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

${\displaystyle E\subseteq \bigcup _{i=1}^{c_{N}}\bigcup _{B\in A_{i}}B.}$

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,

${\displaystyle \mathbf {1} _{E}\leq S_{\mathbf {G} }:=\sum _{B\in \mathbf {G} }\mathbf {1} _{B}\leq b_{N}.}$

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

${\displaystyle f^{*}(x)=\sup _{r>0}{\Bigl (}\mu (B(x,r))^{-1}\int _{B(x,r)}|f(y)|\,d\mu (y){\Bigr )}.}$

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

${\displaystyle \lambda \,\mu {\bigl (}\{x:f^{*}(x)>\lambda \}{\bigr )}\leq b_{N}\,\int |f|\,d\mu .}$

Proof.

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

${\displaystyle \int \mathbf {1} _{B}\,|f|\ d\mu =\int _{B}|f(y)|\,d\mu (y)>\lambda \,\mu (B).}$

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

${\displaystyle {\begin{aligned}\lambda \,\mu (E')&\leq \lambda \,\sum _{B\in \mathbf {G} }\mu (B)\\&\leq \sum _{B\in \mathbf {G} }\int \mathbf {1} _{B}\,|f|\,d\mu =\int S_{\mathbf {G} }\,|f|\,d\mu \leq b_{N}\,\int |f|\,d\mu ,\end{aligned}}}$

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).

• Vitali covering lemma

• Besicovitch, A. S. (1945), "A general form of the covering principle and relative differentiation of additive functions, I", Mathematical Proceedings of the Cambridge Philosophical Society, 41 (02): 103–110, doi:10.1017/S0305004100022453.
  • Besicovitch, A. S. (1946), "A general form of the covering principle and relative differentiation of additive functions, II", Mathematical Proceedings of the Cambridge Philosophical Society, 42: 205–235, doi:10.1017/s0305004100022660.
• DiBenedetto, E (2002), Real analysis, Birkhäuser, ISBN 0-8176-4231-5.
• Füredi, Zoltán; Loeb, Peter A. (1994), "On the best constant for the Besicovitch covering theorem", Proceedings of the American Mathematical Society, 121 (4): 1063–1073, doi:10.1090/S0002-9939-1994-1249875-4, JSTOR 2161215.