Marcinkiewicz interpolation theorem: Difference between revisions

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Revision as of 09:57, 27 January 2017

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Hence [[Parseval's theorem]] easily shows that the Hilbert transform is bounded from <math>L^2</math> to <math>L^2</math>. A much less obvious fact is that it is bounded from <math>L^1</math> to <math>L^{1,w}</math>. Hence Marcinkiewicz's theorem shows that it is bounded from <math>L^p</math> to <math>L^p</math> for any 1 < ''p'' < 2. [[dual space|Duality]] arguments show that it is also bounded for 2 < ''p'' < ∞. In fact, the Hilbert transform is really unbounded for ''p'' equal to 1 or ∞.

Another famous example is the [[Hardy–Littlewood maximal function]], which is only quasilinear[[sublinear operator]] rather than linear. While <math>L^p</math> to <math>L^p</math> bounds can be derived immediately from the <math>L^1</math> to weak <math>L^1</math> estimate by a clever change of variables, Marcinkiewicz interpolation is a more intuitive approach. Since the Hardy–Littlewood Maximal Function is trivially bounded from <math>L^\infty</math> to <math>L^\infty</math>, strong boundedness for all <math>p>1</math> follows immediately from the weak (1,1) estimate and interpolation. The weak (1,1) estimate can be obtained from the [[Vitali covering lemma]].

==History==