Editing Marcinkiewicz interpolation theorem (section)

==Applications and examples==
A famous application example is the [[Hilbert transform]]. Viewed as a [[multiplier (Fourier analysis)|multiplier]], the Hilbert transform of a function ''f'' can be computed by first taking the [[Fourier transform]] of ''f'', then multiplying by the [[sign function]], and finally applying the [[inverse Fourier transform]].

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 [[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]].