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 : Rd → C, where Rd denotes Euclidean space and C denotes the complex numbers, the lemma gives a precise way of partitioning Rd 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 : Rd → C be integrable and α be a positive constant. Then there exists an open set Ω such that:
- (1) Ω is a disjoint union of open cubes, Ω = ∪k Qk, such that for each Qk,
- (2) | f (x)| ≤ α 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, f = g + b. To do this, we define
and let b = f − g. Consequently we have that
for each cube Qj.
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 2dα.
- Singular integral operators of convolution type, for a proof and application of the lemma in one dimension.
- Calderon A. P., Zygmund, A. (1952), "On the existence of certain singular integrals", Acta Math, 88: 85–139
- 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.