In mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, Lorentz spaces, introduced by George Lorentz in the 1950s, are generalisations of the more familiar L^p spaces

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

.

The Lorentz spaces are denoted by L^p,q. Like the L^p spaces, they are characterized by a norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

 (technically a quasinorm) that encodes information about the "size" of a function, just as the L^p 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 L^p norms, by exponentially rescaling the measure in both the range (the p) and the domain (the q). The Lorentz norms, like the L^p norms, are invariant under arbitrary rearrangements of the values of a function.

Definition

The Lorentz space on a measure space (X,μ) is the space of complex-valued measurable function

Measurable function

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

s ƒ on X such that the following quasinorm is finite


where 0 < p < ∞ and 0 < q ≤ ∞. Thus, when q < ∞,


and when q = ∞,


It is also conventional to set L^∞,∞(X,μ) = L^∞(X,μ).

Decreasing rearrangements

The quasinorm is invariant under rearranging the values of the function ƒ, essentially by definition. In particular, given a complex-valued measurable function

Measurable function

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

 ƒ defined on a measure space, (X, μ), its decreasing rearrangement function, can be defined as
where d_ƒ is the so-called distribution function of ƒ, given by
Here, for notational convenience, is defined to be ∞.

Given these definitions, for p, q ∈ (0, ∞), the Lorentz norms are given by

Properties

The Lorentz spaces are genuinely generalisations of the L^p spaces in the sense that for any p, L^p,p = L^p, which follows from Cavalieri's principle

Cavalieri's principle

In geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...

. Further, L^p,∞ coincides with weak L^p. They are quasi-Banach spaces (that is quasi-normed spaces which are also complete) and are normable for p ∈ (1, ∞), q ∈ [1, ∞]. L^p,q is a subspace of L^p,r whenever q < r.