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.
Let be integrable and α be a positive constant. Then there exist an open set Ω such that:
- 1) is a disjoint union of open cubes, , such that for each
- 2) almost everywhere in the complement F of Ω.
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
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 .
- Singular integral operators of convolution type, for a proof and application of the lemma in one dimension.
- Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, ISBN 3-540-52343-X
- Stein, Elias (1970). "Chapters I–II". Singular Integrals and Differentiability Properties of Functions. Princeton University Press.