Whitney covering lemma: Difference between revisions
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⚫ | In [[mathematical analysis]], the '''Whitney covering lemma''' asserts the existence of a certain type of [[partition of a set|partition]] of an [[open set]] in a [[Euclidean space]]. Originally it was employed in the proof of [[Hassler Whitney]]'s [[Whitney extension theorem|extension theorem]]. The [[lemma (mathematics)|lemma]] was subsequently applied to prove generalizations of the [[Calderón–Zygmund lemma|Calderón–Zygmund decomposition]]. |
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⚫ | In [[mathematical analysis]], the '''Whitney covering lemma''', or Whitney decomposition, asserts the existence of a certain type of [[partition of a set|partition]] of an [[open set]] in a [[Euclidean space]]. Originally it was employed in the proof of [[Hassler Whitney]]'s [[Whitney extension theorem|extension theorem]]. The [[lemma (mathematics)|lemma]] was subsequently applied to prove generalizations of the [[Calderón–Zygmund lemma|Calderón–Zygmund decomposition]]. |
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* An open subset ''A'' of '''R'''<sup>''n''</sup> can be written as a disjoint union of countably many [[dyadic cubes]] such that the following inequality holds for all ''j'' ∈ '''N''': |
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'''Whitney Covering Lemma''' {{harv|Grafakos|2008|loc=Appendix J}} |
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Let <math>\Omega</math> be an open non-empty proper subset of <math>\mathbb{R}^n</math>. |
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Then there exists a family of closed cubes <math>\{Q_j\}_j</math> such that |
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* <math>\cup_j Q_j = \Omega</math> and the <math>Q_j</math>'s have disjoint interiors. |
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* If the boundaries of two cubes <math>Q_j</math> and <math>Q_k</math> touch then <math>\frac{1}{4} \leq \frac{\ell(Q_j)}{\ell(Q_k)} \leq 4.</math> |
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* For a given <math>Q_j</math> there exist at most <math>12^n Q_k</math>'s that touch it. |
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Where <math>\ell(Q)</math> denotes the length of a cube <math>Q</math>. |
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==References== |
==References== |
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* {{cite book|last1=Grafakos|first1=Loukas|title=Classical Fourier Analysis|date=2008|publisher=Springer|isbn=978-0-387-09431-1}} |
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* {{citation|title=Real analysis|first=Emmanuele|last=DiBenedetto|publisher=Birkhäuser|year=2002|isbn=0-8176-4231-5}}. |
* {{citation|title=Real analysis|first=Emmanuele|last=DiBenedetto|publisher=Birkhäuser|year=2002|isbn=0-8176-4231-5}}. |
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* {{citation|title= |
* {{citation|title=Singular Integrals and Differentiability Properties of Functions|first=Elias|last=Stein|authorlink=Elias Stein|year=1970|publisher=Princeton University Press}}. |
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* {{citation|title=Analytic extensions of functions defined in closed sets|first=Hassler|last=Whitney|authorlink=Hassler Whitney|journal=Transactions of the American Mathematical Society|year=1934|volume=36|pages=63–89|doi=10.2307/1989708|jstor= |
* {{citation|title=Analytic extensions of functions defined in closed sets|first=Hassler|last=Whitney|authorlink=Hassler Whitney|journal=Transactions of the American Mathematical Society|year=1934|volume=36|pages=63–89|doi=10.2307/1989708|jstor=1989708|issue=1|publisher=American Mathematical Society|doi-access=free}}. |
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[[Category:Covering lemmas]] |
[[Category:Covering lemmas]] |
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Latest revision as of 06:55, 30 October 2024
In mathematical analysis, the Whitney covering lemma, or Whitney decomposition, asserts the existence of a certain type of partition of an open set in a Euclidean space. Originally it was employed in the proof of Hassler Whitney's extension theorem. The lemma was subsequently applied to prove generalizations of the Calderón–Zygmund decomposition.
Roughly speaking, the lemma states that it is possible to decompose an open set by cubes each of whose diameters is proportional, within certain bounds, to its distance from the boundary of the open set. More precisely:
Whitney Covering Lemma (Grafakos 2008, Appendix J)
Let be an open non-empty proper subset of . Then there exists a family of closed cubes such that
- and the 's have disjoint interiors.
- If the boundaries of two cubes and touch then
- For a given there exist at most 's that touch it.
Where denotes the length of a cube .
References
[edit]- Grafakos, Loukas (2008). Classical Fourier Analysis. Springer. ISBN 978-0-387-09431-1.
- DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, ISBN 0-8176-4231-5.
- Stein, Elias (1970), Singular Integrals and Differentiability Properties of Functions, Princeton University Press.
- Whitney, Hassler (1934), "Analytic extensions of functions defined in closed sets", Transactions of the American Mathematical Society, 36 (1), American Mathematical Society: 63–89, doi:10.2307/1989708, JSTOR 1989708.