Lorentz space: Difference between revisions
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For ''p'' > 1, the Lorentz spaces can be equipped with an equivalent norm. When ''p'' = 1, ''L''<sup>1, 1</sup> = ''L''<sup>1</sup> can be equipped with a norm, but this is not possible for ''L''<sup>1,∞</sup>, the weak ''L''<sup>1</sup> space. |
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==Properties== |
==Properties== |
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The Lorentz spaces are genuinely generalisations of the ''L''<sup>''p''</sup> spaces in the sense that for any ''p'', ''L''<sup>''p'',''p''</sup> = ''L''<sup>''p''</sup>, which follows from [[Cavalieri's principle]]. Further, ''L''<sup>''p'',∞</sup> coincides with [[Lp space#Weak Lp|weak ''L''<sup>''p''</sup>]]. They are [[Quasinorm|quasi-Banach spaces]] (that is quasi-normed spaces which are also complete) and are normable for ''p'' ∈ (1, ∞), ''q'' ∈ [1, ∞]. '' |
The Lorentz spaces are genuinely generalisations of the ''L''<sup>''p''</sup> spaces in the sense that for any ''p'', ''L''<sup>''p'',''p''</sup> = ''L''<sup>''p''</sup>, which follows from [[Cavalieri's principle]]. Further, ''L''<sup>''p'',∞</sup> coincides with [[Lp space#Weak Lp|weak ''L''<sup>''p''</sup>]]. They are [[Quasinorm|quasi-Banach spaces]] (that is quasi-normed spaces which are also complete) and are normable for ''p'' ∈ (1, ∞), ''q'' ∈ [1, ∞]. When ''p'' = 1, ''L''<sup>1, 1</sup> = ''L''<sup>1</sup> can be equipped with a norm, but this is not possible for ''L''<sup>1,∞</sup>, the weak ''L''<sup>1</sup> space. |
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The space ''L''<sup>''p'',''q''</sup> is contained in ''L''<sup>''p'',''r''</sup> whenever ''q'' < ''r''. |
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== See also == |
== See also == |
Revision as of 08:57, 25 May 2013
In mathematical analysis, Lorentz spaces, introduced by George 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 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 (the p) and the domain (the q). The Lorentz norms, like the Lp 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 functions 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 ƒ defined on a measure space, (X, μ), its decreasing rearrangement function, can be defined as
where dƒ is the so-called distribution function of ƒ, given by
Here, for notational convenience, is defined to be ∞.
The two functions |f | and f * are equimeasurable, meaning that
where λ is the Lebesgue measure on the real line.
Given these definitions, for p, q ∈ (0, ∞) or q = ∞, the Lorentz quasinorms are given by
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 p ∈ (1, ∞), q ∈ [1, ∞]. When p = 1, L1, 1 = L1 can be equipped with a norm, but this is not possible for L1,∞, the weak L1 space. The space Lp,q is contained in Lp,r whenever q < r.
See also
References
- Grafakos, Loukas (2008), Classical Fourier analysis, Graduate Texts in Mathematics, vol. 249 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09432-8, ISBN 978-0-387-09431-1, MR2445437.