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I'm reading the paper A local smoothing estimate in higher dimensions by Izabella Łaba and Thomas Wolff. The paper can be found at J. Anal. Math. 88 (2002), 149–171, doi: 10.1007/BF02786576, arxiv: math/0205059.

This is an excellent paper, but I am confused at one point. On page 169 (which is page 22 in arxiv version), each cube $\Delta$ is very small with side-length $\delta^{\frac{1}{2}}$ ($\delta$ is a small number). $a_{\Delta}$ is an affine map taking the unit cube to $\Delta$ (see page 152 (which is page 4 in arxiv version) for this defintion). So $a_{\Delta}$ shrinks every geometric object and $a_{\Delta}^{-1}$ should enlarge the measure of each set.

Now look at the first inequality under "Substituting this in (50), we obtain" (still on page 169, which is page 23 in the arxiv version). It seems to me that the factor should be $\delta^{-(d+1)/2}$ instead of $\delta^{(d+1)/2}$, according to (50) and the fact $a_{\Delta}^{-1}$ enlarges the measure.

Am I missing something?

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    $\begingroup$ Have you tried to ask Izabella Łaba? $\endgroup$ – Fedor Petrov May 26 '17 at 5:32
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    $\begingroup$ @FedorPetrov Not yet. I noticed that this paper has been cited many times. So probably many people have read it. $\endgroup$ – Tony B May 28 '17 at 15:05

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