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Whitney covering lemma

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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 Ω {\displaystyle \Omega } be an open non-empty proper subset of R n {\displaystyle \mathbb {R} ^{n}} . Then there exists a family of closed cubes { Q j } j {\displaystyle \{Q_{j}\}_{j}} such that

  • ∪ j Q j = Ω {\displaystyle \cup _{j}Q_{j}=\Omega } and the Q j {\displaystyle Q_{j}} 's have disjoint interiors.
  • n ℓ ( Q j ) ≤ d i s t ( Q j , Ω c ) ≤ 4 n ℓ ( Q j ) . {\displaystyle {\sqrt {n}}\ell (Q_{j})\leq \mathrm {dist} (Q_{j},\Omega ^{c})\leq 4{\sqrt {n}}\ell (Q_{j}).}
  • If the boundaries of two cubes Q j {\displaystyle Q_{j}} and Q k {\displaystyle Q_{k}} touch then 1 4 ≤ ℓ ( Q j ) ℓ ( Q k ) ≤ 4. {\displaystyle {\frac {1}{4}}\leq {\frac {\ell (Q_{j})}{\ell (Q_{k})}}\leq 4.}
  • For a given Q j {\displaystyle Q_{j}} there exist at most 12 n Q k {\displaystyle 12^{n}Q_{k}} 's that touch it.

Where ℓ ( Q ) {\displaystyle \ell (Q)} denotes the length of a cube Q {\displaystyle Q} .

References

  • 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.


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