# 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 .

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).

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