Marcinkiewicz interpolation theorem: Difference between revisions

Revision as of 16:15, 2 December 2022

Revision as of 04:01, 19 January 2023

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:'''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 youone only requirerequires weak boundedness on the extremes ''p'' and ''q'', you still get regular boundedness insidestill 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