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2014-11-14T05:39:41Z

I think this is the right reference

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	~~[[File:L1infin.png\|thumb\|right\|~~The ~~''L''<sup>1,∞</sup> norm~~ of ~~<math>f(x)=1/\|x~~-~~1\|</math> is~~ the ~~area of the largest rectangle with sides parallel to the coordinate axes that~~ can ~~be inscribed in~~ the ~~graph~~.]]		The feeling of freedom and self-control that you feel as you drive through the mountain trails can help you forget all the stress and worries of daily life. For more serious cyclists, next to bicycles themselves, proper shoes are the most important piece of equipment. If you go on a regular mountain biking, you will develop a healthier body and disposition. If you don’t need the overly aggressive styling of the Scapel 5 then you will find the Rush 6 a welcome alternative. That attitude completely misses what is important about Papago Park. <br><br>You can choose from many different suspensions on your bike, make sure the suspension you choose is going to fit the type of cycling you intend for it. Now most people wear gloves whilst they are riding because they keep the hands warm. You can spend under $100 for a bargain bike at a department store, or lay down thousands for a professional model. “Whereas past research emphasized whether or not a relationship existed between bicycle riding on a saddle and erectile dysfunction, Schrader now says that the next step of contemporary research on the subject should focus on intervention. Titanium is very light and gives you an advantage and is usually used in racing bicycles. <br><br>Technological know-how is advancing very rapidly and manufactures have observed ways to make batteries much more efficient and Eco-pleasant at the very same time along with photo voltaic PV cells getting additional and a lot more effortless and economical to make. Equally expensive can be the tag attached to a lighter-weighted bike. The tape (or string against the ruler) will indicate what your helmet size is. The bicycle you have to choose should be under the budget, saving enough for you to buy the necessary accessories for your men's mountain bike, like helmet, gloves, eye gear, etc. The users can enjoy these stills when travelling and offline as well. <br><br>The tracks are mostly dirt paths on outdoor scenery. In between, the biker can perform whatever new trick he or she has learnt up and this trick is known as the gap jump. As the tracks are very rough it is far more difficult on a hardtail bike. The internet will do just fine, maybe even better, as there are countless of online stores and biking websites from which you may get the information you need. To find out the latest News and Tips on folding bikes, visit Best Folding Bikes at bestfoldingbikes. <br><br>So I should be looking for an 18 to 19 inch bike frame. Why pollute the air when you can experience an electrical folding bike. If you cherished this post and you would like to obtain far more info about [http://www.silverstoregames.com/profile/ardeasey Choosing the right ride for you mountain bike sizing.] kindly check out our own web site. The price may generally differ depending on the condition of the bike and on how new the model really is. I've always been curious about this aspect of dating, because very few women have comparable experiences. Some switchbacks can be technically challenging and the narrow, winding trails can make for a few surprises when encountering other riders, but much of the system is gentle, rolling and easily managed by novices.
	~~In [[mathematical analysis]]~~, ~~Lorentz spaces~~, ~~introduced by [[George Lorentz]] in~~ the ~~1950s~~,~~<ref>G~~. ~~Lorentz, "Some new function spaces", ''Annals~~ of ~~Mathematics'' '''51''' (1950), pp~~. ~~37-55~~.<~~/ref~~><~~ref~~>~~G. Lorentz~~, ~~"On~~ the ~~theory~~ of ~~spaces Λ", ''Pacific Journal of Mathematics'' '''1''' (1951), pp~~. ~~411-429.</ref>~~ are ~~generalisations of~~ the ~~more familiar [[Lp space\|''L''<sup>''p''</sup> spaces]]~~.

	~~The Lorentz spaces are denoted by ''L''<sup>''p''~~,~~''q''</sup>~~. ~~Like the ''L''<sup>''p''</sup> spaces, they are characterized by~~ a ~~[[norm (mathematics)\|norm]] (technically~~ a ~~[[quasinorm]])~~ that ~~encodes information about~~ the ~~"size"~~ of ~~a function, just as~~ the ~~''L''<sup>''p''</sup> 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''~~<~~sup~~>~~''p''~~<~~/sup~~> ~~norms, by exponentially rescaling~~ the ~~measure in both the range (the ''p'')~~ and the ~~domain~~ (the ~~''q''~~). The ~~Lorentz norms, like~~ the ~~''L''<sup>''p''</sup> 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]]s ''&fnof;'' on ''X'' such that the following [[quasinorm]] is finite~~

	~~:<math>\\|f\\|_{L^{p~~,~~q}(X~~,~~\mu)} = p^{1/q}\\|t\mu\{\|f\|\ge t\}^{1/p}\\|_{L^q(\mathbb{R}^+~~,~~\frac{dt}{t})}</math>~~

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

	~~:<math>\\|f\\|_{L^{p,q}(X,\mu)}=p^{1/q}\left(\int_0^\infty t^q \mu\left\{x\mid \|f(x)\| \ge t\right\}^{q/p}\,\frac{dt}{t}\right)^{1/q}~~.<~~/math~~>

	~~and when ''q'' = ∞,~~

	:<~~math~~>~~\\|f\\|_{L^{p,\infty}(X,\mu)}^p = \sup_{t>0}\left(t^p\mu\left\{x\mid \|f(x)\|>t\right\}\right)~~.~~</math>~~

	~~It is also conventional to set L<sup>∞~~,~~∞</sup>(''X'',μ) = L<sup>∞</sup>(''X'',μ).~~

	~~==Decreasing rearrangements==~~
	~~The quasinorm~~ is ~~invariant under rearranging~~ the ~~values of~~ the ~~function ''&fnof;'', essentially by definition. In particular, given a complex-valued [[measurable function]] ''&fnof;'' defined~~ on a ~~measure space~~, ~~''(X~~, ~~μ)'', its '''decreasing rearrangement''' function, <math>f^{*}: [0, \infty) \rightarrow [0, \infty]</math> can be defined~~ as
	~~:<math>f^{*}(t) = \inf\{\alpha \in \mathbb{R}^{+}: d_f(\alpha) \leq t\}</math>~~
	~~where ''d''<sub>''&fnof;''</sub> is~~ the ~~so-called distribution function of ''&fnof;''~~, ~~given by~~
	:<~~math>d_f(\alpha) = \mu(\{x \in X : \|f(x)\|~~ > ~~\alpha\}).~~<~~/math~~>
	~~Here,~~ for ~~notational convenience, <math>\inf \emptyset</math> is defined~~ to ~~be ∞~~.

	~~The two functions \|''f'' \| and {{nowrap\|''f'' *}} are '''equimeasurable''', meaning that~~
	~~:<math> \mu \bigl( \{ x \in X : \|f(x)\| > \alpha\} \bigr) = \lambda \bigl( \{ t > 0 : f^*(t) > \alpha\} \bigr), \quad \alpha > 0, </math>~~
	~~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~~
	:~~<math>t \in \mathbf{R} \ \longrightarrow \ f^*(\|t\|)~~ / 2.</~~math>~~

	~~Given these definitions, for ''p'', ''q'' ∈ (0, ∞) or ''q'' = ∞,~~ the ~~Lorentz quasinorms are given by~~
	~~:<math>\\| f \\|_{L^{p, q}} = \left\{~~
	~~\begin{array}{l l}~~
	~~\left( \int_0^{\infty} (t^{\frac{1}{p}} f^{*}(t))^q \, \frac{dt}{t} \~~right~~)^{\frac{1}{q}} & q \in (0, \infty),\\~~
	~~\displaystyle \sup_{t > 0} \, t^{\frac{1}{p}} f^{*}(t) & q = \infty~~.
	~~\end{array}~~
	~~\right~~.~~</math>~~

	~~==Properties==~~
	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> is equipped with~~ a ~~norm~~, but ~~it is not possible to define a norm equivalent to the quasinorm~~ of ~~''L''<sup>1,∞</sup>, the weak ''L''<sup>1</sup> space. As a concrete example that the triangle inequality fails in ''L''<sup>1,∞</sup>, consider~~

	~~:<math>f(x) = \tfrac{1}{x} \chi_{(0,1)}(x)\ \ \text{and} \ \ g(x) = \tfrac{1}{1-x} \chi_{(0,1)}(x),</math>~~

	~~whose ''L''<sup>1,∞</sup> quasi-norm equals one, whereas~~ the ~~quasi-norm of their sum {{nowrap\|''f'' + ''g''}} equals four.~~

	~~The space ''L''<sup>''p'',''q''</sup>~~ is ~~contained in ''L''<sup>''p''~~,~~''r''</sup> whenever {{nowrap\|''q'' < ''r''}}. The Lorentz spaces are real [[interpolation space]]s between ''L''<sup>1</sup>~~ and ~~''L''<sup>∞</sup>~~.

	~~== See also ==~~
	* [[Interpolation space]]

	~~==References==~~
	*{{Citation \| last1=Grafakos \| first1=Loukas \| title=Classical Fourier analysis \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=2nd \| series=Graduate Texts in Mathematics \| isbn=978-0-387-09431-1 \| doi= 10.1007/978-0-387-09432-8 \| id={{MathSciNet \| id = 2445437}} \| year=2008 \| volume=249}}.

	~~==Notes==~~
	~~<references/>~~

	~~[[Category:Banach spaces]]~~

74.129.51.198: /* Definition */

2012-06-19T23:49:05Z

Definition

New page

[[File:L1infin.png|thumb|right|The ''L''1,∞ norm of <math>f(x)=1/|x-1|</math> 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,<ref>G. Lorentz, "Some new function spaces", ''Annals of Mathematics'' '''51''' (1950), pp. 37-55.</ref><ref>G. Lorentz, "On the theory of spaces Λ", ''Pacific Journal of Mathematics'' '''1''' (1951), pp. 411-429.</ref> are generalisations of the more familiar [[Lp space|''L''''p'' spaces]].

The Lorentz spaces are denoted by ''L''''p'',''q''. Like the ''L''''p'' spaces, they are characterized by a [[norm (mathematics)|norm]] (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]]s ''&fnof;'' on ''X'' such that the following [[quasinorm]] is finite

:<math>\|f\|_{L^{p,q}(X,\mu)} = p^{1/q}\|t\mu\{|f|\ge t\}^{1/p}\|_{L^q(\mathbb{R}^+,\frac{dt}{t})}</math>

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

:<math>\|f\|_{L^{p,q}(X,\mu)}=p^{1/q}\left(\int_0^\infty t^q \mu\left\{x\mid |f(x)| \ge t\right\}^{q/p}\,\frac{dt}{t}\right)^{1/q}.</math>

and when ''q'' = ∞,

:<math>\|f\|_{L^{p,\infty}(X,\mu)}^p = \sup_{t>0}\left(t^p\mu\left\{x\mid |f(x)|>t\right\}\right).</math>

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 ''&fnof;'', essentially by definition. In particular, given a complex-valued [[measurable function]] ''&fnof;'' defined on a measure space, ''(X, μ)'', its '''decreasing rearrangement''' function, <math>f^{*}: [0, \infty) \rightarrow [0, \infty]</math> can be defined as
:<math>f^{*}(t) = \inf\{\alpha \in \mathbb{R}^{+}: d_f(\alpha) \leq t\}</math>
where ''d''''&fnof;'' is the so-called distribution function of ''&fnof;'', given by
:<math>d_f(\alpha) = \mu(\{x \in X : |f(x)| > \alpha\}).</math>
Here, for notational convenience, <math>\inf \emptyset</math> is defined to be ∞.

The two functions |''f'' | and {{nowrap|''f'' *}} are '''equimeasurable''', meaning that
:<math> \mu \bigl( \{ x \in X : |f(x)| > \alpha\} \bigr) = \lambda \bigl( \{ t > 0 : f^*(t) > \alpha\} \bigr), \quad \alpha > 0, </math>
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
:<math>t \in \mathbf{R} \ \longrightarrow \ f^*(|t|) / 2.</math>

Given these definitions, for ''p'', ''q'' ∈ (0, ∞) or ''q'' = ∞, the Lorentz quasinorms are given by
:<math>\| f \|_{L^{p, q}} = \left\{
\begin{array}{l l}
\left( \int_0^{\infty} (t^{\frac{1}{p}} f^{*}(t))^q \, \frac{dt}{t} \right)^{\frac{1}{q}} & q \in (0, \infty),\\
\displaystyle \sup_{t > 0} \, t^{\frac{1}{p}} f^{*}(t) & q = \infty.
\end{array}
\right.</math>

==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]]. Further, ''L''''p'',∞ coincides with [[Lp space#Weak Lp|weak ''L''''p'']]. 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''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

:<math>f(x) = \tfrac{1}{x} \chi_{(0,1)}(x)\ \ \text{and} \ \ g(x) = \tfrac{1}{1-x} \chi_{(0,1)}(x),</math>

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

The space ''L''''p'',''q'' is contained in ''L''''p'',''r'' whenever {{nowrap|''q'' < ''r''}}. The Lorentz spaces are real [[interpolation space]]s between ''L''1 and ''L''∞.

== See also ==
* [[Interpolation space]]

==References==
*{{Citation | last1=Grafakos | first1=Loukas | title=Classical Fourier analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-09431-1 | doi= 10.1007/978-0-387-09432-8 | id={{MathSciNet | id = 2445437}} | year=2008 | volume=249}}.

==Notes==
<references/>

[[Category:Banach spaces]]