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I am looking for the following version of Calderon-Zygmund decomposition, consider an function $f \in L^1(R^{d+1})$ and cylinders of the form $Q_{R,R^p}$ for some fixed $p \in (0,\infty)$, The cylinders are cubes of radius $R$ in $R^d$ and of height $R^p$ in the time direction.

Question: Can I find a sequence of cubes $\{Q_i\}$ such that Calderon Zygmund decomposition holds and each cube $\{Q_i\}$ has scaling of the form $(r,r^p)$?

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  • $\begingroup$ Just a simple remark, without having checked the details. If you subdivide into $k$ parts in $x$ and $h$ parts in $t$ you should get any $p=(\log k)/(\log h)$ with $k,h \geq 2$ integers (the case $k=2, h=4$ gives parabolic cylinders). Is this true? $\endgroup$
    – Giorgio Metafune
    Commented Sep 13, 2022 at 17:15
  • $\begingroup$ When k and h are integers, then the CZ theory works. In other cases, the subdivision will produce cylinders that could go over or stay below the previous generation. So I'm trying to understand how to subdivide in t direction when p isn't an integer or rational number. $\endgroup$
    – Adi
    Commented Sep 13, 2022 at 19:34
  • $\begingroup$ As an alternative, you could use Whitney balls which work well in homogenuous spaces and get somewhat weaker results for every $p$. Maybe they suffice. $\endgroup$
    – Giorgio Metafune
    Commented Sep 13, 2022 at 19:58
  • $\begingroup$ Unfortunately that isn't enough for my needs because that result is a bit of an existential one. In the sense, there exists a covering with CZ type properties, but the problem is that i don't know which balls are at which scale nor do i have any effective estimates on their locations. $\endgroup$
    – Adi
    Commented Sep 13, 2022 at 20:52
  • $\begingroup$ That is strange. Don't you have the same problem with dyadic cubes? Usually you don't know at which scale they are selected nor which centers. $\endgroup$
    – Giorgio Metafune
    Commented Sep 13, 2022 at 21:00

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