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. Covering lemma 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 , (2) almost everywhere in the complement of . Calderón–Zygmund decomposition Given as above, we may write as the sum of a "good" function and a "bad" function , . To do this, we define and let . Consequently we have that for each cube . 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 . See also Singular integral operators of convolution type, for a proof and application of the lemma in one dimension. Rising sun lemma References Theorems in Fourier analysis Lemmas in analysis