Besicovitch covering theorem

In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space R^N by balls such that each point of E is the center of some ball in the cover. The Besicovitch covering theorem asserts that there exists a constant c_N depending only on the dimension N with the following property:

• Given any Besicovitch cover F of a bounded set E, there are c_N subcollections of balls A₁ = {B_n1}, …, A_cN = {B_ncN} contained in F such that each collection A_i consists of disjoint balls, and

[math]\displaystyle{ E \subseteq \bigcup_{i=1}^{c_N} \bigcup_{B\in A_i} B. }[/math]

Let G denote the subcollection of F consisting of all balls from the c_N disjoint families A₁,...,A_cN. The less precise following statement is clearly true: every point x ∈ R^N belongs to at most c_N different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from the subcollection G). This property gives actually an equivalent form for the theorem (except for the value of the constant).

• There exists a constant b_N depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there is a subcollection G of F such that G is a cover of the set E and every point x ∈ E belongs to at most b_N different balls from the subcover G.

In other words, the function S_G equal to the sum of the indicator functions of the balls in G is larger than 1_E and bounded on R^N by the constant b_N,

[math]\displaystyle{ \mathbf{1}_E \le S_{\mathbf {G}} := \sum_{B \in \mathbf{G}} \mathbf{1}_B \le b_N. }[/math]

Application to maximal functions and maximal inequalities

Let μ be a Borel non-negative measure on R^N, finite on compact subsets and let [math]\displaystyle{ f }[/math] be a [math]\displaystyle{ \mu }[/math]-integrable function. Define the maximal function [math]\displaystyle{ f^* }[/math] by setting for every [math]\displaystyle{ x }[/math] (using the convention [math]\displaystyle{ \infty \times 0 = 0 }[/math])

[math]\displaystyle{ f^*(x) = \sup_{r \gt 0} \Bigl( \mu(B(x, r))^{-1} \int_{B(x, r)} |f(y)| \, d\mu(y) \Bigr). }[/math]

This maximal function is lower semicontinuous, hence measurable. The following maximal inequality is satisfied for every λ > 0 :

[math]\displaystyle{ \lambda \, \mu \bigl( \{ x : f^*(x) \gt \lambda \} \bigr) \le b_N \, \int |f| \, d\mu. }[/math]

Proof.

The set E_λ of the points x such that [math]\displaystyle{ f^*(x) \gt \lambda }[/math] clearly admits a Besicovitch cover F_λ by balls B such that

[math]\displaystyle{ \int \mathbf{1}_B \, |f| \ d\mu = \int_{B} |f(y)| \, d\mu(y) \gt \lambda \, \mu(B). }[/math]

For every bounded Borel subset E´ of E_λ, one can find a subcollection G extracted from F_λ that covers E´ and such that S_G ≤ b_N, hence

[math]\displaystyle{ \begin{align} \lambda \, \mu(E') &\le \lambda \, \sum_{B \in \mathbf{G}} \mu(B)\\ &\le \sum_{B \in \mathbf{G}} \int \mathbf{1}_B \, |f| \, d\mu = \int S_{\mathbf {G}} \, |f| \, d\mu \le b_N \, \int |f| \, d\mu, \end{align} }[/math]

which implies the inequality above.

When dealing with the Lebesgue measure on R^N, it is more customary to use the easier (and older) Vitali covering lemma in order to derive the previous maximal inequality (with a different constant).

See also

• Vitali covering lemma

References

• Besicovitch, A. S. (1945), "A general form of the covering principle and relative differentiation of additive functions, I", Proceedings of the Cambridge Philosophical Society 41 (02): 103–110, doi:10.1017/S0305004100022453 .
  • "A general form of the covering principle and relative differentiation of additive functions, II", Proceedings of the Cambridge Philosophical Society 42: 205–235, 1946, doi:10.1017/s0305004100022660 .
• DiBenedetto, E (2002), Real analysis, Birkhäuser, ISBN 0-8176-4231-5 .
• Füredi, Z; Loeb, P.A. (1994), "On the best constant for the Besicovitch covering theorem", Proceedings of the American Mathematical Society 121 (4): 1063–1073, doi:10.2307/2161215 .