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.

This is an excellent paper, but I am confused at one point. On page 169, 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 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). 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?