# Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,^{[1]}^{[2]} are generalisations of the more familiar spaces.

The Lorentz spaces are denoted by . Like the spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the norm does. The two basic qualitative notions of "size" of a function are: how tall is the graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the norms, by exponentially rescaling the measure in both the range () and the domain (). The Lorentz norms, like the norms, are invariant under arbitrary rearrangements of the values of a function.

## DefinitionEdit

The Lorentz space on a measure space is the space of complex-valued measurable functions on *X* such that the following quasinorm is finite

where and . Thus, when ,

and, when ,

It is also conventional to set .

## Decreasing rearrangementsEdit

The quasinorm is invariant under rearranging the values of the function , essentially by definition. In particular, given a complex-valued measurable function defined on a measure space, , its **decreasing rearrangement** function, can be defined as

where is the so-called **distribution function** of , given by

Here, for notational convenience, is defined to be .

The two functions and are **equimeasurable**, meaning that

where is the Lebesgue measure on the real line. The related symmetric decreasing rearrangement function, which is also equimeasurable with , would be defined on the real line by

Given these definitions, for and , the Lorentz quasinorms are given by

## Lorentz sequence spacesEdit

When (the counting measure on ), the resulting Lorentz space is a sequence space. However, in this case it is convenient to use different notation.

### Definition.Edit

for (or in the complex case), let denote the p-norm for and the ∞-norm. Denote by the Banach space of all sequences with finite p-norm. Let the Banach space of all sequences satisfying , endowed with the ∞-norm. Denote by the normed space of all sequences with only finitely many nonzero entries. These spaces all play a role in the definition of the Lorentz sequence spaces below.

Let be a sequence of positive real numbers satisfying , and define the norm . The *Lorentz sequence space* is defined as the Banach space of all sequences where this norm is finite. Equivalently, we can define as the completion of under .

## PropertiesEdit

The Lorentz spaces are genuinely generalisations of the spaces in the sense that, for any , , which follows from Cavalieri's principle. Further, coincides with weak . They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for and . When , is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of , the weak space. As a concrete example that the triangle inequality fails in , consider

whose quasi-norm equals one, whereas the quasi-norm of their sum equals four.

The space is contained in whenever . The Lorentz spaces are real interpolation spaces between and .

## See alsoEdit

## ReferencesEdit

- Grafakos, Loukas (2008),
*Classical Fourier analysis*, Graduate Texts in Mathematics,**249**(2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09432-8, ISBN 978-0-387-09431-1, MR 2445437.