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 , where denotes Euclidean space and denotes the complex numbers, the lemma gives a precise way of partitioning 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.
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Let be integrable and α be a positive constant. Then there exist sets F and such that:
- 1) with
- 2) almost everywhere in F;
- 3) is a union of cubes, , whose interiors are mutually disjoint, and so that for each
Given f as above, we may write f as the sum of a "good" function g and a "bad" function b, . To do this, we define
where denotes the interior of , and let . Consequently we have that
- for each cube
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 for almost every x in F, and on each cube in , g is equal to the average value of f over that cube, which by the covering chosen is not more than .