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.

Contents

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 %2B 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(t)\,dt, & 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