3
$\begingroup$

In the paper by Nazarov, Treil and Volberg: Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators on non-homogeneous spaces 1998, Int. Math. Res. Not. (DOI), on the page 6, they stated a version of Vitali covering theorem for a separable metric space $X$ endowed with a measure $\mu$ “not necessarily having the doubling property”:

Claim: Let $E \subset X$ be any set and $(B(x,r_x))_{x\in E}$ be a family of balls with uniformly bounded radii. Then there always exists a “countable” subfamily of disjoint balls whose triple extensions cover $E$.

The fact “countable subfamily” is strange to me. The general covering theorem for metric spaces seems to claim the existence of a subfamily without mentioning its countability. If the space is equiped with a doubling measure then it’s well known that such a countable subfamily exists. Where can I find a reference for this claim of their paper? Do the main ingredient of this countable property come from the hypothesis of “separable” metric spaces?

$\endgroup$
1
  • 5
    $\begingroup$ Is your doubt abut the countability of this subfamily of disjoint open balls? But any family of disjoint nonempty open sets of a separable space must be countable (there is a countable set that meets each member of the family). $\endgroup$ Jul 13, 2018 at 14:53

1 Answer 1

8
$\begingroup$

The covering lemma as stated here is true in any separable metric space. No measure is needed at all.

Theorem 1. Let $\mathcal{B}$ be a family either of closed balls or open balls from a separable metric space such that $$ \sup\{\operatorname{diam}(B):B\in\mathcal{B}\}<\infty. $$ Then there is a finite or countable sequence $\{ B_i\}_{i\in I}$ of pairwise disjoint balls such that $$ \bigcup_{B\in\mathcal B} B\subset\bigcup_{i\in I} 5 B_i. $$

This is (verbatim) Theorem 2.2 in [2]. For a proof, see page 47 in [1] or almost any book on geometric measure theory.

The "$3r"$ case is true, at least in the case of finite families of balls. The following statement is Theorem 2.1 taken verbatim from [2]:

Theorem 2. Let $\mathcal{B}$ be a finite family either of closed balls or open balls from a metric space. Then there exists a finite subfamily $\{ B_i\}_{i\in I}\subset\mathcal{B} $ of pairwise disjoint balls such that $$ \bigcup_{B\in\mathcal B} B\subset\bigcup_{i\in I} 3 B_i. $$

Without separability there are easy counterexamples to Theorem 1: uncountable space with the discrete metric, covered by balls of radii $1/10$.

I am not sure if Theorem 1 is true with $5$ replaced by $3$ as otherwise Tolsa would state it with $3$ instead of $5$. However, in all applications the actual constant $3$ or $5$ is not important.

In fact Nazarov, Treil and Volberg assume that the space is separable.

[1] http://www.pitt.edu/~hajlasz/Notatki/Analysis%20I.pdf

[2] X. Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón–Zygmund theory. Progress in Mathematics, 307. Birkhäuser/Springer, Cham, 2014.

$\endgroup$
1
  • $\begingroup$ One may add that one can always extract a subfamily of disjoint balls with the stated property; separability of the ambient space implies then that this family is countable. $\endgroup$ Apr 11, 2020 at 10:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.