Marcinkiewicz space

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)