Marcinkiewicz interpolation theorem

In mathematics, the Marcinkiewitz theorem is a theorem that allows to interpolate between $L^{p}$ spaces. It is similar in spirit to the Riesz-Thorin theorem, but can be used in certain situations where the Riesz-Thorin theorem cannot.

You might want to read Riesz-Thorin theorem first, since it covers a similar, but conceptually simpler topic. More useful background can be found in Fourier series, operator norm and $L^{p}$ space.

Preliminaries

A function f on a measure space (X, F, ω) is called weak $L^{1}$ if it satisfies the following inequality

$\omega (\{x:|f(x)|>N\})\leq {\frac {C}{N}}.$

The smallest constant C in the inequality above is called the weak $L^{1}$ norm and is usually denoted by ||f||_1,w or ||f||_1,∞. Similarly the space is usually denoted by L^1,w or L^1,∞

Any $L^{1}$ function belongs to L^1,w and in addition one has the inequality

$||f||_{1,w}\leq ||f||_{1}.$

This is nothing but Markov's inequality.