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

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

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 ∈ R^N 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,

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

Application to maximal functions and maximal inequalities

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

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

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

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

Proof.

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

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

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

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

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).

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.