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.