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The [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m0623301.png" /> of all (classes of) functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m0623302.png" /> measurable on the half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m0623303.png" /> and having a finite norm
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<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/m/m062/m062330/m0623304.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m0623305.png" /> is a rearrangement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m0623306.png" />, that is, the non-increasing left-continuous function equimeasurable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m0623307.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m0623308.png" /> is a positive non-decreasing function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m0623309.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233010.png" /> does not increase (in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233011.png" /> is a non-decreasing concave function). The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233012.png" /> was introduced by J. Marcinkiewicz [[#References|[1]]].
+
The [[Banach space|Banach space]]  $  M _  \psi  $
 +
of all (classes of) functions  $  x $
 +
measurable on the half-line  $  ( 0 , \infty ) $
 +
and having a finite norm
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233013.png" /> is bounded from below and from above by positive constants, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233014.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233015.png" />. In all other cases it is not separable. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233016.png" /> is an interpolating space (see [[Interpolation of operators|Interpolation of operators]]) between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233018.png" /> with interpolation constant 1.
+
$$ \tag{1 }
 +
\| x \| _ {M _  \psi  }  = \
 +
\sup _ {0 < h < \infty } \
  
On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233019.png" /> the functional
+
\frac{1}{\psi ( h) }
  
<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/m/m062/m062330/m06233020.png" /></td> </tr></table>
+
\int\limits _ { 0 } ^ { h }  x  ^ {*} ( s)  d s ,
 +
$$
  
is defined; its norm does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233021.png" />. The functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233022.png" /> does not have the properties of a norm; it is equivalent to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233023.png" /> if and only if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233024.png" />,
+
where  $  x  ^ {*} ( s) $
 +
is a rearrangement of $  x ( t) $,
 +
that is, the non-increasing left-continuous function equimeasurable with  $  | x ( t) | $,
 +
and  $  \psi ( t) $
 +
is a positive non-decreasing function on  $  ( 0 , \infty ) $
 +
for which  $  \psi ( t) / t $
 +
does not increase (in particular,  $  \psi ( t) $
 +
is a non-decreasing concave function). The space  $  M _  \psi  $
 +
was introduced by J. Marcinkiewicz [[#References|[1]]].
  
<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/m/m062/m062330/m06233025.png" /></td> </tr></table>
+
If  $  \psi ( t) $
 +
is bounded from below and from above by positive constants, then  $  M _  \psi  $
 +
is isomorphic to  $  L _ {1} $.  
 +
In all other cases it is not separable. The space  $  M _  \psi  $
 +
is an interpolating space (see [[Interpolation of operators|Interpolation of operators]]) between  $  L _ {1} $
 +
and  $  L _  \infty  $
 +
with interpolation constant 1.
  
(In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233027.png" />.)
+
On  $  M _  \psi  $
 +
the functional
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233028.png" /> first arose in the interpolation theorem of Marcinkiewicz (with the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233029.png" />) and is connected with [[Interpolation of operators|interpolation of operators]] of weak type. It has the following extremal property: It is the broadest among the symmetric spaces the fundamental function of which coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233030.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233032.png" /> is the characteristic function of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233033.png" />. If
+
$$
 +
F ( x) = \sup _ {0 < t < \infty } \
  
<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/m/m062/m062330/m06233034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{t}{\psi ( t) }
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233035.png" /> is isomorphic (isometric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233036.png" /> is concave) to the dual space of the Lorentz space with the norm
+
x  ^ {*} ( t)
 +
$$
  
<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/m/m062/m062330/m06233037.png" /></td> </tr></table>
+
is defined; its norm does not exceed  $  \| x \| _ {M _  \psi  } $.
 +
The functional  $  F ( x) $
 +
does not have the properties of a norm; it is equivalent to the norm  $  \| x \| _ {M _  \psi  } $
 +
if and only if for  $  s > 1 $,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233038.png" /> is the least concave majorant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233039.png" />. Under condition (2) there is a distinguished subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233041.png" />, consisting of all functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233042.png" /> for which
+
$$
 +
\inf _
 +
{0 < t < \infty } \
  
<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/m/m062/m062330/m06233043.png" /></td> </tr></table>
+
\frac{\psi ( s t ) }{\psi ( t) }
 +
  > 1 .
 +
$$
  
If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233045.png" /> is the closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233046.png" /> of the set of all bounded functions of compact support. In this case the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233047.png" /> is isomorphic to the Lorentz space and, consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233048.png" /> is isomorphic to the second dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233049.png" />.
+
(In particular, for  $  \psi ( t) = t  ^  \alpha  $
 +
if  $  0 \leq  \alpha \leq  1 $.)
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233050.png" /> is a space with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233051.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233052.png" /> defined on its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233053.png" />-algebra of measurable sets, then for each measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233054.png" /> its rearrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233056.png" />, is defined, which makes it possible to introduce the Marcinkiewicz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233057.png" /> with the norm (1).
+
The space  $  M _  \psi  $
 +
first arose in the interpolation theorem of Marcinkiewicz (with the functional  $  F ( x) $)
 +
and is connected with [[Interpolation of operators|interpolation of operators]] of weak type. It has the following extremal property: It is the broadest among the symmetric spaces the fundamental function of which coincides with  $  h / \psi ( h) $,
 +
that is,  $  \| \chi _ {( 0 , h ) }  \| _ {E} = h / \psi ( h) $,
 +
where  $  \chi _ {( 0 , h ) }  $
 +
is the characteristic function of the interval  $  ( 0 , h ) $.
 +
If
 +
 
 +
$$ \tag{2 }
 +
\psi ( + 0 )  =  0 ,\ \
 +
\psi ( \infty )  = \infty ,
 +
$$
 +
 
 +
then  $  M _  \psi  $
 +
is isomorphic (isometric if  $  \psi $
 +
is concave) to the dual space of the Lorentz space with the norm
 +
 
 +
$$
 +
\| y \| _ {\Lambda _  \psi  }  = \
 +
\int\limits _ { 0 } ^  \infty  y  ^ {*} ( s)  d \widetilde \psi  ( s) ,
 +
$$
 +
 
 +
where  $  \widetilde \psi  ( s) $
 +
is the least concave majorant of  $  \psi ( s) $.
 +
Under condition (2) there is a distinguished subspace  $  M _  \psi  ^ {0} $
 +
in  $  M _  \psi  $,
 +
consisting of all functions from  $  M _  \psi  $
 +
for which
 +
 
 +
$$
 +
\lim\limits _ {h \rightarrow 0 , \infty } \
 +
 
 +
\frac{1}{\psi ( h) }
 +
 
 +
\int\limits _ { 0 } ^ { h }  x  ^ {*} ( t)  d t  = 0 .
 +
$$
 +
 
 +
If, in addition,  $  \lim\limits _ {t \rightarrow 0 }  \psi ( t) / t = \infty $,
 +
then  $  M _  \psi  ^ {0} $
 +
is the closure in  $  M _  \psi  $
 +
of the set of all bounded functions of compact support. In this case the dual of  $  M _  \psi  ^ {0} $
 +
is isomorphic to the Lorentz space and, consequently,  $  M _  \psi  $
 +
is isomorphic to the second dual space of  $  M _  \psi  ^ {0} $.
 +
 
 +
If  $  \mathfrak M $
 +
is a space with a $  \sigma $-
 +
finite measure $  \mu $
 +
defined on its $  \sigma $-
 +
algebra of measurable sets, then for each measurable function $  x ( m) $
 +
its rearrangement $  x  ^ {*} ( s) $,
 +
$  0 < s < \infty $,  
 +
is defined, which makes it possible to introduce the Marcinkiewicz space $  M _  \psi  ( \mathfrak M , \mu ) $
 +
with the norm (1).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Marcinkiewicz,  "Sur l'interpolation d'opérations"  ''C.R. Acad. Sci. Paris'' , '''208'''  (1939)  pp. 1272–1273</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Marcinkiewicz,  "Sur l'interpolation d'opérations"  ''C.R. Acad. Sci. Paris'' , '''208'''  (1939)  pp. 1272–1273</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233058.png" /> be a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233059.png" />. The left-continuous decreasing rearrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233060.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233061.png" /> is defined by the properties:
+
Let $  f $
 +
be a continuous function on $  [ 0 , 1 ] $.  
 +
The left-continuous decreasing rearrangement $  f ^ { * } $
 +
of $  f $
 +
is defined by the properties:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233062.png" /> is decreasing (i.e. non-increasing);
+
i) $  f ^ { * } $
 +
is decreasing (i.e. non-increasing);
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233063.png" /> is left-continuous;
+
ii) $  f ^ { * } $
 +
is left-continuous;
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233065.png" /> have the same measure for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233066.png" />.
+
iii) $  \{ {x } : {f ( x) > s } \} $
 +
and $  \{ {x } : {f ^ { * } ( x) > s } \} $
 +
have the same measure for all $  s $.
  
Alternatively one considers left-continuous or right-continuous decreasing (or increasing) rearrangements. The right-continuous decreasing rearrangement can be described as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233067.png" /> be the measure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233068.png" />. Then
+
Alternatively one considers left-continuous or right-continuous decreasing (or increasing) rearrangements. The right-continuous decreasing rearrangement can be described as follows. Let m ( y) $
 +
be the measure of the set $  \{ {u } : {f ( u) > y } \} $.  
 +
Then
  
<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/m/m062/m062330/m06233069.png" /></td> </tr></table>
+
$$
 +
f ^ { * } ( x)  = \sup  \{ {y } : {m ( y)  > x } \}
 +
.
 +
$$
  
 
The notion is a continuous analogue of putting a finite sequence of real numbers in decreasing (or increasing) order. This last construction is of importance in the context of the [[Majorization ordering|majorization ordering]] and there are in fact various continuous analogues of results connected with that ordering, such as the Muirhead inequalities and the result linking the majorization ordering and doubly-stochastic matrices, cf. [[#References|[a1]]]–[[#References|[a3]]].
 
The notion is a continuous analogue of putting a finite sequence of real numbers in decreasing (or increasing) order. This last construction is of importance in the context of the [[Majorization ordering|majorization ordering]] and there are in fact various continuous analogues of results connected with that ordering, such as the Muirhead inequalities and the result linking the majorization ordering and doubly-stochastic matrices, cf. [[#References|[a1]]]–[[#References|[a3]]].
  
The Lorentz space defined by means of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233070.png" /> as above is the space of all measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233071.png" /> such that
+
The Lorentz space defined by means of a function $  \psi $
 +
as above is the space of all 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/m/m062/m062330/m06233072.png" /></td> </tr></table>
+
$$
 +
\| f \| _  \psi  = \int\limits _ { 0 } ^  \infty  f ^ { * } ( t) \
 +
d \widetilde \psi  ( t)  < \infty ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233073.png" /> is the decreasing rearrangement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233075.png" /> is the least concave majorant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233076.png" />. More generally one also considers Lorentz spaces based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233077.png" /> norms (instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233078.png" /> like above).
+
where $  f ^ { * } $
 +
is the decreasing rearrangement of $  | f | $
 +
and $  \widetilde \psi  $
 +
is the least concave majorant of $  \psi $.  
 +
More generally one also considers Lorentz spaces based on $  L _ {p} $
 +
norms (instead of $  L _ {1} $
 +
like above).
  
The analogous Lorentz sequence spaces are defined as follows. For every non-increasing sequence of positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233079.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233080.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233081.png" /> be the space of all sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233082.png" /> of scalars for which
+
The analogous Lorentz sequence spaces are defined as follows. For every non-increasing sequence of positive numbers $  w = ( w _ {n} ) _ {n=} 1  ^  \infty  $
 +
and every $  p \geq  1 $,  
 +
let $  d ( w , p ) $
 +
be the space of all sequences $  x = ( a _ {i} ) $
 +
of scalars for which
  
<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/m/m062/m062330/m06233083.png" /></td> </tr></table>
+
$$
 +
\| x \| _ {(} w,p)  = \left (
 +
\sum _ { n= } 1 ^  \infty  | a _ {\pi ( n) }
 +
|  ^ {p} w _ {n} \right )  ^ {1/p}  < \infty ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233084.png" /> is a permutation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233085.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233086.png" /> is a non-increasing sequence. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233087.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233088.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233089.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233090.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062330/m06233091.png" />. These two  "trivial"  cases are sometimes excluded. For a great deal of material on Lorentz sequence spaces see [[#References|[a4]]].
+
where $  \pi $
 +
is a permutation of $  \{ 1 , 2 ,\dots \} $
 +
such that $  ( | a _ {\pi ( n) }  | ) _ {n=} 1  ^  \infty  $
 +
is a non-increasing sequence. If $  \inf  w _ {n} > 0 $,  
 +
then $  d ( w , p ) $
 +
is isomorphic to $  l _ {p} $,  
 +
and if $  \sum _ {n=} 1  ^  \infty  w _ {n} < \infty $,  
 +
then $  d ( w , p ) \simeq l _  \infty  $.  
 +
These two  "trivial"  cases are sometimes excluded. For a great deal of material on Lorentz sequence spaces see [[#References|[a4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.W. Marshall,  J. Olkin,  "Inequalities: theory of majorization and its applications" , Acad. Press  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.S. Bullen,  D.S. Mitrinović,  P.M. Vasić,  "Means and their inequalities" , Reidel  (1988)  pp. 27ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces" , '''1. Sequence spaces''' , Springer  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.W. Marshall,  J. Olkin,  "Inequalities: theory of majorization and its applications" , Acad. Press  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.S. Bullen,  D.S. Mitrinović,  P.M. Vasić,  "Means and their inequalities" , Reidel  (1988)  pp. 27ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces" , '''1. Sequence spaces''' , Springer  (1977)</TD></TR></table>

Revision as of 07:59, 6 June 2020


The Banach space $ M _ \psi $ of all (classes of) functions $ x $ measurable on the half-line $ ( 0 , \infty ) $ and having a finite norm

$$ \tag{1 } \| x \| _ {M _ \psi } = \ \sup _ {0 < h < \infty } \ \frac{1}{\psi ( h) } \int\limits _ { 0 } ^ { h } x ^ {*} ( s) d s , $$

where $ x ^ {*} ( s) $ is a rearrangement of $ x ( t) $, that is, the non-increasing left-continuous function equimeasurable with $ | x ( t) | $, and $ \psi ( t) $ is a positive non-decreasing function on $ ( 0 , \infty ) $ for which $ \psi ( t) / t $ does not increase (in particular, $ \psi ( t) $ is a non-decreasing concave function). The space $ M _ \psi $ was introduced by J. Marcinkiewicz [1].

If $ \psi ( t) $ is bounded from below and from above by positive constants, then $ M _ \psi $ is isomorphic to $ L _ {1} $. In all other cases it is not separable. The space $ M _ \psi $ is an interpolating space (see Interpolation of operators) between $ L _ {1} $ and $ L _ \infty $ with interpolation constant 1.

On $ M _ \psi $ the functional

$$ F ( x) = \sup _ {0 < t < \infty } \ \frac{t}{\psi ( t) } x ^ {*} ( t) $$

is defined; its norm does not exceed $ \| x \| _ {M _ \psi } $. The functional $ F ( x) $ does not have the properties of a norm; it is equivalent to the norm $ \| x \| _ {M _ \psi } $ if and only if for $ s > 1 $,

$$ \inf _ {0 < t < \infty } \ \frac{\psi ( s t ) }{\psi ( t) } > 1 . $$

(In particular, for $ \psi ( t) = t ^ \alpha $ if $ 0 \leq \alpha \leq 1 $.)

The space $ M _ \psi $ first arose in the interpolation theorem of Marcinkiewicz (with the functional $ F ( x) $) and is connected with interpolation of operators of weak type. It has the following extremal property: It is the broadest among the symmetric spaces the fundamental function of which coincides with $ h / \psi ( h) $, that is, $ \| \chi _ {( 0 , h ) } \| _ {E} = h / \psi ( h) $, where $ \chi _ {( 0 , h ) } $ is the characteristic function of the interval $ ( 0 , h ) $. If

$$ \tag{2 } \psi ( + 0 ) = 0 ,\ \ \psi ( \infty ) = \infty , $$

then $ M _ \psi $ is isomorphic (isometric if $ \psi $ is concave) to the dual space of the Lorentz space with the norm

$$ \| y \| _ {\Lambda _ \psi } = \ \int\limits _ { 0 } ^ \infty y ^ {*} ( s) d \widetilde \psi ( s) , $$

where $ \widetilde \psi ( s) $ is the least concave majorant of $ \psi ( s) $. Under condition (2) there is a distinguished subspace $ M _ \psi ^ {0} $ in $ M _ \psi $, consisting of all functions from $ M _ \psi $ for which

$$ \lim\limits _ {h \rightarrow 0 , \infty } \ \frac{1}{\psi ( h) } \int\limits _ { 0 } ^ { h } x ^ {*} ( t) d t = 0 . $$

If, in addition, $ \lim\limits _ {t \rightarrow 0 } \psi ( t) / t = \infty $, then $ M _ \psi ^ {0} $ is the closure in $ M _ \psi $ of the set of all bounded functions of compact support. In this case the dual of $ M _ \psi ^ {0} $ is isomorphic to the Lorentz space and, consequently, $ M _ \psi $ is isomorphic to the second dual space of $ M _ \psi ^ {0} $.

If $ \mathfrak M $ is a space with a $ \sigma $- finite measure $ \mu $ defined on its $ \sigma $- algebra of measurable sets, then for each measurable function $ x ( m) $ its rearrangement $ x ^ {*} ( s) $, $ 0 < s < \infty $, is defined, which makes it possible to introduce the Marcinkiewicz space $ M _ \psi ( \mathfrak M , \mu ) $ with the norm (1).

References

[1] J. Marcinkiewicz, "Sur l'interpolation d'opérations" C.R. Acad. Sci. Paris , 208 (1939) pp. 1272–1273
[2] S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)
[3] E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)

Comments

Let $ f $ be a continuous function on $ [ 0 , 1 ] $. The left-continuous decreasing rearrangement $ f ^ { * } $ of $ f $ is defined by the properties:

i) $ f ^ { * } $ is decreasing (i.e. non-increasing);

ii) $ f ^ { * } $ is left-continuous;

iii) $ \{ {x } : {f ( x) > s } \} $ and $ \{ {x } : {f ^ { * } ( x) > s } \} $ have the same measure for all $ s $.

Alternatively one considers left-continuous or right-continuous decreasing (or increasing) rearrangements. The right-continuous decreasing rearrangement can be described as follows. Let $ m ( y) $ be the measure of the set $ \{ {u } : {f ( u) > y } \} $. Then

$$ f ^ { * } ( x) = \sup \{ {y } : {m ( y) > x } \} . $$

The notion is a continuous analogue of putting a finite sequence of real numbers in decreasing (or increasing) order. This last construction is of importance in the context of the majorization ordering and there are in fact various continuous analogues of results connected with that ordering, such as the Muirhead inequalities and the result linking the majorization ordering and doubly-stochastic matrices, cf. [a1][a3].

The Lorentz space defined by means of a function $ \psi $ as above is the space of all measurable functions $ f $ such that

$$ \| f \| _ \psi = \int\limits _ { 0 } ^ \infty f ^ { * } ( t) \ d \widetilde \psi ( t) < \infty , $$

where $ f ^ { * } $ is the decreasing rearrangement of $ | f | $ and $ \widetilde \psi $ is the least concave majorant of $ \psi $. More generally one also considers Lorentz spaces based on $ L _ {p} $ norms (instead of $ L _ {1} $ like above).

The analogous Lorentz sequence spaces are defined as follows. For every non-increasing sequence of positive numbers $ w = ( w _ {n} ) _ {n=} 1 ^ \infty $ and every $ p \geq 1 $, let $ d ( w , p ) $ be the space of all sequences $ x = ( a _ {i} ) $ of scalars for which

$$ \| x \| _ {(} w,p) = \left ( \sum _ { n= } 1 ^ \infty | a _ {\pi ( n) } | ^ {p} w _ {n} \right ) ^ {1/p} < \infty , $$

where $ \pi $ is a permutation of $ \{ 1 , 2 ,\dots \} $ such that $ ( | a _ {\pi ( n) } | ) _ {n=} 1 ^ \infty $ is a non-increasing sequence. If $ \inf w _ {n} > 0 $, then $ d ( w , p ) $ is isomorphic to $ l _ {p} $, and if $ \sum _ {n=} 1 ^ \infty w _ {n} < \infty $, then $ d ( w , p ) \simeq l _ \infty $. These two "trivial" cases are sometimes excluded. For a great deal of material on Lorentz sequence spaces see [a4].

References

[a1] A.W. Marshall, J. Olkin, "Inequalities: theory of majorization and its applications" , Acad. Press (1979)
[a2] P.S. Bullen, D.S. Mitrinović, P.M. Vasić, "Means and their inequalities" , Reidel (1988) pp. 27ff
[a3] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1952)
[a4] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1. Sequence spaces , Springer (1977)
How to Cite This Entry:
Marcinkiewicz space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Marcinkiewicz_space&oldid=47761
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article