Calderón–Zygmund theory

In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund.

Given an integrable function  f  : R^d → C, where R^d denotes Euclidean space and C denotes the complex numbers, the lemma gives a precise way of partitioning R^d into two sets: one where  f  is essentially small; the other a countable collection of cubes where  f  is essentially large, but where some control of the function is retained.

This leads to the associated Calderón–Zygmund decomposition of  f , wherein  f  is written as the sum of "good" and "bad" functions, using the above sets.

Let  f  : R^d → C be integrable and α be a positive constant. Then there exists an open set Ω such that:

Given  f  as above, we may write  f  as the sum of a "good" function g and a "bad" function b,  f  = g + b. To do this, we define

and let b =  f  − g. Consequently we have that

The function b is thus supported on a collection of cubes where  f  is allowed to be "large", but has the beneficial property that its average value is zero on each of these cubes. Meanwhile, |g(x)| ≤ α for almost every x in F, and on each cube in Ω, g is equal to the average value of  f  over that cube, which by the covering chosen is not more than 2^dα.

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