In mathematics, the Calder贸n鈥揨ygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calder贸n and Antoni Zygmund.
Given an integrable function f : Rd 鈫 C, 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 Calder贸n鈥揨ygmund decomposition of f , wherein f is written as the sum of "good" and "bad" functions, using the above sets.
Let f : Rd 鈫 C be integrable and 伪 be a positive constant. Then there exists an open set 惟 such that:
- (1) 惟 is a disjoint union of open cubes, 惟 = 鈭k Qk, such that for each Qk,
- (2) | f (x)| 鈮 伪 almost everywhere in the complement F of 惟.
Here, denotes the measure of the set .
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
and let b = f 鈭 g. Consequently we have that
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)| 鈮 伪 for almost every x in F, and on each cube in 惟, g is equal to the average value of f over that cube, which by the covering chosen is not more than 2d伪.
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citation}}
: CS1 maint: multiple names: authors list (
link)In mathematics, the Calder贸n鈥揨ygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calder贸n and Antoni Zygmund.
Given an integrable function f : Rd 鈫 C, 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 Calder贸n鈥揨ygmund decomposition of f , wherein f is written as the sum of "good" and "bad" functions, using the above sets.
Let f : Rd 鈫 C be integrable and 伪 be a positive constant. Then there exists an open set 惟 such that:
- (1) 惟 is a disjoint union of open cubes, 惟 = 鈭k Qk, such that for each Qk,
- (2) | f (x)| 鈮 伪 almost everywhere in the complement F of 惟.
Here, denotes the measure of the set .
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
and let b = f 鈭 g. Consequently we have that
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)| 鈮 伪 for almost every x in F, and on each cube in 惟, g is equal to the average value of f over that cube, which by the covering chosen is not more than 2d伪.
{{
citation}}
: CS1 maint: multiple names: authors list (
link)