Calderón–Zygmund lemma

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  : R^d → C, where R^d denotes Euclidean space and C denotes the complex numbers, the lemma gives a precise way of partitioning R^d 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.