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.

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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.

NotesEdit

  1. ^ G. Lorentz, "Some new function spaces", Annals of Mathematics 51 (1950), pp. 37-55.
  2. ^ G. Lorentz, "On the theory of spaces Λ", Pacific Journal of Mathematics 1 (1951), pp. 411-429.