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→Applications and examples: wiki-linking "Vitali covering lemma" |
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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 you only require weak boundedness on the extremes ''p'' and ''q'', you still get regular boundedness inside. 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
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