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

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 ,

::$\alpha\le \frac \int_ , f(x), \, dx \leq 2^d \alpha.$

:(2) almost everywhere in the complement of .

Here, $m(Q_k)$ denotes the measure of the set $Q_k$.

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

:$g(x) = \beginf(x), & x \in F, \\ \frac\int_f(t)\,dt, & x \in Q_j,\end$

and let . Consequently we have that

:$b(x) = 0,\ x\in F$
:$\frac\int_ b(x)\, dx = 0$

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

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{{DEFAULTSORT:Calderon-Zygmund Lemma
Theorems in Fourier analysis
Lemmas in mathematical analysis