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Establishing [[Imbedding theorems|imbedding theorems]] goes back to T.K. Donaldson and N.S. Trudinger, and R.A. Adams (see [[#References|[a4]]], [[#References|[a10]]], [[#References|[a2]]]). For the sake of simplicity, consider the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200101.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200102.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200103.png" /> (cf. [[Orlicz–Sobolev space|Orlicz–Sobolev space]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200104.png" /> is sufficiently smooth (a Lipschitz boundary, for instance; see [[#References|[a1]]]). Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200106.png" />. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200107.png" /> corresponds to the sublimiting case for usual Sobolev spaces (cf. also [[Sobolev space|Sobolev space]]) and in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200108.png" /> is imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i1200109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001011.png" />. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001012.png" /> corresponds to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001014.png" />. If one defines the generalized Hölder spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001016.png" /> is a increasing continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001018.png" />, as the space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001019.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001020.png" />, then the target space for the imbeddings is a [[Hölder space|Hölder space]] of this type with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001021.png" />.
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A (partial) ordering of such functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001022.png" /> (and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001023.png" />-functions) can be introduced in the following way: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001025.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001026.png" />-functions, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001027.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001028.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001029.png" />. Further, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001030.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001031.png" /> is a function of the same type. If now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001032.png" /> is imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001035.png" /> is compactly imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001036.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001037.png" /> is imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001040.png" /> is compactly imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001041.png" />.
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The sublimiting case was handled also by the method of Fourier analysis (cf. e.g. [[#References|[a9]]], [[#References|[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., [[#References|[a3]]]). Recently (1998), logarithmic Sobolev spaces, which are nothing but Orlicz–Sobolev spaces with generating function of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120010/i12001042.png" />, 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., [[#References|[a7]]], [[#References|[a5]]], [[#References|[a6]]]).
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Establishing [[Imbedding theorems|imbedding theorems]] goes back to T.K. Donaldson and N.S. Trudinger, and R.A. Adams (see [[#References|[a4]]], [[#References|[a10]]], [[#References|[a2]]]). For the sake of simplicity, consider the space $W ^ { 1 } L _ { \Phi } ( \Omega )$ with an $N$-function $\Phi$ (cf. [[Orlicz–Sobolev space|Orlicz–Sobolev space]]), where $\partial \Omega$ is sufficiently smooth (a Lipschitz boundary, for instance; see [[#References|[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|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|Hölder space]] of this type with $\sigma ( t ) = \int _ { t ^ { - n }} g_ {\Phi }^ { \infty } ( s ) d s$.
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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 $C ^ { 0 , \sigma _ { 1 } ( t ) } ( \Omega )$.
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 +
The sublimiting case was handled also by the method of Fourier analysis (cf. e.g. [[#References|[a9]]], [[#References|[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., [[#References|[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., [[#References|[a7]]], [[#References|[a5]]], [[#References|[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.
 
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.A. Adams,  "Sobolev spaces" , Acad. Press  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.A. Adams,  "General logarithmic Sobolev inequalities and Orlicz imbeddings"  ''J. Funct. Anal.'' , '''34'''  (1979)  pp. 292–303</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Cianchi,  "A sharp embedding theorem for Orlicz–Sobolev spaces"  ''Indiana Univ. Math. J.'' , '''45'''  (1996)  pp. 39–65</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T.K. Donaldson,  N.S. Trudinger,  "Orlicz–Sobolev spaces and imbedding theorems"  ''J. Funct. Anal.'' , '''8'''  (1971)  pp. 52–75</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.E. Edmunds,  M. Krbec,  "Two limiting cases of Sobolev imbeddings"  ''Houston J. Math.'' , '''21'''  (1995)  pp. 119–128</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D.E. Edmunds,  H. Triebel,  "Logarithmic Sobolev spaces and their applications to spectral theory"  ''Proc. London Math. Soc.'' , '''71''' :  3  (1995)  pp. 333–371</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  N. Fusco,  P.L. Lions,  C. Sbordone,  "Sobolev imbedding theorems in borderline case"  ''Proc. Amer. Math. Soc.'' , '''124'''  (1996)  pp. 562–565</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  V. Kokilashvili,  M. Krbec,  "Weighted inequalities in Lorentz and Orlicz spaces" , World Sci.  (1991)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A. Torchinsky,  "Interpolation of operators and Orlicz classes"  ''Studia Math.'' , '''59'''  (1976)  pp. 177–207</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  N. Trudinger,  "On imbeddings into Orlicz spaces and some applications"  ''J. Math. Mech.'' , '''17'''  (1967)  pp. 473–483</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  R.A. Adams,  "Sobolev spaces" , Acad. Press  (1975)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.A. Adams,  "General logarithmic Sobolev inequalities and Orlicz imbeddings"  ''J. Funct. Anal.'' , '''34'''  (1979)  pp. 292–303</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A. Cianchi,  "A sharp embedding theorem for Orlicz–Sobolev spaces"  ''Indiana Univ. Math. J.'' , '''45'''  (1996)  pp. 39–65</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  T.K. Donaldson,  N.S. Trudinger,  "Orlicz–Sobolev spaces and imbedding theorems"  ''J. Funct. Anal.'' , '''8'''  (1971)  pp. 52–75</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  D.E. Edmunds,  M. Krbec,  "Two limiting cases of Sobolev imbeddings"  ''Houston J. Math.'' , '''21'''  (1995)  pp. 119–128</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  D.E. Edmunds,  H. Triebel,  "Logarithmic Sobolev spaces and their applications to spectral theory"  ''Proc. London Math. Soc.'' , '''71''' :  3  (1995)  pp. 333–371</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  N. Fusco,  P.L. Lions,  C. Sbordone,  "Sobolev imbedding theorems in borderline case"  ''Proc. Amer. Math. Soc.'' , '''124'''  (1996)  pp. 562–565</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  V. Kokilashvili,  M. Krbec,  "Weighted inequalities in Lorentz and Orlicz spaces" , World Sci.  (1991)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  A. Torchinsky,  "Interpolation of operators and Orlicz classes"  ''Studia Math.'' , '''59'''  (1976)  pp. 177–207</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  N. Trudinger,  "On imbeddings into Orlicz spaces and some applications"  ''J. Math. Mech.'' , '''17'''  (1967)  pp. 473–483</td></tr></table>

Latest revision as of 16:08, 27 January 2024

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
How to Cite This Entry:
Imbedding theorems for Orlicz-Sobolev spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_theorems_for_Orlicz-Sobolev_spaces&oldid=12069
This article was adapted from an original article by Miroslav Krbec (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article