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 , 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

<blockquote>Let be integrable and be a positive constant. Then there exists an open set such that:

Here, denotes the measure of the set . </blockquote>

Calderón–Zygmund decomposition

<blockquote>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 .</blockquote>

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