Английская Википедия:Calderón–Zygmund lemma

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

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 Шаблон:Math, where Шаблон:Math denotes Euclidean space and Шаблон:Math denotes the complex numbers, the lemma gives a precise way of partitioning Шаблон:Math into two sets: one where Шаблон:Math is essentially small; the other a countable collection of cubes where Шаблон:Math is essentially large, but where some control of the function is retained.

This leads to the associated Calderón–Zygmund decomposition of Шаблон:Math, wherein Шаблон:Math is written as the sum of "good" and "bad" functions, using the above sets.

Covering lemma

Let Шаблон:Math be integrable and Шаблон:Mvar be a positive constant. Then there exists an open set Шаблон:Math such that:

(1) Шаблон:Math is a disjoint union of open cubes, Шаблон:Math, such that for each Шаблон:Math,
<math>\alpha\le \frac{1}{m(Q_k)} \int_{Q_k} |f(x)| \, dx \leq 2^d \alpha.</math>
(2) Шаблон:Math almost everywhere in the complement Шаблон:Mvar of Шаблон:Math.

Here, <math>m(Q_k)</math> denotes the measure of the set <math>Q_k</math>.

Calderón–Zygmund decomposition

Given Шаблон:Math as above, we may write Шаблон:Math as the sum of a "good" function Шаблон:Mvar and a "bad" function Шаблон:Mvar, Шаблон:Math. To do this, we define

<math>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}</math>

and let Шаблон:Math. Consequently we have that

<math>b(x) = 0,\ x\in F</math>
<math>\frac{1}{m(Q_j)}\int_{Q_j} b(x)\, dx = 0</math>

for each cube Шаблон:Math.

The function Шаблон:Mvar is thus supported on a collection of cubes where Шаблон:Math is allowed to be "large", but has the beneficial property that its average value is zero on each of these cubes. Meanwhile, Шаблон:Math for almost every Шаблон:Mvar in Шаблон:Mvar, and on each cube in Шаблон:Math, Шаблон:Mvar is equal to the average value of Шаблон:Math over that cube, which by the covering chosen is not more than Шаблон:Math.

See also

References

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Calderón–Zygmund_lemma&oldid=13728519