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.

Covering lemma

Let  f  : Rd → C be integrable and α be a positive constant. Then there exists an open set Ω such that: Ω is a disjoint union of open cubes, , such that for each Qk , almost everywhere in the complement F of Ω . Here, m(Q_k) denotes the measure of the set Q_k.

Calderón–Zygmund decomposition

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 . 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, 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 .