 34 (1979) pp. 292–303
[a3]	A. Cianchi, "A sharp embedding theorem for Orlicz–Sobolev spaces" Indiana Univ. Math. J. , 45 (1996) pp. 39–65
[a4]	T.K. Donaldson, N.S. Trudinger, "Orlicz–Sobolev spaces and imbedding theorems" J. Funct. Anal. , 8 (1971) pp. 52–75
[a5]	D.E. Edmunds, M. Krbec, "Two limiting cases of Sobolev imbeddings" Houston J. Math. , 21 (1995) pp. 119–128
[a6]	D.E. Edmunds, H. Triebel, "Logarithmic Sobolev spaces and their applications to spectral theory" Proc. London Math. Soc. , 71 : 3 (1995) pp. 333–371
[a7]	N. Fusco, P.L. Lions, C. Sbordone, "Sobolev imbedding theorems in borderline case" Proc. Amer. Math. Soc. , 124 (1996) pp. 562–565
[a8]	V. Kokilashvili, M. Krbec, "Weighted inequalities in Lorentz and Orlicz spaces" , World Sci. (1991)
[a9]	A. Torchinsky, "Interpolation of operators and Orlicz classes" Studia Math. , 59 (1976) pp. 177–207
[a10]	N. Trudinger, "On imbeddings into Orlicz spaces and some applications" J. Math. Mech. , 17 (1967) pp. 473–483