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 is essentially small; the other a countable collection of cubes where is essentially large, but where some control of the function is retained.
This leads to the associated Calderón–Zygmund decomposition of, wherein is written as the sum of "good" and "bad" functions, using the above sets.
Let be integrable and be a positive constant. Then there exists an open set such that:(1) is a disjoint union of open cubes,, such that for each,
\alpha\le
1 m(Qk)
\int Qk |f(x)|dx\leq2d\alpha.
(2) almost everywhere in the complement of .
Here,
denotes the measure of the setm(Qk)
.Qk
Given as above, we may write as the sum of a "good" function and a "bad" function, . To do this, we defineg(x)=\begin{cases}f(x),&x\inF,\
1 m(Qj)
\int Qj f(t)dt,&x\inQj,\end{cases}
and let . Consequently we have that
b(x)=0, x\inF
for each cube .
1 m(Qj)
\int Qj b(x)dx=0
The function is thus supported on a collection of cubes where 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 in, and on each cube in, is equal to the average value of over that cube, which by the covering chosen is not more than .
. Elias Stein. Chapters I–II. Singular Integrals and Differentiability Properties of Functions . registration . Princeton University Press . 1970. 9780691080796 .