Marcinkiewicz interpolation theorem

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In mathematics, the Marcinkiewicz theorem, discovered by Józef Marcinkiewicz, is a result about interpolation of operators acting on L^p spaces and related spaces. Interpolation of operators should not be confused with somewhat different mathematical procedure of interpolation of functions.

Marcinkiewicz' theorem 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.

Contents 1 Preliminaries 2 Formulation 3 Applications and examples 4 History

[edit] Preliminaries

A function f on a measure space (X, F, ω) is called weak L¹ if it satisfies the following inequality

The smallest constant C in the inequality above is called the weak L¹ 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,∞.

(Note: This terminology is a bit misleading since the weak norm does not satisfy the triangle inequality as one can see by considering the sum of the functions on (0,1) given by 1 / x and 1 / (1 − x), which has norm 4 not 2.)

Any L¹ function belongs to L^1,w and in addition one has the inequality

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 L^p space as the space of all functions f such that | f | ^p belong to L^1,w, and the weak L^p norm using

[edit] Formulation

Informally, Marcinkiewicz's theorem is

Theorem: Let T be a bounded linear operator from L^p to L^p,w and at the same time from L^q to L^q,w. Then T is also a bounded operator from L^r to L^r 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 L^r norm of T but this bound increases to infinity as r converges to either p or q.

[edit] Applications and examples

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 L² to L². A much less obvious fact is that it is bounded from L¹ to L^1,w. Hence Marcinkiewicz's theorem shows that it is bounded from L^p to L^p 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 ∞.

Another famous example is the Hardy-Littlewood Maximal Function. While L^p to L^p bounds can be derived immediately from the L¹ to weak L¹ estimate by a clever change of variables, Marcinkiewicz interpolation is a more intuitive approach. Since the Hardy-Littlewood Maximal Function is trivially bounded from to , strong boundedness for all p > 1 follows immediately from the weak (1,1) estimate and interpolation.

[edit] History

Marcinkiewicz had originally told this result to Antoni Zygmund shortly before he died in World War II. The theorem was almost forgotten by Zygmund, and was absent from his original works on the theory of Singular Integral Operators. Later Zygmund realized that Marcinkiewicz's result could greatly simplify his work, at which time he published his former student's theorem together with a generalization of his own.