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''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]]).

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

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

for some $\lambda &gt; 0$, 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 $t &gt; 0$,

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

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

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

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

\begin{equation*} \| f \| = \operatorname { inf } \{ \epsilon &gt; 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: $f \leq g$ whenever $f ( t ) \leq g ( t )$ almost everywhere. If $w ( t ) \equiv 1$, then the Orlicz–Lorentz space becomes an Orlicz space, and if $\varphi ( u ) = u ^ { p }$, $1 \leq p &lt; \infty$, then it becomes a Lorentz space [[#References|[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 $\operatorname { inf } _ { t &gt; 0 } S ( 2 t ) / S ( t ) &gt; 1$ and a Young function $\varphi$ satisfies condition $\Delta _ { 2 }$ whenever $\varphi ( 2 u ) \leq K \varphi ( u )$ for all $u \geq 0$ and some $K &gt; 0$. 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 $\Lambda _ { \varphi , w }$ [[#References|[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|Separable space]]).

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

2) The function $\psi ( v ) = \operatorname { sup } _ { u &gt; 0 } \{ u v - \varphi ( u ) \}$ 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 $1 &lt; p , q &lt; \infty$ 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 &lt; \infty, \end{equation*}

and the dual norm is equivalent to the quasi-norm $\operatorname { inf } \{ \lambda &gt; 0 : \int \psi ( f ^ { * } / \lambda w ) w &lt; \infty \}$. 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 $\varphi$ 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. $\operatorname { sup } _ { u &gt; 0 } \varphi ^ { \prime } ( a u ) / \varphi ^ { \prime } ( u ) &lt; 1$ for every $0 &lt; a &lt; 1$, where $\varphi ^ { \prime }$ is the right derivative of $\varphi$ (cf. also [[Differentiation|Differentiation]]).

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 $\sigma$-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====
<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>