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A natural generalization of the notion of a [[Sobolev space|Sobolev space]], where the underlying role of the Lebesgue spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o1200601.png" /> is played by the more general Orlicz spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o1200602.png" /> (cf. also [[Lebesgue space|Lebesgue space]]; [[Orlicz space|Orlicz space]]). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o1200603.png" /> is a Young function (cf. [[Orlicz–Lorentz space|Orlicz–Lorentz space]]) or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o1200604.png" />-function, depending on the desired generality of the concept. Various authors use different definitions of the generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o1200605.png" />.
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A natural generalization of the notion of a [[Sobolev space]], where the underlying role of the Lebesgue spaces $L _ { p }$ is played by the more general Orlicz spaces $L _ { \Phi }$ (cf. also [[Lebesgue space]]; [[Orlicz space]]). Here, $\Phi$ is a Young function (cf. [[Orlicz–Lorentz space|Orlicz–Lorentz space]]) or an $N$-function, depending on the desired generality of the concept. Various authors use different definitions of the generating function $\Phi$.
  
 
The classical setting of the theory of Orlicz spaces can be found, e.g., in [[#References|[a19]]], [[#References|[a11]]]; the theory of more general modular spaces goes back to H. Nakano [[#References|[a16]]] and it has been systematically developed by the Poznań school in the framework of Orlicz–Musielak spaces, see [[#References|[a15]]].
 
The classical setting of the theory of Orlicz spaces can be found, e.g., in [[#References|[a19]]], [[#References|[a11]]]; the theory of more general modular spaces goes back to H. Nakano [[#References|[a16]]] and it has been systematically developed by the Poznań school in the framework of Orlicz–Musielak spaces, see [[#References|[a15]]].
  
Since the 1960s, the need to use spaces of functions with generalized derivatives in Orlicz spaces came from various applications in integral equations, boundary value problems, etc., whenever it is useful to consider wider scales of spaces than those based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o1200606.png" /> (see, e.g. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a14]]], [[#References|[a18]]] and references therein).
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Since the 1960s, the need to use spaces of functions with generalized derivatives in Orlicz spaces came from various applications in integral equations, boundary value problems, etc., whenever it is useful to consider wider scales of spaces than those based on $L _ { p }$ (see, e.g. [[#References|[a3]]], [[#References|[a4]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a14]]], [[#References|[a18]]] and references therein).
  
The standard construction of an Orlicz–Sobolev space is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o1200607.png" /> be a domain, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o1200608.png" /> be a classical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006010.png" />-function, that is, real-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006011.png" />, continuous, increasing, convex, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006013.png" />. (Observe that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006014.png" />, then only the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006015.png" /> at infinity matters.) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006016.png" />. Then the Orlicz–Sobolev space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006017.png" /> is the set of measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006019.png" /> with generalized (weak) derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006020.png" /> up to the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006021.png" /> (cf. [[Weak derivative|Weak derivative]]; [[Generalized function|Generalized function]]), and equipped with the finite norm
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The standard construction of an Orlicz–Sobolev space is as follows. Let $\Omega \subset \mathbf{R} ^ { n }$ be a domain, let $\Phi$ be a classical $N$-function, that is, real-valued function on $[ 0 , + \infty )$, continuous, increasing, convex, and such that $\operatorname { lim } _ { t \rightarrow 0 } \Phi ( t ) / t = 0$, $\operatorname { lim } _ { t \rightarrow + \infty } \Phi ( t ) / t = + \infty$. (Observe that if $| \Omega | < \infty$, then only the behaviour of $\Phi$ at infinity matters.) Let $k \in \mathbf{N}$. Then the Orlicz–Sobolev space $W ^ { k } L _ { \Phi } ( \Omega )$ is the set of measurable functions $f$ on $\Omega$ with generalized (weak) derivatives $D ^ { \alpha } f$ up to the order $k$ (cf. [[Weak derivative|Weak derivative]]; [[Generalized function|Generalized function]]), and equipped with the finite norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \left\| f |_ { W ^{k} L _ { \Phi } ( \Omega ) } \right\| = \sum _ { | \alpha | \leq k } \| D ^ { \alpha } f \| _ { L _ { \Phi } ( \Omega ) }. \end{equation}
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006023.png" /> is the [[Luxemburg norm|Luxemburg norm]] in the Orlicz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006024.png" />. Alternatively, in (a1) one can use the Orlicz norm and any expression equivalent to the summation (such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006025.png" />). Also, it is possible to define anisotropic spaces, considering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006027.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006028.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006029.png" />-functions. Moreover, one can also consider mixed norms, replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006030.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006031.png" />-tuples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006032.png" />-functions and various combinations. It is easy to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006033.png" /> is a closed subset of a product of the Orlicz spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006034.png" />, hence it is a [[Banach space|Banach space]].
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Here $\| . \| _ { L _ { \Phi }  ( \Omega )}$ is the [[Luxemburg norm|Luxemburg norm]] in the Orlicz space $L _ { \Phi } ( \Omega )$. Alternatively, in (a1) one can use the Orlicz norm and any expression equivalent to the summation (such as $\operatorname{max}$). Also, it is possible to define anisotropic spaces, considering $\| D ^ { \alpha } f |_{L _ { \Phi _ { \alpha } }} ( \Omega ) \|$, $\alpha \leq k$, where the $\Phi _ { \alpha }$ are $N$-functions. Moreover, one can also consider mixed norms, replacing $\Phi _ { \alpha }$ by $n$-tuples of $N$-functions and various combinations. It is easy to see that $W ^ { k } L _ { \Phi } ( \Omega )$ is a closed subset of a product of the Orlicz spaces $L _ { \Phi } ( \Omega )$, hence it is a [[Banach space|Banach space]].
  
Elementary properties of Orlicz spaces imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006035.png" /> is separable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006036.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006038.png" />-condition (cf. [[Orlicz–Lorentz space|Orlicz–Lorentz space]]). Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006039.png" /> is reflexive if and only if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006040.png" /> and the complementary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006041.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006042.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006043.png" />) satisfy the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006044.png" />-condition. It is useful to introduce the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006045.png" />, consisting of the functions with generalized derivatives in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006047.png" /> is the closure of the set of bounded functions with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006048.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006049.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006050.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006051.png" />; further, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006052.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006053.png" />. Analogues of density theorems known from the theory of Sobolev spaces, with various assumptions about the smoothness of the boundary of the domain in question, can be proved too, thus making it possible to consider alternative definitions. This is important for the standard proof techniques, which work with smooth functions only. (See [[#References|[a1]]], [[#References|[a6]]], [[#References|[a10]]], [[#References|[a12]]].)
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Elementary properties of Orlicz spaces imply that $W ^ { k } L _ { \Phi } ( \Omega )$ is separable if and only if $\Phi$ satisfies the $\Delta _ { 2 }$-condition (cf. [[Orlicz–Lorentz space|Orlicz–Lorentz space]]). Furthermore, $W ^ { k } L _ { \Phi } ( \Omega )$ is reflexive if and only if both $\Phi$ and the complementary function $\tilde { \Phi }$ to $\Phi$ (that is, $\tilde { \Phi } ( s ) = \operatorname { sup } \{ | s | t - \Phi ( t ) : t \geq 0 \}$) satisfy the $\Delta _ { 2 }$-condition. It is useful to introduce the space $W ^ { k } E _ { \Phi } ( \Omega )$, consisting of the functions with generalized derivatives in $E _ { \Phi } ( \Omega )$, where $E _ { \Phi } ( \Omega )$ is the closure of the set of bounded functions with compact support in $\Omega$. If $\Omega = {\bf R} ^ { n }$, then $C _ { 0 } ^ { \infty }$ is dense in $W ^ { k } E _ { \Phi } ( \Omega )$; further, $C ^ { \infty } ( \Omega ) \cap W ^ { k } E _ { \Phi } ( \Omega )$ is dense in $W ^ { k } E _ { \Phi } ( \Omega )$. Analogues of density theorems known from the theory of Sobolev spaces, with various assumptions about the smoothness of the boundary of the domain in question, can be proved too, thus making it possible to consider alternative definitions. This is important for the standard proof techniques, which work with smooth functions only. (See [[#References|[a1]]], [[#References|[a6]]], [[#References|[a10]]], [[#References|[a12]]].)
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006054.png" />, then one can define the Orlicz–Sobolev potential spaces in a manner quite analogous to the Sobolev case as well as to the case of potential spaces on sufficiently smooth domains. This is also a natural way to define Orlicz–Sobolev spaces of fractional order. The situation here, however, is more delicate than in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006055.png" />-based case. Interpolation properties of Orlicz spaces (see, e.g., [[#References|[a9]]]) guarantee that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006056.png" /> (and consequently also spaces on domains with the extension property) can be obtained by interpolation (cf. [[Interpolation of operators|Interpolation of operators]]) from a suitable couple of Sobolev spaces provided that both the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006057.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006058.png" /> and its complementary function satisfy the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006059.png" />-condition (or, equivalently, that each [[Boyd index|Boyd index]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006060.png" /> (see [[#References|[a2]]]) lies strictly between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006062.png" />). Namely, in this case a Mikhlin-type multiplier theorem holds. Furthermore, the known trace theorems (see [[#References|[a13]]], [[#References|[a17]]], for instance), identifying the trace space for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006064.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006065.png" />-condition, with the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006066.png" /> of functions with finite norm
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If $\Omega = {\bf R} ^ { n }$, then one can define the Orlicz–Sobolev potential spaces in a manner quite analogous to the Sobolev case as well as to the case of potential spaces on sufficiently smooth domains. This is also a natural way to define Orlicz–Sobolev spaces of fractional order. The situation here, however, is more delicate than in the $L _ { p }$-based case. Interpolation properties of Orlicz spaces (see, e.g., [[#References|[a9]]]) guarantee that $W ^ { k } E _ { \Phi } (  \mathbf{R} ^ { n } )$ (and consequently also spaces on domains with the extension property) can be obtained by interpolation (cf. [[Interpolation of operators|Interpolation of operators]]) from a suitable couple of Sobolev spaces provided that both the $N$-function $\Phi$ and its complementary function satisfy the $\Delta _ { 2 }$-condition (or, equivalently, that each [[Boyd index|Boyd index]] of $\Phi$ (see [[#References|[a2]]]) lies strictly between $1$ and $\infty$). Namely, in this case a Mikhlin-type multiplier theorem holds. Furthermore, the known trace theorems (see [[#References|[a13]]], [[#References|[a17]]], for instance), identifying the trace space for $W ^ { k } L _ { \Phi } ( \Omega )$, where $\tilde { \Phi }$ satisfies the $\Delta _ { 2 }$-condition, with the subspace of $W ^ { k - 1 } L _ { \Phi } ( \partial \Omega )$ of functions with finite norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006067.png" /></td> </tr></table>
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\begin{equation*} \| f \| _ { W ^ { k - 1 }  L _ { \Phi } ( \partial \Omega )}  + \textbf { inf }  \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006068.png" /></td> </tr></table>
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\begin{equation*} \left\{ \lambda > 0 : \sum _ { | \alpha | = k - 1 } \int _ { \partial \Omega \times \partial \Omega } \Phi \left( \frac { \Delta_{ y - x} F ( x ) } { | y - x | } \right) \eta ( x , y ) \leq 1 \right\}, \end{equation*}
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006069.png" /></td> </tr></table>
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\begin{equation*} \Delta _ { h } F ( x ) = F ( x + h ) - F ( x ), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006070.png" /></td> </tr></table>
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\begin{equation*} \eta ( x , y ) = | y - x | ^ { 2 - n } d x d y, \end{equation*}
  
 
indicate how another  "natural"  interpolation scale (this time with respect to the smoothness) should look like. The theory, however, is not yet (1998) complete in this area.
 
indicate how another  "natural"  interpolation scale (this time with respect to the smoothness) should look like. The theory, however, is not yet (1998) complete in this area.
  
In connection with certain applications (imbeddings in finer scales of spaces, entropy and approximation numbers, etc.), attention has been given to logarithmic Orlicz and Orlicz–Sobolev spaces, where the generating function is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006071.png" /> for large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006073.png" />. It turns out that these can also be handled using extrapolation of Sobolev spaces, making it thus possible to employ various techniques (e.g. an analytical Fourier approach) known from the theory of Sobolev spaces. A basic reference is [[#References|[a5]]].
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In connection with certain applications (imbeddings in finer scales of spaces, entropy and approximation numbers, etc.), attention has been given to logarithmic Orlicz and Orlicz–Sobolev spaces, where the generating function is equivalent to $t ^ { p } \operatorname { log } ^ { \sigma } t$ for large values of $t$ and $1 < p < \infty$. It turns out that these can also be handled using extrapolation of Sobolev spaces, making it thus possible to employ various techniques (e.g. an analytical Fourier approach) known from the theory of Sobolev spaces. A basic reference is [[#References|[a5]]].
  
For a survey of the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006074.png" />, see [[#References|[a1]]] and [[#References|[a12]]].
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For a survey of the properties of $W ^ { k } L _ { \Phi } ( \Omega )$, see [[#References|[a1]]] and [[#References|[a12]]].
  
 
====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">  D.W. Boyd,  "Indices for the Orlicz spaces"  ''Pacific J. Math.'' , '''38'''  (1971)  pp. 315–323</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Donaldson,  "Nonlinear elliptic boundary-value problems in Orlicz–Sobolev spaces"  ''J. Diff. Eq.'' , '''10'''  (1971)  pp. 507–528</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Donaldson,  "Inhomogeneous Orlicz–Sobolev spaces and nonlinear parabolic initial value problems"  ''J. Diff. Eq.'' , '''16'''  (1974)  pp. 201–256</TD></TR><TR><TD valign="top">[a5]</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">[a6]</TD> <TD valign="top">  A. Fougères,  "Approximation dans les espaces de Sobolev et de Sobolev–Orlicz"  ''C.R. Acad. Sci. Paris Ser. A'' , '''274'''  (1972)  pp. 479–482</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.-P. Gossez,  "Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients"  ''Trans. Amer. Math. Soc.'' , '''190'''  (1974)  pp. 163–205</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.-P. Gossez,  V. Mustonen,  "Variational inequalities in Orlicz–Sobolev spaces"  ''Nonlinear Anal. Theory Meth. Appl.'' , '''11'''  (1987)  pp. 379–392</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J. Gustavsson,  J. Peetre,  "Interpolation of Orlicz spaces"  ''Studia Math.'' , '''60'''  (1977)  pp. 33–59</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H. Hudzik,  "The problems of separability, duality, reflexivity and of comparison for generalized Orlicz–Sobolev spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120060/o12006075.png" />"  ''Comment. Math. Helvetici'' , '''21'''  (1979)  pp. 315–324</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  M.A. Krasnosel'skii,  Ya.B. Rutitskii,  "Convex functions and Orlicz spaces" , Noordhoff  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A. Kufner,  O. John,  S. Fučik,  "Function spaces" , Acad. Prague  (1977)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  M.-Th. Lacroix,  "Espaces d'interpolation et de traces des espaces de Sobolev–Orlicz d'ordre 1"  ''C.R. Acad. Sci. Paris Ser. A'' , '''280'''  (1975)  pp. 271–274</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  R. Landes,  V. Mustonen,  "Pseudo-monotone mappings in Sobolev–Orlicz spaces and nonlinear boundary value problems on unbounded domains"  ''J. Math. Anal. Appl.'' , '''88'''  (1982)  pp. 25–36</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  J. Musielak,  "Orlicz spaces and modular spaces" , ''Lecture Notes Math.'' , '''1034''' , Springer  (1983)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  H. Nakano,  "Modulared semi-ordered linear spaces" , Maruzen  (1950)</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  G. Palmieri,  "Alcune disuguaglianze per derivate intermedie negli spazi di Orlicz–Sobolev e applicazioni"  ''Rend. Accad. Sci. Fis. Mat. IV. Ser., Napoli'' , '''46'''  (1979)  pp. 633–652  (In Italian)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  P.A. Vuillermot,  "Hölder-regularity for the solutions of strongly nonlinear eigenvalue problems on Orlicz–Sobolev spaces"  ''Houston J. Math.'' , '''13'''  (1987)  pp. 281–287</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  A.C. Zaanen,  "Linear analysis" , Noordhoff  (1953)</TD></TR></table>
+
<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">  D.W. Boyd,  "Indices for the Orlicz spaces"  ''Pacific J. Math.'' , '''38'''  (1971)  pp. 315–323</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  T. Donaldson,  "Nonlinear elliptic boundary-value problems in Orlicz–Sobolev spaces"  ''J. Diff. Eq.'' , '''10'''  (1971)  pp. 507–528</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  T. Donaldson,  "Inhomogeneous Orlicz–Sobolev spaces and nonlinear parabolic initial value problems"  ''J. Diff. Eq.'' , '''16'''  (1974)  pp. 201–256</td></tr><tr><td valign="top">[a5]</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">[a6]</td> <td valign="top">  A. Fougères,  "Approximation dans les espaces de Sobolev et de Sobolev–Orlicz"  ''C.R. Acad. Sci. Paris Ser. A'' , '''274'''  (1972)  pp. 479–482</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J.-P. Gossez,  "Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients"  ''Trans. Amer. Math. Soc.'' , '''190'''  (1974)  pp. 163–205</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  J.-P. Gossez,  V. Mustonen,  "Variational inequalities in Orlicz–Sobolev spaces"  ''Nonlinear Anal. Theory Meth. Appl.'' , '''11'''  (1987)  pp. 379–392</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J. Gustavsson,  J. Peetre,  "Interpolation of Orlicz spaces"  ''Studia Math.'' , '''60'''  (1977)  pp. 33–59</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  H. Hudzik,  "The problems of separability, duality, reflexivity and of comparison for generalized Orlicz–Sobolev spaces $W_ { m } ^ { k } ( \Omega )$"  ''Comment. Math. Helvetici'' , '''21'''  (1979)  pp. 315–324</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  M.A. Krasnosel'skii,  Ya.B. Rutitskii,  "Convex functions and Orlicz spaces" , Noordhoff  (1961)  (In Russian)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  A. Kufner,  O. John,  S. Fučik,  "Function spaces" , Acad. Prague  (1977)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  M.-Th. Lacroix,  "Espaces d'interpolation et de traces des espaces de Sobolev–Orlicz d'ordre 1"  ''C.R. Acad. Sci. Paris Ser. A'' , '''280'''  (1975)  pp. 271–274</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  R. Landes,  V. Mustonen,  "Pseudo-monotone mappings in Sobolev–Orlicz spaces and nonlinear boundary value problems on unbounded domains"  ''J. Math. Anal. Appl.'' , '''88'''  (1982)  pp. 25–36</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  J. Musielak,  "Orlicz spaces and modular spaces" , ''Lecture Notes Math.'' , '''1034''' , Springer  (1983)</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  H. Nakano,  "Modulared semi-ordered linear spaces" , Maruzen  (1950)</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  G. Palmieri,  "Alcune disuguaglianze per derivate intermedie negli spazi di Orlicz–Sobolev e applicazioni"  ''Rend. Accad. Sci. Fis. Mat. IV. Ser., Napoli'' , '''46'''  (1979)  pp. 633–652  (In Italian)</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  P.A. Vuillermot,  "Hölder-regularity for the solutions of strongly nonlinear eigenvalue problems on Orlicz–Sobolev spaces"  ''Houston J. Math.'' , '''13'''  (1987)  pp. 281–287</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  A.C. Zaanen,  "Linear analysis" , Noordhoff  (1953)</td></tr></table>

Latest revision as of 19:25, 12 February 2024

A natural generalization of the notion of a Sobolev space, where the underlying role of the Lebesgue spaces $L _ { p }$ is played by the more general Orlicz spaces $L _ { \Phi }$ (cf. also Lebesgue space; Orlicz space). Here, $\Phi$ is a Young function (cf. Orlicz–Lorentz space) or an $N$-function, depending on the desired generality of the concept. Various authors use different definitions of the generating function $\Phi$.

The classical setting of the theory of Orlicz spaces can be found, e.g., in [a19], [a11]; the theory of more general modular spaces goes back to H. Nakano [a16] and it has been systematically developed by the Poznań school in the framework of Orlicz–Musielak spaces, see [a15].

Since the 1960s, the need to use spaces of functions with generalized derivatives in Orlicz spaces came from various applications in integral equations, boundary value problems, etc., whenever it is useful to consider wider scales of spaces than those based on $L _ { p }$ (see, e.g. [a3], [a4], [a7], [a8], [a14], [a18] and references therein).

The standard construction of an Orlicz–Sobolev space is as follows. Let $\Omega \subset \mathbf{R} ^ { n }$ be a domain, let $\Phi$ be a classical $N$-function, that is, real-valued function on $[ 0 , + \infty )$, continuous, increasing, convex, and such that $\operatorname { lim } _ { t \rightarrow 0 } \Phi ( t ) / t = 0$, $\operatorname { lim } _ { t \rightarrow + \infty } \Phi ( t ) / t = + \infty$. (Observe that if $| \Omega | < \infty$, then only the behaviour of $\Phi$ at infinity matters.) Let $k \in \mathbf{N}$. Then the Orlicz–Sobolev space $W ^ { k } L _ { \Phi } ( \Omega )$ is the set of measurable functions $f$ on $\Omega$ with generalized (weak) derivatives $D ^ { \alpha } f$ up to the order $k$ (cf. Weak derivative; Generalized function), and equipped with the finite norm

\begin{equation} \tag{a1} \left\| f |_ { W ^{k} L _ { \Phi } ( \Omega ) } \right\| = \sum _ { | \alpha | \leq k } \| D ^ { \alpha } f \| _ { L _ { \Phi } ( \Omega ) }. \end{equation}

Here $\| . \| _ { L _ { \Phi } ( \Omega )}$ is the Luxemburg norm in the Orlicz space $L _ { \Phi } ( \Omega )$. Alternatively, in (a1) one can use the Orlicz norm and any expression equivalent to the summation (such as $\operatorname{max}$). Also, it is possible to define anisotropic spaces, considering $\| D ^ { \alpha } f |_{L _ { \Phi _ { \alpha } }} ( \Omega ) \|$, $\alpha \leq k$, where the $\Phi _ { \alpha }$ are $N$-functions. Moreover, one can also consider mixed norms, replacing $\Phi _ { \alpha }$ by $n$-tuples of $N$-functions and various combinations. It is easy to see that $W ^ { k } L _ { \Phi } ( \Omega )$ is a closed subset of a product of the Orlicz spaces $L _ { \Phi } ( \Omega )$, hence it is a Banach space.

Elementary properties of Orlicz spaces imply that $W ^ { k } L _ { \Phi } ( \Omega )$ is separable if and only if $\Phi$ satisfies the $\Delta _ { 2 }$-condition (cf. Orlicz–Lorentz space). Furthermore, $W ^ { k } L _ { \Phi } ( \Omega )$ is reflexive if and only if both $\Phi$ and the complementary function $\tilde { \Phi }$ to $\Phi$ (that is, $\tilde { \Phi } ( s ) = \operatorname { sup } \{ | s | t - \Phi ( t ) : t \geq 0 \}$) satisfy the $\Delta _ { 2 }$-condition. It is useful to introduce the space $W ^ { k } E _ { \Phi } ( \Omega )$, consisting of the functions with generalized derivatives in $E _ { \Phi } ( \Omega )$, where $E _ { \Phi } ( \Omega )$ is the closure of the set of bounded functions with compact support in $\Omega$. If $\Omega = {\bf R} ^ { n }$, then $C _ { 0 } ^ { \infty }$ is dense in $W ^ { k } E _ { \Phi } ( \Omega )$; further, $C ^ { \infty } ( \Omega ) \cap W ^ { k } E _ { \Phi } ( \Omega )$ is dense in $W ^ { k } E _ { \Phi } ( \Omega )$. Analogues of density theorems known from the theory of Sobolev spaces, with various assumptions about the smoothness of the boundary of the domain in question, can be proved too, thus making it possible to consider alternative definitions. This is important for the standard proof techniques, which work with smooth functions only. (See [a1], [a6], [a10], [a12].)

If $\Omega = {\bf R} ^ { n }$, then one can define the Orlicz–Sobolev potential spaces in a manner quite analogous to the Sobolev case as well as to the case of potential spaces on sufficiently smooth domains. This is also a natural way to define Orlicz–Sobolev spaces of fractional order. The situation here, however, is more delicate than in the $L _ { p }$-based case. Interpolation properties of Orlicz spaces (see, e.g., [a9]) guarantee that $W ^ { k } E _ { \Phi } ( \mathbf{R} ^ { n } )$ (and consequently also spaces on domains with the extension property) can be obtained by interpolation (cf. Interpolation of operators) from a suitable couple of Sobolev spaces provided that both the $N$-function $\Phi$ and its complementary function satisfy the $\Delta _ { 2 }$-condition (or, equivalently, that each Boyd index of $\Phi$ (see [a2]) lies strictly between $1$ and $\infty$). Namely, in this case a Mikhlin-type multiplier theorem holds. Furthermore, the known trace theorems (see [a13], [a17], for instance), identifying the trace space for $W ^ { k } L _ { \Phi } ( \Omega )$, where $\tilde { \Phi }$ satisfies the $\Delta _ { 2 }$-condition, with the subspace of $W ^ { k - 1 } L _ { \Phi } ( \partial \Omega )$ of functions with finite norm

\begin{equation*} \| f \| _ { W ^ { k - 1 } L _ { \Phi } ( \partial \Omega )} + \textbf { inf } \end{equation*}

\begin{equation*} \left\{ \lambda > 0 : \sum _ { | \alpha | = k - 1 } \int _ { \partial \Omega \times \partial \Omega } \Phi \left( \frac { \Delta_{ y - x} F ( x ) } { | y - x | } \right) \eta ( x , y ) \leq 1 \right\}, \end{equation*}

where

\begin{equation*} \Delta _ { h } F ( x ) = F ( x + h ) - F ( x ), \end{equation*}

\begin{equation*} \eta ( x , y ) = | y - x | ^ { 2 - n } d x d y, \end{equation*}

indicate how another "natural" interpolation scale (this time with respect to the smoothness) should look like. The theory, however, is not yet (1998) complete in this area.

In connection with certain applications (imbeddings in finer scales of spaces, entropy and approximation numbers, etc.), attention has been given to logarithmic Orlicz and Orlicz–Sobolev spaces, where the generating function is equivalent to $t ^ { p } \operatorname { log } ^ { \sigma } t$ for large values of $t$ and $1 < p < \infty$. It turns out that these can also be handled using extrapolation of Sobolev spaces, making it thus possible to employ various techniques (e.g. an analytical Fourier approach) known from the theory of Sobolev spaces. A basic reference is [a5].

For a survey of the properties of $W ^ { k } L _ { \Phi } ( \Omega )$, see [a1] and [a12].

References

[a1] R.A. Adams, "Sobolev spaces" , Acad. Press (1975)
[a2] D.W. Boyd, "Indices for the Orlicz spaces" Pacific J. Math. , 38 (1971) pp. 315–323
[a3] T. Donaldson, "Nonlinear elliptic boundary-value problems in Orlicz–Sobolev spaces" J. Diff. Eq. , 10 (1971) pp. 507–528
[a4] T. Donaldson, "Inhomogeneous Orlicz–Sobolev spaces and nonlinear parabolic initial value problems" J. Diff. Eq. , 16 (1974) pp. 201–256
[a5] D.E. Edmunds, H. Triebel, "Logarithmic Sobolev spaces and their applications to spectral theory" Proc. London Math. Soc. , 71 : 3 (1995) pp. 333–371
[a6] A. Fougères, "Approximation dans les espaces de Sobolev et de Sobolev–Orlicz" C.R. Acad. Sci. Paris Ser. A , 274 (1972) pp. 479–482
[a7] J.-P. Gossez, "Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients" Trans. Amer. Math. Soc. , 190 (1974) pp. 163–205
[a8] J.-P. Gossez, V. Mustonen, "Variational inequalities in Orlicz–Sobolev spaces" Nonlinear Anal. Theory Meth. Appl. , 11 (1987) pp. 379–392
[a9] J. Gustavsson, J. Peetre, "Interpolation of Orlicz spaces" Studia Math. , 60 (1977) pp. 33–59
[a10] H. Hudzik, "The problems of separability, duality, reflexivity and of comparison for generalized Orlicz–Sobolev spaces $W_ { m } ^ { k } ( \Omega )$" Comment. Math. Helvetici , 21 (1979) pp. 315–324
[a11] M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (In Russian)
[a12] A. Kufner, O. John, S. Fučik, "Function spaces" , Acad. Prague (1977)
[a13] M.-Th. Lacroix, "Espaces d'interpolation et de traces des espaces de Sobolev–Orlicz d'ordre 1" C.R. Acad. Sci. Paris Ser. A , 280 (1975) pp. 271–274
[a14] R. Landes, V. Mustonen, "Pseudo-monotone mappings in Sobolev–Orlicz spaces and nonlinear boundary value problems on unbounded domains" J. Math. Anal. Appl. , 88 (1982) pp. 25–36
[a15] J. Musielak, "Orlicz spaces and modular spaces" , Lecture Notes Math. , 1034 , Springer (1983)
[a16] H. Nakano, "Modulared semi-ordered linear spaces" , Maruzen (1950)
[a17] G. Palmieri, "Alcune disuguaglianze per derivate intermedie negli spazi di Orlicz–Sobolev e applicazioni" Rend. Accad. Sci. Fis. Mat. IV. Ser., Napoli , 46 (1979) pp. 633–652 (In Italian)
[a18] P.A. Vuillermot, "Hölder-regularity for the solutions of strongly nonlinear eigenvalue problems on Orlicz–Sobolev spaces" Houston J. Math. , 13 (1987) pp. 281–287
[a19] A.C. Zaanen, "Linear analysis" , Noordhoff (1953)
How to Cite This Entry:
Orlicz-Sobolev space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orlicz-Sobolev_space&oldid=18323
This article was adapted from an original article by Miroslav Krbec (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article