The classical Calderon-Zygmund decomposition says that if $f\geq 0$ is $L^1$ on a cubes $B$, with average value $\alpha$, then there is a sequence of disjoint cubes $B\_j$, such that the average of $f$ on each $B\_j$ is in between $\alpha$ and $2^n \alpha$, and $f\leq \alpha$ a.e. away from $\bigcup\_j B\_j$.


I am wondering if there is a similar result on (compact) manifolds, for example with a volume doubling condition. If one uses a Vitali type argument, it seems there is no control on the radius (they could be too small), so no upper bound in the average value of $f$ on (small) balls obtained by Vitali - in contrast to the $2^n\alpha$ upper bound of Calderon-Zygmund.



Calderon-Zygmund decomposition on manifolds?
Calderon-Zygmund theory generalises without much difficulty to doubling metric measure spaces (or more generally to "spaces of homogeneous type"). See for instance Chapter 1 of


*Stein, Elias M.*, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy, Princeton Mathematical Series. 43. Princeton, NJ: Princeton University Press. xiii, 695 pp. (1993). [ZBL0821.42001](https://zbmath.org/?q=an:0821.42001).


The generalisation of the Calderon-Zygmund decomposition in this setting is given in Section 1.4.


In more recent years, most of Calderon-Zygmund theory has also been extended to the non-doubling case, though one has to make some natural modifications to the statements in order to avoid trivial counterexamples. One reference is Chapter 2 of


*Tolsa, Xavier*, [**Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory**](http://dx.doi.org/10.1007/978-3-319-00596-6), Progress in Mathematics 307. Cham: Birkhäuser/Springer (ISBN 978-3-319-00595-9/hbk; 978-3-319-00596-6/ebook). xiii, 396 p. (2014). [ZBL1290.42002](https://zbmath.org/?q=an:1290.42002).


The Calderon-Zygmund decomposition is given as Lemma 2.14 of that book, though the statement may look slightly different from the classical one.


One common trick in this subject is to replace balls by roughly equivalent "dyadic cubes" that have good geometric properties, such as nesting. See for instance


*Hytönen, Tuomas; Kairema, Anna*, [**What is a cube?**](http://dx.doi.org/10.5186/aasfm.2013.3838), Ann. Acad. Sci. Fenn., Math. 38, No. 2, 405-412 (2013). [ZBL1288.30066](https://zbmath.org/?q=an:1288.30066).


for a brief introduction to this topic.


