Whitney covering lemma

In mathematical analysis, the Whitney covering lemma 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,

• An open subset A of Rⁿ can be written as a disjoint union of countably many dyadic cubes such that the following inequality holds for all j ∈ N:

$\mathrm {diam} (Q_{j})\leq {\text{dist}}(Q_{j},\partial A)\leq 4\,{\text{diam}}(Q_{j}).\,$

References

• 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.