Lorentz space

Short description: Function space

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

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

## Contents

* 1 Definition
* 2 Decreasing rearrangements
* 3 Lorentz sequence spaces
* 3.1 Definition.
* 4 Properties
* 4.1 Hölder's inequality
* 4.2 Dual space
* 4.3 Atomic decomposition
* 5 See also
* 6 References
* 7 Notes

## Definition

The Lorentz space on a measure space (X,μ) is the space of complex-valued measurable functions f:X->C‾ such that the following quasinorm is finite

‖f‖Lp,q(X,μ)=p1q‖tμ{|f|≥t}1p‖Lq(𝐑+,dtt)

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

‖f‖Lp,q(X,μ)=p1q(∫0∞tqμ{x:|f(x)|≥t}qpdtt)1q=(∫0∞(τμ{x:|f(x)|p≥τ})qpdττ)1q.

and, when q=∞,

‖f‖Lp,∞(X,μ)p=supt>0(tpμ{x:|f(x)|>t}).

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, f∗:[0,∞)->[0,∞] can be defined as

f∗(t)=inf⁡{α∈𝐑+:df(α)≤t}

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

df(α)=μ({x∈X:|f(x)|>α}).

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

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

μ({x∈X:|f(x)|>α})=λ({t>0:f∗(t)>α}),α>0,

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

𝐑∋t↦12f∗(|t|).

Given these definitions, for 0<p<∞ and 0<q≤∞, the Lorentz quasinorms are given by

‖f‖Lp,q={(∫0∞(t1pf∗(t))qdtt)1qq∈(0,∞),sup⁡t>0t1pf∗(t)q=∞.

## Lorentz sequence spaces

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

### Definition.

For (an)n=1∞∈ℝℕ (or ℂℕ in the complex case), let ‖(an)n=1∞‖p=(∑n=1∞|an|p)1/p denote the p-norm for 1≤p<∞ and ‖(an)n=1∞‖∞=supn∈ℕ|an| the ∞-norm. Denote by ℓp the Banach space of all sequences with finite p-norm. Let c0 the Banach space of all sequences satisfying limn->∞an=0, endowed with the ∞-norm. Denote by c00 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 d(w,p) below.

Let w=(wn)n=1∞∈c0∖ℓ1 be a sequence of positive real numbers satisfying 1=w1≥w2≥w3≥⋯, and define the norm ‖(an)n=1∞‖d(w,p)=supσ∈Π‖(aσ(n)wn1/p)n=1∞‖p. The Lorentz sequence space d(w,p) is defined as the Banach space of all sequences where this norm is finite. Equivalently, we can define d(w,p) as the completion of c00 under ‖⋅‖d(w,p).

## Properties

The Lorentz spaces are genuinely generalisations of the Lp spaces in the sense that, for any p, Lp,p=Lp, which follows from Cavalieri's principle. Further, Lp,∞ coincides with weak Lp. They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for 1<p<∞ and 1≤q≤∞. When p=1, L1,1=L1 is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of L1,∞, the weak L1 space. As a concrete example that the triangle inequality fails in L1,∞, consider

f(x)=1xχ(0,1)(x)andg(x)=11−xχ(0,1)(x),

whose L1,∞ quasi-norm equals one, whereas the quasi-norm of their sum f+g equals four.

The space Lp,q is contained in Lp,r whenever q<r. The Lorentz spaces are real interpolation spaces between L1 and L∞.

### Hölder's inequality

‖fg‖Lp,q≤Ap1,p2,q1,q2‖f‖Lp1,q1‖g‖Lp2,q2 where 0<p,p1,p2<∞, 0<q,q1,q2≤∞, 1/p=1/p1+1/p2, and 1/q=1/q1+1/q2.

### Dual space

If (X,μ) is a nonatomic σ-finite measure space, then
(i) (Lp,q)*={0} for 0<p<1, or 1=p<q<∞;
(ii) (Lp,q)*=Lp′,q′ for 1<p<∞,0<q≤∞, or 0<q≤p=1;
(iii) (Lp,∞)*≠{0} for 1≤p≤∞.
Here p′=p/(p−1) for 1<p<∞, p′=∞ for 0<p≤1, and ∞′=1.

### Atomic decomposition

The following are equivalent for 0<p≤∞,1≤q≤∞.
(i) ‖f‖Lp,q≤Ap,qC.
(ii) f=∑n∈ℤfn where fn has disjoint support, with measure ≤2n, on which 0<Hn+1≤|fn|≤Hn almost everywhere, and ‖Hn2n/p‖ℓq(ℤ)≤Ap,qC.
(iii) |f|≤∑n∈ℤHnχEn almost everywhere, where μ(En)≤Ap′,q2n and ‖Hn2n/p‖ℓq(ℤ)≤Ap,qC.
(iv) f=∑n∈ℤfn where fn has disjoint support En, with nonzero measure, on which B02n≤|fn|≤B12n almost everywhere, B0,B1 are positive constants, and ‖2nμ(En)1/p‖ℓq(ℤ)≤Ap,qC.
(v) |f|≤∑n∈ℤ2nχEn almost everywhere, where ‖2nμ(En)1/p‖ℓq(ℤ)≤Ap,qC.

## See also

* Interpolation space
* Hardy–Littlewood inequality

## References

* 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 .

## Notes

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.

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