Marcinkiewicz theorem

In mathematics, the Marcinkiewicz theorem, discovered by Józef Marcinkiewicz, allows one 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 <math>L^1<math> if it satisfies the following inequality

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

The smallest constant C in the inequality above is called the weak <math>L^1<math> 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 <math>L^1<math> function belongs to L^1,w and in addition one has the inequality

<math>||f||_{1,w}\leq ||f||_1.<math>

This is nothing but Markov's inequality. The converse is not true. For example, the function 1/x belongs to L^1,w but not to L¹.

Similarly, one may define the weak <math>L^p<math> space as the space of all functions f such that <math>|f|^p<math> belong to L^1,w, and the weak <math>L^p<math> norm using

<math>||f||_{p,w}=||\,|f|^p ||_{1,w}^{1/p}.<math>

Formulation

Informally, Marcinkiewicz's theorem is

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 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.

Application example

A famous application example is the Hilbert transform. Viewed as a multiplier, the Hilbert transform is

Fourier/multiplying by the sign function/Inverse Fourier.

Hence Parseval's theorem easily shows that it 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. 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 ∞.