Calderón–Zygmund lemma

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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  : RdC, 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[edit]

Let f  : RdC 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[edit]

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[edit]

References[edit]

  • Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, ISBN 3-540-52343-X 
  • Stein, Elias (1970). "Chapters I–II". Singular Integrals and Differentiability Properties of Functions. Princeton University Press.