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 $f:\; mathbf\{R\}^\{d\}\; o\; mathbf\{C\}$, 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 $f:\; mathbf\{R\}^\{d\}\; o\; mathbf\{C\}$ 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 $Fcap\; Omega\; =\; varnothing;$

: 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 $Q\_k,$

::

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

::

where $Q\_j^o$ denotes the interior of $Q\_j$, and let $b\; =\; f\; -\; g$. Consequently we have that

::$b(x)\; =\; 0,\; xin\; F$

:: $int\_\{Q\_j\}\; b(x),\; dx\; =\; 0$ 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

* cite book
last = Stein | first = Elias | authorlink = Elias Stein
chapter = Chapters I-II
title = Singular Integrals and Differentiability Properties of Functions
publisher = Princeton University Press | year = 1970