Lorentz space

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The norm of is the area of the largest rectangle with sides parallel to the coordinate axes that can be inscribed in the graph.

In mathematical analysis, Lorentz spaces, introduced by George 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 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.

Definition[edit]

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 rearrangements[edit]

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 spaces[edit]

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 .

Properties[edit]

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 also[edit]

References[edit]

Notes[edit]

  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.