Marcinkiewicz theorem

In mathematics, the Marcinkiewicz interpolation theorem, discovered by Józef Marcinkiewicz (1939), is a result bounding the norms of non-linear operators acting on L^p spaces.

Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators.

Let f be a measurable function with real or complex values, defined on a measure space (X, F, ω). The distribution function of f is defined by

Then f is called weak ${\displaystyle L^{1}}$ if there exists a constant C such that the distribution of f satisfies the following inequality for all t > 0:

The smallest constant C in the inequality above is called the weak ${\displaystyle L^{1}}$ 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,∞.

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