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The more general formulation needs p≤q; I'm not sure however, how "it follows from the former through an application of Hölder's inequality and a duality argument". |
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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''<sub>0</sub>, ''q''<sub>0</sub>) and of weak type (''p''<sub>1</sup>, ''q''<sub>1</sub>) where ''q''<sub>0</sub> ≠ ''q''<sub>1</sub>, then for each θ ∈ (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.
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