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  : 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.

Covering lemma

Let  f  : Rd → C be integrable and α be a positive constant. Then there exists an open set Ω such that:
(1) Ω is a disjoint union of open cubes, Ω = ∪k Qk, such that for each Qk,
\alpha\le \frac{1}{m(Q_k)} \int_{Q_k} |f(x)| \, dx \leq 2^d \alpha.
(2) | f (x)| ≤ α almost everywhere in the complement F of Ω.

Calderón–Zygmund decomposition

Given  f  as above, we may write  f  as the sum of a "good" function g and a "bad" function b,  f  = g + b. To do this, we define
g(x) = \begin{cases}f(x), & x \in F, \\ \frac{1}{m(Q_j)}\int_{Q_j}f(t)\,dt, & x \in Q_j,\end{cases}

and let b =  f  − g. Consequently we have that

b(x) = 0,\ x\in F
\frac{1}{m(Q_j)}\int_{Q_j} b(x)\, dx = 0
for each cube Qj.

The function b is thus supported on a collection of cubes where  f  is allowed to be "large", but has the beneficial property that its average value is zero on each of these cubes. Meanwhile |g(x)| ≤ α for almost every x in F, and on each cube in Ω, g is equal to the average value of  f  over that cube, which by the covering chosen is not more than 2dα.

See also

References

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