Editing Lp space (section)

===Atomic decomposition===


If <math>1 \leq p < \infty</math> then every non-negative <math>f \in L^p(\mu)</math> has an {{em|atomic decomposition}},{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}} meaning that there exist a sequence <math>(r_n)_{n \in \Z}</math> of non-negative real numbers and a sequence of non-negative functions <math>(f_n)_{n \in \Z},</math> called {{em|the atoms}}, whose supports <math>\left(\operatorname{supp} f_n\right)_{n \in \Z}</math> are [[Disjoint sets|pairwise disjoint sets]] of measure <math>\mu\left(\operatorname{supp} f_n\right) \leq 2^{n+1},</math> such that
<math display=block>f ~=~ \sum_{n \in \Z} r_n \, f_n \, ,</math>
and for every integer <math>n \in \Z,</math> 
<math display=block>\|f_n\|_\infty ~\leq~ 2^{-\tfrac{n}{p}} \, ,</math> 
and
<math display=block>\tfrac{1}{2} \|f\|_p^p ~\leq~ \sum_{n \in \Z} r_n^p ~\leq~ 2 \|f\|^p_p \, ,</math>
and where moreover, the sequence of functions <math>(r_n f_n)_{n \in\Z}</math> depends only on <math>f</math> (it is independent of <math>p</math>).{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}} 
These inequalities guarantee that <math>\|f_n\|_p^p \leq 2</math> for all integers <math>n</math> while the supports of <math>(f_n)_{n \in \Z}</math> being pairwise disjoint implies{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}} 
<math display=block>\|f\|_p^p ~=~ \sum_{n \in \Z} r_n^p \, \|f_n\|^p_p \, .</math>

An atomic decomposition can be explicitly given by first defining for every integer <math>n \in \Z,</math>{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}}<ref group=note>This [[infimum]] is attained by <math>t_n;</math> that is, <math>\mu(f > t_n) < 2^n</math> holds.</ref> 
<math display=block>t_n = \inf \{t \in \Reals : \mu(f > t) < 2^n\}</math>
and then letting
<math display=block>r_n ~=~ 2^{n/p} \, t_n ~ \text{ and } \quad f_n ~=~ \frac{f}{r_n} \, \mathbf{1}_{( t_{n+1} < f \leq t_n )}</math>
where <math>\mu(f > t) = \mu(\{s : f(s) > t\})</math> denotes the measure of the set <math>(f > t) := \{s \in S : f(s) > t\}</math> and <math>\mathbf{1}_{(t_{n+1} < f \leq t_n)}</math> denotes the [[indicator function]] of the set <math>(t_{n+1} < f \leq t_n) := \{s \in S : t_{n+1} < f(s) \leq t_n\}.</math> 
The sequence <math>(t_n)_{n \in \Z}</math> is decreasing and converges to <math>0</math> as <math>n \to \infty.</math>{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}} Consequently, if <math>t_n = 0</math> then <math>t_{n+1} = 0</math> and <math>(t_{n+1} < f \leq t_n) = \varnothing</math> so that <math>f_n = \frac{1}{r_n} \, f \,\mathbf{1}_{(t_{n+1} < f \leq t_n)}</math> is identically equal to <math>0</math> (in particular, the division <math>\tfrac{1}{r_n}</math> by <math>r_n = 0</math> causes no issues). 

The [[complementary cumulative distribution function]] <math>t \in \Reals \mapsto \mu(|f| > t)</math> of <math>|f| = f</math> that was used to define the <math>t_n</math> also appears in the definition of the weak <math>L^p</math>-norm (given below) and can be used to express the <math>p</math>-norm <math>\|\cdot\|_p</math> (for <math>1 \leq p < \infty</math>) of <math>f \in L^p(S, \mu)</math> as the integral{{sfn|Bahouri|Chemin|Danchin|2011|pp=7–8}}
<math display=block>\|f\|_p^p ~=~ p \, \int_0^\infty t^{p-1} \mu(|f| > t) \, \mathrm{d} t \, ,</math>
where the integration is with respect to the usual Lebesgue measure on <math>(0, \infty).</math>