# Lorentz space

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

The Lorentz spaces are denoted by *L*^{p,q}. Like the *L ^{p}* spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the

*L*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

^{p}*L*norms, by exponentially rescaling the measure in both the range (Template:Mvar) and the domain (Template:Mvar). The Lorentz norms, like the

^{p}*L*norms, are invariant under arbitrary rearrangements of the values of a function.

^{p}## Definition

The Lorentz space on a measure space (*X*, *μ*) is the space of complex-valued measurable functions *f* 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 *f*, essentially by definition. In particular, given a complex-valued measurable function *f* defined on a measure space, (*X*, *μ*), its **decreasing rearrangement** function, can be defined as

where *d _{f}* is the so-called distribution function of

*f*, given by

Here, for notational convenience, inf ∅ is defined to be ∞.

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

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

Given these definitions, for *p*, *q* ∈ (0, ∞) or *q* = ∞, the Lorentz quasinorms are given by

## Properties

The Lorentz spaces are genuinely generalisations of the *L ^{p}* spaces in the sense that for any Template:Mvar,

*L*

^{p,p}=

*L*, which follows from Cavalieri's principle. Further,

^{p}*L*

^{p,∞}coincides with weak

*L*. They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for

^{p}*p*∈ (1, ∞),

*q*∈ [1, ∞]. When

*p*= 1,

*L*

^{1,1}=

*L*

^{1}is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of

*L*

^{1,∞}, the weak

*L*

^{1}space. As a concrete example that the triangle inequality fails in

*L*

^{1,∞}, consider

whose *L*^{1,∞} quasi-norm equals one, whereas the quasi-norm of their sum *f* + *g* equals four.

The space *L*^{p,q} is contained in *L*^{p,r} whenever *q* < *r*. The Lorentz spaces are real interpolation spaces between *L*^{1} and *L*^{∞}.

## See also

## References

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