Orlicz-Lorentz space

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)