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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Calderón–Zygmund lemma is a fundamental result in
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
,
harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
, and
singular integral In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, wh ...
s. It is named for the mathematicians
Alberto Calderón Alberto Pedro Calderón (September 14, 1920 – April 16, 1998) was an Argentine mathematician. His name is associated with the University of Buenos Aires, but first and foremost with the University of Chicago, where Calderón and his mentor, th ...
and
Antoni Zygmund Antoni Zygmund (December 26, 1900 – May 30, 1992) was a Polish-American mathematician. He worked mostly in the area of mathematical analysis, including harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...
. Given an
integrable function In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Inte ...
, where denotes
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
and denotes the
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
, the lemma gives a precise way of partitioning into two
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s: one where is essentially small; the other a
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
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

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