Whitney covering lemma
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
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