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''Lorentz–Orlicz space''
 
''Lorentz–Orlicz space''
  
 
There are several definitions of Orlicz–Lorentz spaces known in the literature ([[#References|[a6]]], [[#References|[a11]]], [[#References|[a10]]], [[#References|[a2]]]). All of them are generalizations of both Orlicz and Lorentz spaces (cf. [[Orlicz space|Orlicz space]]; for Lorentz space, see [[Marcinkiewicz space|Marcinkiewicz space]]). The Orlicz–Lorentz space presented below arises naturally as an intermediate space between ordinary Lorentz space and the space of bounded functions in the Calderón–Lozanovskii method of interpolation ([[#References|[a4]]]; cf. also [[Interpolation of operators|Interpolation of operators]]).
 
There are several definitions of Orlicz–Lorentz spaces known in the literature ([[#References|[a6]]], [[#References|[a11]]], [[#References|[a10]]], [[#References|[a2]]]). All of them are generalizations of both Orlicz and Lorentz spaces (cf. [[Orlicz space|Orlicz space]]; for Lorentz space, see [[Marcinkiewicz space|Marcinkiewicz space]]). The Orlicz–Lorentz space presented below arises naturally as an intermediate space between ordinary Lorentz space and the space of bounded functions in the Calderón–Lozanovskii method of interpolation ([[#References|[a4]]]; cf. also [[Interpolation of operators|Interpolation of operators]]).
  
Given a Young function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o1200501.png" />, i.e. a convex function (cf. also [[Convex function (of a real variable)|Convex function (of a real variable)]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o1200502.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o1200503.png" />, the Orlicz–Lorentz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o1200504.png" /> ([[#References|[a6]]], [[#References|[a11]]]) is the collection of all real-valued Lebesgue-measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o1200505.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o1200506.png" /> (cf. also [[Lebesgue integral|Lebesgue integral]]) such that
+
Given a Young function $\varphi$, i.e. a convex function (cf. also [[Convex function (of a real variable)|Convex function (of a real variable)]]) $\varphi : \mathbf{R} _ { + } \rightarrow \mathbf{R} _ { + }$ such that $\varphi ( 0 ) = 0$, the Orlicz–Lorentz space $\Lambda _ { \varphi , w }$ ([[#References|[a6]]], [[#References|[a11]]]) is the collection of all real-valued Lebesgue-measurable functions $f$ on $\mathbf{R} _ { + }$ (cf. also [[Lebesgue integral|Lebesgue integral]]) such that
  
<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/o120050/o1200507.png" /></td> </tr></table>
+
\begin{equation*} I ( \lambda f ) : = \int _ { 0 } ^ { \infty } \varphi ( \lambda f ^ { * } ( s ) ) w ( s ) d s < \infty \end{equation*}
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o1200508.png" />, where the so-called weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o1200509.png" /> is positive, non-increasing and such that
+
for some $\lambda > 0$, where the so-called weight function $w : \mathbf{R} _ { + } \rightarrow \mathbf{R} _ { + }$ is positive, non-increasing and such that
  
<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/o120050/o12005010.png" /></td> </tr></table>
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\begin{equation*} \int _ { 0 } ^ { \infty } w ( s ) d s = \infty, \end{equation*}
  
and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005011.png" />,
+
and for all $t > 0$,
  
<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/o120050/o12005012.png" /></td> </tr></table>
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\begin{equation*} S ( t ) : = \int _ { 0 } ^ { t } w ( s ) d s < \infty. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005013.png" /> denotes the non-increasing rearrangement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005014.png" />, that is,
+
Here, $f ^ { * }$ denotes the non-increasing rearrangement of $f$, that is,
  
<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/o120050/o12005015.png" /></td> </tr></table>
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\begin{equation*} f ^ { * } ( t ) = \operatorname { inf } \{ s > 0 : d_f ( s ) \leq t \} \end{equation*}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005018.png" /> is the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005019.png" />. The Orlicz–Lorentz space, equipped with the norm
+
for $t \geq 0$, where $d_f ( t ) = m ( \{ s > 0 : | f ( s ) | > t \} )$ and $m$ is the Lebesgue measure on $\mathbf{R}$. The Orlicz–Lorentz space, equipped with the 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/o120050/o12005020.png" /></td> </tr></table>
+
\begin{equation*} \| f \| = \operatorname { inf } \{ \epsilon > 0 : I ( f / \epsilon ) \leq 1 \} \end{equation*}
  
is a Banach function lattice (cf. also [[Banach lattice|Banach lattice]]; [[Banach space|Banach space]]) with ordering: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005021.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005022.png" /> almost everywhere. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005023.png" />, then the Orlicz–Lorentz space becomes an Orlicz space, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005025.png" />, then it becomes a Lorentz space [[#References|[a9]]].
+
is a Banach function lattice (cf. also [[Banach lattice|Banach lattice]]; [[Banach space|Banach space]]) with ordering: $f \leq g$ whenever $f ( t ) \leq g ( t )$ almost everywhere. If $w ( t ) \equiv 1$, then the Orlicz–Lorentz space becomes an Orlicz space, and if $\varphi ( u ) = u ^ { p }$, $1 \leq p < \infty$, then it becomes a Lorentz space [[#References|[a9]]].
  
Many properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005026.png" /> have been described in terms of growth conditions imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005028.png" />. The most common growth conditions are regularity of the weight and condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005029.png" /> of a Young function. It is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005030.png" /> is regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005031.png" /> and a Young function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005032.png" /> satisfies condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005034.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005036.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005037.png" />. The methods applied in the theory of Orlicz–Lorentz spaces are derived from those for both Orlicz and Lorentz spaces.
+
Many properties of $\Lambda _ { \varphi , w }$ have been described in terms of growth conditions imposed on $\varphi$ and $w$. The most common growth conditions are regularity of the weight and condition $\Delta _ { 2 }$ of a Young function. It is said that $w$ is regular if $\operatorname { inf } _ { t > 0 } S ( 2 t ) / S ( t ) > 1$ and a Young function $\varphi$ satisfies condition $\Delta _ { 2 }$ whenever $\varphi ( 2 u ) \leq K \varphi ( u )$ for all $u \geq 0$ and some $K > 0$. The methods applied in the theory of Orlicz–Lorentz spaces are derived from those for both Orlicz and Lorentz spaces.
  
Some results on isomorphic or isometric properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005038.png" /> are as follows.
+
Some results on isomorphic or isometric properties of $\Lambda _ { \varphi , w }$ are as follows.
  
1) Condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005040.png" /> is equivalent to the following properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005041.png" /> [[#References|[a6]]].
+
1) Condition $\Delta _ { 2 }$ on $\varphi$ is equivalent to the following properties of $\Lambda _ { \varphi , w }$ [[#References|[a6]]].
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005042.png" /> does not contain an isometric copy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005043.png" />.
+
$\Lambda _ { \varphi , w }$ does not contain an isometric copy of $l ^ { \infty }$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005044.png" /> does not contain an isomorphic copy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005045.png" />.
+
$\Lambda _ { \varphi , w }$ does not contain an isomorphic copy of $l ^ { \infty }$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005046.png" /> does not contain an isomorphic copy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005047.png" />.
+
$\Lambda _ { \varphi , w }$ does not contain an isomorphic copy of $c_0$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005048.png" /> is separable (cf. also [[Separable space|Separable space]]).
+
$\Lambda _ { \varphi , w }$ is separable (cf. also [[Separable space|Separable space]]).
  
The norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005049.png" /> is absolutely continuous (cf. also [[Absolute continuity|Absolute continuity]]).
+
The norm in $\Lambda _ { \varphi , w }$ is absolutely continuous (cf. also [[Absolute continuity|Absolute continuity]]).
  
2) The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005051.png" /> is called the Young conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005052.png" /> (cf. also [[Conjugate function|Conjugate function]]). The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005054.png" /> are conjugate to each other and this duality is analogous to the duality between power functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005056.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005058.png" />. The dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005059.png" /> can be described in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005060.png" />. In fact, if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005062.png" /> satisfy condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005064.png" /> is regular, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005065.png" /> is the family of all Lebesgue-measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005066.png" /> such that
+
2) The function $\psi ( v ) = \operatorname { sup } _ { u > 0 } \{ u v - \varphi ( u ) \}$ on $\mathbf{R} _ { + }$ is called the Young conjugate to $\varphi$ (cf. also [[Conjugate function|Conjugate function]]). The functions $\varphi$ and $\psi$ are conjugate to each other and this duality is analogous to the duality between power functions $u ^ { p }$ and $u ^ { q }$ with $1 < p , q < \infty$ and $1 / p + 1 / q = 1$. The dual space $\Lambda _ { \varphi , w } ^ { * }$ can be described in terms of $\psi$. In fact, if both $\varphi$ and $\psi$ satisfy condition $\Delta _ { 2 }$ and $w$ is regular, then $\Lambda _ { \varphi , w } ^ { * }$ is the family of all Lebesgue-measurable functions $f$ such that
  
<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/o120050/o12005067.png" /></td> </tr></table>
+
\begin{equation*} \int _ { 0 } ^ { \infty } \psi ( f ^ { * } ( s ) / w ( s ) ) w ( s ) d s < \infty, \end{equation*}
  
and the dual norm is equivalent to the quasi-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005068.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005069.png" /> is reflexive (cf. also [[Reflexive space|Reflexive space]]) if and only if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005071.png" /> satisfy condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005072.png" />. Superreflexivity (cf. also [[Reflexive space|Reflexive space]]) requires, in addition, the assumption of regularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005073.png" /> [[#References|[a4]]].
+
and the dual norm is equivalent to the quasi-norm $\operatorname { inf } \{ \lambda > 0 : \int \psi ( f ^ { * } / \lambda w ) w < \infty \}$. The space $\Lambda _ { \varphi , w }$ is reflexive (cf. also [[Reflexive space|Reflexive space]]) if and only if both $\varphi$ and $\psi$ satisfy condition $\Delta _ { 2 }$. Superreflexivity (cf. also [[Reflexive space|Reflexive space]]) requires, in addition, the assumption of regularity of $w$ [[#References|[a4]]].
  
3) A number of geometric properties, like uniform convexity, rotundity, extreme points, local uniform convexity, and normal and uniform normal structure in Orlicz–Lorentz spaces have been characterized in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005075.png" /> ([[#References|[a3]]], [[#References|[a4]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a5]]]). For instance, necessary and sufficient conditions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005076.png" /> to be uniformly convex are that both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005078.png" /> satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005079.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005080.png" /> is regular and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005081.png" /> is uniformly convex, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005082.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005083.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005084.png" /> is the right derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005085.png" /> (cf. also [[Differentiation|Differentiation]]).
+
3) A number of geometric properties, like uniform convexity, rotundity, extreme points, local uniform convexity, and normal and uniform normal structure in Orlicz–Lorentz spaces have been characterized in terms of $\varphi$ and $w$ ([[#References|[a3]]], [[#References|[a4]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a5]]]). For instance, necessary and sufficient conditions for $\Lambda _ { \varphi , w }$ to be uniformly convex are that both $\varphi$ and $\varphi ^ { * }$ satisfy $\Delta _ { 2 }$, that $w$ is regular and that $\varphi$ is uniformly convex, i.e. $\operatorname { sup } _ { u > 0 } \varphi ^ { \prime } ( a u ) / \varphi ^ { \prime } ( u ) < 1$ for every $0 < a < 1$, where $\varphi ^ { \prime }$ is the right derivative of $\varphi$ (cf. also [[Differentiation]]).
  
 
Order convexity and concavity as well as Boyd indices in Orlicz–Lorentz spaces have been studied in, e.g., [[#References|[a10]]], [[#References|[a1]]].
 
Order convexity and concavity as well as Boyd indices in Orlicz–Lorentz spaces have been studied in, e.g., [[#References|[a10]]], [[#References|[a1]]].
  
In the definition of Orlicz–Lorentz space one can replace Lebesgue-measurable functions by measurable functions with respect to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005086.png" />-finite measure space. All results stated above remain the same in the case of a non-atomic infinite measure. For other measure spaces, different versions of condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120050/o12005087.png" /> are applied ([[#References|[a3]]], [[#References|[a6]]], [[#References|[a8]]]).
+
In the definition of Orlicz–Lorentz space one can replace Lebesgue-measurable functions by measurable functions with respect to a $\sigma$-finite measure space. All results stated above remain the same in the case of a non-atomic infinite measure. For other measure spaces, different versions of condition $\Delta _ { 2 }$ are applied ([[#References|[a3]]], [[#References|[a6]]], [[#References|[a8]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y. Raynaud,  "On Lorentz–Sharpley spaces" , ''Proc. Workshop on Interpolation Spaces and Related Topics'' , ''Israel Math. Conf. Proc.'' , '''5'''  (1992)  pp. 207–228</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Torchinsky,  "Interplation of operators and Orlicz classes"  ''Studia Math.'' , '''59'''  (1976)  pp. 177–207</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Cerda,  H. Hudzik,  A. Kamińska,  M. Mastyło,  "Geometric properties of symmetric spaces with applications to Orlicz–Lorentz spaces"  ''Positivity'' , '''2'''  (1998)  pp. 311–337</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Hudzik,  A. Kamińska,  M. Mastyło,  "Geometric properties of some Calderón–Lozanovskii and Orlicz–Lorentz spaces"  ''Houston J. Math.'' , '''22'''  (1996)  pp. 639–663</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Kamińska,  P.K. Lin,  H. Sun,  "Uniformly normal structure of Orlicz–Lorentz spaces" , ''Interaction Between Functional Analysis, Harmonic Analysis, and Probability (Columbia, Missouri, 1994)'' , ''Lecture Notes Pure Appl. Math.'' , '''175''' , M. Dekker  (1996)  pp. 229–238</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Kamińska,  "Some remarks on Orlicz–Lorentz spaces"  ''Math. Nachr.'' , '''147'''  (1990)  pp. 29–38</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Kamińska,  "Extreme points in Orlicz–Lorentz spaces"  ''Arch. Math.'' , '''55'''  (1990)  pp. 173–180</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Kamińska,  "Uniform convexity of generalized Lorentz spaces"  ''Arch. Math.'' , '''56'''  (1991)  pp. 181–188</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces I–II" , Springer  (1977–1979)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  St. Montgomery-Smith,  "Boyd indices of Orlicz–Lorentz spaces" , ''Function Spaces (Edwardsville, IL, 1994)'' , ''Lecture Notes Pure Appl. Math.'' , '''172''' , M. Dekker  (1995)  pp. 321–334</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  L. Maligranda,  "Indices and interpolation"  ''Dissert. Math.'' , '''234'''  (1985)</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  Y. Raynaud,  "On Lorentz–Sharpley spaces" , ''Proc. Workshop on Interpolation Spaces and Related Topics'' , ''Israel Math. Conf. Proc.'' , '''5'''  (1992)  pp. 207–228</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Torchinsky,  "Interplation of operators and Orlicz classes"  ''Studia Math.'' , '''59'''  (1976)  pp. 177–207</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Cerda,  H. Hudzik,  A. Kamińska,  M. Mastyło,  "Geometric properties of symmetric spaces with applications to Orlicz–Lorentz spaces"  ''Positivity'' , '''2'''  (1998)  pp. 311–337</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H. Hudzik,  A. Kamińska,  M. Mastyło,  "Geometric properties of some Calderón–Lozanovskii and Orlicz–Lorentz spaces"  ''Houston J. Math.'' , '''22'''  (1996)  pp. 639–663</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Kamińska,  P.K. Lin,  H. Sun,  "Uniformly normal structure of Orlicz–Lorentz spaces" , ''Interaction Between Functional Analysis, Harmonic Analysis, and Probability (Columbia, Missouri, 1994)'' , ''Lecture Notes Pure Appl. Math.'' , '''175''' , M. Dekker  (1996)  pp. 229–238</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Kamińska,  "Some remarks on Orlicz–Lorentz spaces"  ''Math. Nachr.'' , '''147'''  (1990)  pp. 29–38</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A. Kamińska,  "Extreme points in Orlicz–Lorentz spaces"  ''Arch. Math.'' , '''55'''  (1990)  pp. 173–180</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  A. Kamińska,  "Uniform convexity of generalized Lorentz spaces"  ''Arch. Math.'' , '''56'''  (1991)  pp. 181–188</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces I–II" , Springer  (1977–1979)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  St. Montgomery-Smith,  "Boyd indices of Orlicz–Lorentz spaces" , ''Function Spaces (Edwardsville, IL, 1994)'' , ''Lecture Notes Pure Appl. Math.'' , '''172''' , M. Dekker  (1995)  pp. 321–334</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  L. Maligranda,  "Indices and interpolation"  ''Dissert. Math.'' , '''234'''  (1985)</td></tr>
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</table>

Latest revision as of 19:10, 22 January 2024

Lorentz–Orlicz space

There are several definitions of Orlicz–Lorentz spaces known in the literature ([a6], [a11], [a10], [a2]). All of them are generalizations of both Orlicz and Lorentz spaces (cf. Orlicz space; for Lorentz space, see Marcinkiewicz space). The Orlicz–Lorentz space presented below arises naturally as an intermediate space between ordinary Lorentz space and the space of bounded functions in the Calderón–Lozanovskii method of interpolation ([a4]; cf. also Interpolation of operators).

Given a Young function $\varphi$, i.e. a convex function (cf. also Convex function (of a real variable)) $\varphi : \mathbf{R} _ { + } \rightarrow \mathbf{R} _ { + }$ such that $\varphi ( 0 ) = 0$, the Orlicz–Lorentz space $\Lambda _ { \varphi , w }$ ([a6], [a11]) is the collection of all real-valued Lebesgue-measurable functions $f$ on $\mathbf{R} _ { + }$ (cf. also Lebesgue integral) such that

\begin{equation*} I ( \lambda f ) : = \int _ { 0 } ^ { \infty } \varphi ( \lambda f ^ { * } ( s ) ) w ( s ) d s < \infty \end{equation*}

for some $\lambda > 0$, where the so-called weight function $w : \mathbf{R} _ { + } \rightarrow \mathbf{R} _ { + }$ is positive, non-increasing and such that

\begin{equation*} \int _ { 0 } ^ { \infty } w ( s ) d s = \infty, \end{equation*}

and for all $t > 0$,

\begin{equation*} S ( t ) : = \int _ { 0 } ^ { t } w ( s ) d s < \infty. \end{equation*}

Here, $f ^ { * }$ denotes the non-increasing rearrangement of $f$, that is,

\begin{equation*} f ^ { * } ( t ) = \operatorname { inf } \{ s > 0 : d_f ( s ) \leq t \} \end{equation*}

for $t \geq 0$, where $d_f ( t ) = m ( \{ s > 0 : | f ( s ) | > t \} )$ and $m$ is the Lebesgue measure on $\mathbf{R}$. The Orlicz–Lorentz space, equipped with the norm

\begin{equation*} \| f \| = \operatorname { inf } \{ \epsilon > 0 : I ( f / \epsilon ) \leq 1 \} \end{equation*}

is a Banach function lattice (cf. also Banach lattice; Banach space) with ordering: $f \leq g$ whenever $f ( t ) \leq g ( t )$ almost everywhere. If $w ( t ) \equiv 1$, then the Orlicz–Lorentz space becomes an Orlicz space, and if $\varphi ( u ) = u ^ { p }$, $1 \leq p < \infty$, then it becomes a Lorentz space [a9].

Many properties of $\Lambda _ { \varphi , w }$ have been described in terms of growth conditions imposed on $\varphi$ and $w$. The most common growth conditions are regularity of the weight and condition $\Delta _ { 2 }$ of a Young function. It is said that $w$ is regular if $\operatorname { inf } _ { t > 0 } S ( 2 t ) / S ( t ) > 1$ and a Young function $\varphi$ satisfies condition $\Delta _ { 2 }$ whenever $\varphi ( 2 u ) \leq K \varphi ( u )$ for all $u \geq 0$ and some $K > 0$. The methods applied in the theory of Orlicz–Lorentz spaces are derived from those for both Orlicz and Lorentz spaces.

Some results on isomorphic or isometric properties of $\Lambda _ { \varphi , w }$ are as follows.

1) Condition $\Delta _ { 2 }$ on $\varphi$ is equivalent to the following properties of $\Lambda _ { \varphi , w }$ [a6].

$\Lambda _ { \varphi , w }$ does not contain an isometric copy of $l ^ { \infty }$.

$\Lambda _ { \varphi , w }$ does not contain an isomorphic copy of $l ^ { \infty }$.

$\Lambda _ { \varphi , w }$ does not contain an isomorphic copy of $c_0$.

$\Lambda _ { \varphi , w }$ is separable (cf. also Separable space).

The norm in $\Lambda _ { \varphi , w }$ is absolutely continuous (cf. also Absolute continuity).

2) The function $\psi ( v ) = \operatorname { sup } _ { u > 0 } \{ u v - \varphi ( u ) \}$ on $\mathbf{R} _ { + }$ is called the Young conjugate to $\varphi$ (cf. also Conjugate function). The functions $\varphi$ and $\psi$ are conjugate to each other and this duality is analogous to the duality between power functions $u ^ { p }$ and $u ^ { q }$ with $1 < p , q < \infty$ and $1 / p + 1 / q = 1$. The dual space $\Lambda _ { \varphi , w } ^ { * }$ can be described in terms of $\psi$. In fact, if both $\varphi$ and $\psi$ satisfy condition $\Delta _ { 2 }$ and $w$ is regular, then $\Lambda _ { \varphi , w } ^ { * }$ is the family of all Lebesgue-measurable functions $f$ such that

\begin{equation*} \int _ { 0 } ^ { \infty } \psi ( f ^ { * } ( s ) / w ( s ) ) w ( s ) d s < \infty, \end{equation*}

and the dual norm is equivalent to the quasi-norm $\operatorname { inf } \{ \lambda > 0 : \int \psi ( f ^ { * } / \lambda w ) w < \infty \}$. The space $\Lambda _ { \varphi , w }$ is reflexive (cf. also Reflexive space) if and only if both $\varphi$ and $\psi$ satisfy condition $\Delta _ { 2 }$. Superreflexivity (cf. also Reflexive space) requires, in addition, the assumption of regularity of $w$ [a4].

3) A number of geometric properties, like uniform convexity, rotundity, extreme points, local uniform convexity, and normal and uniform normal structure in Orlicz–Lorentz spaces have been characterized in terms of $\varphi$ and $w$ ([a3], [a4], [a6], [a7], [a8], [a5]). For instance, necessary and sufficient conditions for $\Lambda _ { \varphi , w }$ to be uniformly convex are that both $\varphi$ and $\varphi ^ { * }$ satisfy $\Delta _ { 2 }$, that $w$ is regular and that $\varphi$ is uniformly convex, i.e. $\operatorname { sup } _ { u > 0 } \varphi ^ { \prime } ( a u ) / \varphi ^ { \prime } ( u ) < 1$ for every $0 < a < 1$, where $\varphi ^ { \prime }$ is the right derivative of $\varphi$ (cf. also Differentiation).

Order convexity and concavity as well as Boyd indices in Orlicz–Lorentz spaces have been studied in, e.g., [a10], [a1].

In the definition of Orlicz–Lorentz space one can replace Lebesgue-measurable functions by measurable functions with respect to a $\sigma$-finite measure space. All results stated above remain the same in the case of a non-atomic infinite measure. For other measure spaces, different versions of condition $\Delta _ { 2 }$ are applied ([a3], [a6], [a8]).

References

[a1] Y. Raynaud, "On Lorentz–Sharpley spaces" , Proc. Workshop on Interpolation Spaces and Related Topics , Israel Math. Conf. Proc. , 5 (1992) pp. 207–228
[a2] A. Torchinsky, "Interplation of operators and Orlicz classes" Studia Math. , 59 (1976) pp. 177–207
[a3] J. Cerda, H. Hudzik, A. Kamińska, M. Mastyło, "Geometric properties of symmetric spaces with applications to Orlicz–Lorentz spaces" Positivity , 2 (1998) pp. 311–337
[a4] H. Hudzik, A. Kamińska, M. Mastyło, "Geometric properties of some Calderón–Lozanovskii and Orlicz–Lorentz spaces" Houston J. Math. , 22 (1996) pp. 639–663
[a5] A. Kamińska, P.K. Lin, H. Sun, "Uniformly normal structure of Orlicz–Lorentz spaces" , Interaction Between Functional Analysis, Harmonic Analysis, and Probability (Columbia, Missouri, 1994) , Lecture Notes Pure Appl. Math. , 175 , M. Dekker (1996) pp. 229–238
[a6] A. Kamińska, "Some remarks on Orlicz–Lorentz spaces" Math. Nachr. , 147 (1990) pp. 29–38
[a7] A. Kamińska, "Extreme points in Orlicz–Lorentz spaces" Arch. Math. , 55 (1990) pp. 173–180
[a8] A. Kamińska, "Uniform convexity of generalized Lorentz spaces" Arch. Math. , 56 (1991) pp. 181–188
[a9] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces I–II" , Springer (1977–1979)
[a10] St. Montgomery-Smith, "Boyd indices of Orlicz–Lorentz spaces" , Function Spaces (Edwardsville, IL, 1994) , Lecture Notes Pure Appl. Math. , 172 , M. Dekker (1995) pp. 321–334
[a11] L. Maligranda, "Indices and interpolation" Dissert. Math. , 234 (1985)
How to Cite This Entry:
Orlicz-Lorentz space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orlicz-Lorentz_space&oldid=16855
This article was adapted from an original article by Anna Kamińska (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article