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 $f:\mathbf {R} ^{d}\to \mathbf {C}$, where $\mathbf {R} ^{d}$ denotes Euclidean space and $\mathbf {C}$ denotes the complex numbers, the lemma gives a precise way of partitioning $\mathbf {R} ^{d}$ 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.

Calderón–Zygmund lemma

Covering lemma

Let $f:\mathbf {R} ^{d}\to \mathbf {C}$ be integrable and α be a positive constant. Then there exist sets F and $\Omega$ such that:

1) $\mathbf {R} ^{d}=F\cup \Omega$ with $F\cap \Omega =\varnothing ;$

2) $|f(x)|\leq \alpha$ almost everywhere in F;

3) $\Omega$ is a union of cubes, $\Omega =\cup _{k}Q_{k}$, whose interiors are mutually disjoint, and so that for each $Q_{k},$

$\alpha <{\frac {1}{m(Q_{k})}}\int _{Q_{k}}f(x)\,dx\leq 2^{d}\alpha .$

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

$g(x)=\left\{{\begin{array}{cc}f(x),&x\in F,\\{\frac {1}{m(Q_{j})}}\int _{Q_{j}}f(x)\,dx,&x\in Q_{j}^{o},\end{array}}\right.$

where $Q_{j}^{o}$ denotes the interior of $Q_{j}$, and let $b=f-g$. Consequently we have that

$b(x)=0,\ x\in F$

$\int _{Q_{j}}b(x)\,dx=0$ for each cube $Q_{j}.$

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 $|g(x)|\leq \alpha$ for almost every x in F, and on each cube in $\Omega$, g is equal to the average value of f over that cube, which by the covering chosen is not more than $2^{d}\alpha$.

References

• Stein, Elias (1970). "Chapters I–II". Singular Integrals and Differentiability Properties of Functions. Princeton University Press.