Editing Marcinkiewicz interpolation theorem (section)

==Preliminaries==
Let ''f'' be a [[measurable function]] with real or complex values, defined on a [[measure space]] (''X'',&nbsp;''F'',&nbsp;ω).  The [[cumulative distribution function|distribution function]] of ''f'' is defined by

:<math>\lambda_f(t) = \omega\left\{x\in X\mid |f(x)| > t\right\}.</math>

Then ''f'' is called '''weak <math>L^1</math>''' if there exists a constant ''C'' such that the distribution function of ''f'' satisfies the following inequality for all ''t''&nbsp;>&nbsp;0:

:<math>\lambda_f(t)\leq \frac{C}{t}.</math>

The smallest constant ''C'' in the inequality above is called the '''weak <math>L^1</math> norm''' and is usually denoted by <math>\|f\|_{1,w}</math> or <math>\|f\|_{1,\infty}.</math> Similarly the space is usually denoted by ''L''<sup>1,''w''</sup> or ''L''<sup>1,∞</sup>.

(Note: This terminology is a bit misleading since the weak norm does not satisfy the triangle inequality as one can see by considering the sum of the functions on <math> (0,1) </math> given by <math> 1/x </math>  and <math> 1/(1-x) </math>, which has norm 4 not 2.)

Any <math>L^1</math> function belongs to ''L''<sup>1,''w''</sup> and in addition one has the inequality

:<math>\|f\|_{1,w}\leq \|f\|_1.</math>

This is nothing but [[Markov's inequality]] (aka [[Chebyshev's Inequality]]). The converse is not true. For example, the function 1/''x'' belongs to ''L''<sup>1,''w''</sup> but not to ''L''<sup>1</sup>.

Similarly, one may define the [[Lp space#Weak Lp|'''weak <math>L^p</math> space''']] as the space of all functions ''f'' such that <math>|f|^p</math> belong to ''L''<sup>1,''w''</sup>, and the '''weak <math>L^p</math> norm''' using

:<math>\|f\|_{p,w}= \left \||f|^p  \right \|_{1,w}^{\frac{1}{p}}.</math>

More directly, the ''L''<sup>''p'',''w''</sup> norm is defined as the best constant ''C'' in the inequality

:<math>\lambda_f(t) \le \frac{C^p}{t^p}</math>

for all ''t''&nbsp;>&nbsp;0.