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Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$.

I'm looking to construct some kind of dyadic cube decomposition or bounded overlap covering of the region $\{f > 0 \}$. I would like it to be the case that for a cube $Q$, we have $$ (\text{edge length of}\ Q)^2 \approx \frac{1}{|Q|}\int_Q f^2 $$

Can I do something like this? (The ''$\approx$'' should ideally mean within fixed dimensional constants of one another.)


The idea is that we want $$ \frac{1}{e(Q)^{n+2}}\int_Q f^2 $$ to be basically the same on all cubes. Since $f$ is pointwise very small, we expect that for large cubes this quantity is very small. But as the cube size decreases, can we expect that this quantity increases? As an example, if the function $f$ is constant, $f \equiv c$ say, then $$ \frac{1}{e(Q)^{n+2}}\int_Q f^2 = \frac{c^2}{e(Q)^2}, $$

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    $\begingroup$ That is not possible. If $f=c$ is constant, then the integral average over any cube equals to the constant $c^2$ which can be arbitrarily large. $\endgroup$ Aug 7, 2019 at 20:47
  • $\begingroup$ Hmm good point. I guess I know in fact more than I said. $\endgroup$
    – SBK
    Aug 7, 2019 at 20:53
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    $\begingroup$ Even with the modified version, if $f=0$, then it is still not possible. $\endgroup$ Aug 7, 2019 at 21:32
  • $\begingroup$ I see... Maybe I'll try one more time before I give up $\endgroup$
    – SBK
    Aug 8, 2019 at 8:38

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