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{{Orphan|date=October 2024}}

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]].

Roughly speaking, the lemma states that it is possible to decompose an open set by cubes each of whose [[diameter]]s is proportional, within certain bounds, to its distance from the [[boundary (topology)|boundary]] of the open set.  More precisely:

'''Whitney Covering Lemma''' {{harv|Grafakos|2008|loc=Appendix J}}

Let <math>\Omega</math> be an open non-empty proper subset of <math>\mathbb{R}^n</math>.
Then there exists a family of closed cubes <math>\{Q_j\}_j</math> such that

* <math>\cup_j Q_j = \Omega</math> and the <math>Q_j</math>'s have disjoint interiors.
* <math>\sqrt{n} \ell(Q_j) \leq \mathrm{dist}(Q_j, \Omega^c) \leq 4 \sqrt{n} \ell(Q_j).</math>
* 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>
* For a given <math>Q_j</math> there exist at most <math>12^n Q_k</math>'s that touch it.

Where <math>\ell(Q)</math> denotes the length of a cube <math>Q</math>.

==References==

* {{cite book|last1=Grafakos|first1=Loukas|title=Classical Fourier Analysis|date=2008|publisher=Springer|isbn=978-0-387-09431-1}}
* {{citation|title=Real analysis|first=Emmanuele|last=DiBenedetto|publisher=Birkhäuser|year=2002|isbn=0-8176-4231-5}}.
* {{citation|title=Singular Integrals and Differentiability Properties of Functions|first=Elias|last=Stein|authorlink=Elias Stein|year=1970|publisher=Princeton University Press}}.
* {{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}}.

[[Category:Covering lemmas]]


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