Marcinkiewicz interpolation theorem: Difference between revisions

Revision as of 14:21, 3 February 2014

Revision as of 17:25, 25 February 2014

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:<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''^''p'' to ''L''^''p'',''w'' is at most ''N''<sub>''p''</supsub>, and the operator norm of ''T'' from ''L''^''q'' to ''L''^''q'',''w'' is at most ''N''<sub>''q''</supsub>. Then the following '''interpolation inequality''' holds for all ''r'' between ''p'' and ''q'' and all ''f''&nbsp;∈&nbsp;''L''^''r'':

:<math>\|Tf\|_r\le \gamma N_p^\delta N_q^{1-\delta}\|f\|_r</math>

where

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:<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''₀, ''q''₀) and of weak type (''p''<sub>1</supsub>, ''q''₁) where ''q''₀&nbsp;≠&nbsp;''q''₁, 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.