Editing Marcinkiewicz interpolation theorem (section)

==Formulation==
Informally, Marcinkiewicz's theorem is

:'''Theorem.''' Let ''T'' be a [[bounded linear operator]] from <math>L^p</math> to <math>L^{p,w}</math> and at the same time from <math>L^q</math> to <math>L^{q,w}</math>. Then ''T'' is also a bounded operator from <math>L^r</math> to <math>L^r</math> for any ''r'' between ''p'' and ''q''.
 
In other words, even if one only requires weak boundedness on the extremes ''p'' and ''q'', regular boundedness still holds. To make this more formal, one has to explain that ''T'' is bounded only on a [[Dense set|dense]] subset and can be completed. See [[Riesz-Thorin theorem]] for these details.

Where Marcinkiewicz's theorem is weaker than the Riesz-Thorin theorem is in the estimates of the norm. The theorem gives bounds for the <math>L^r</math> norm of ''T'' but this bound increases to infinity as ''r'' converges to either ''p'' or ''q''.  Specifically {{harv|DiBenedetto|2002|loc=Theorem VIII.9.2}}, suppose that

:<math>\|Tf\|_{p,w} \le N_p\|f\|_p,</math>
:<math>\|Tf\|_{q,w} \le N_q\|f\|_q,</math>

so that the [[operator norm]] of ''T'' from ''L''<sup>''p''</sup> to ''L''<sup>''p'',''w''</sup> is at most ''N''<sub>''p''</sub>, and the operator norm of ''T'' from ''L''<sup>''q''</sup> to ''L''<sup>''q'',''w''</sup> is at most ''N''<sub>''q''</sub>.  Then the following '''interpolation inequality''' holds for all ''r'' between ''p'' and ''q'' and all ''f''&nbsp;∈&nbsp;''L''<sup>''r''</sup>:
:<math>\|Tf\|_r\le \gamma N_p^\delta N_q^{1-\delta}\|f\|_r</math>
where
:<math>\delta=\frac{p(q-r)}{r(q-p)}</math>
and
:<math>\gamma=2\left(\frac{r(q-p)}{(r-p)(q-r)}\right)^{1/r}.</math>
The constants δ and γ can also be given for ''q''&nbsp;=&nbsp;∞ by passing to the limit.

A version of the theorem also holds more generally if ''T'' is only assumed to be a quasilinear operator  in the following sense: there exists a constant ''C''&nbsp;>&nbsp;0 such that ''T'' satisfies

:<math>|T(f+g)(x)| \le C(|Tf(x)|+|Tg(x)|)</math>

for [[almost everywhere|almost every]] ''x''. The theorem holds precisely as stated, except with γ replaced by

:<math>\gamma=2C\left(\frac{r(q-p)}{(r-p)(q-r)}\right)^{1/r}.</math>

An operator ''T'' (possibly quasilinear) satisfying an estimate of the form

:<math>\|Tf\|_{q,w}\le C\|f\|_p</math>

is said to be of '''weak type (''p'',''q'')'''.  An operator is simply of type (''p'',''q'') if ''T'' is a bounded transformation from ''L<sup>p</sup>'' to ''L<sup>q</sup>'':

:<math>\|Tf\|_q\le C\|f\|_p.</math>

A more general formulation of the interpolation theorem is as follows:

* If ''T'' is a quasilinear operator of weak type (''p''<sub>0</sub>, ''q''<sub>0</sub>) and of weak type (''p''<sub>1</sub>, ''q''<sub>1</sub>) where ''q''<sub>0</sub>&nbsp;≠&nbsp;''q''<sub>1</sub>, then for each θ&nbsp;∈&nbsp;(0,1), ''T'' is of type (''p'',''q''), for ''p'' and ''q'' with ''p'' ≤ ''q'' of the form
::<math>\frac{1}{p} = \frac{1-\theta}{p_0}+\frac{\theta}{p_1},\quad \frac{1}{q} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}.</math>

The latter formulation follows from the former through an application of [[Hölder's inequality]] and a duality argument.{{Citation needed|reason=How to use Hölder's inequality and the special case?|date=June 2016}}