ScalingStacks

On $\tilde{B}_{t}$ we have

$$\frac{C^{-1}}{|\log|t||^{n}}\mu_{t}\leqslant\tilde{\omega}_{t}^{n}\leqslant\frac{C}{|\log|t||^{n}}\mu_{t},$$

and again by direct computation in polar coordinates (as in [3]), thanks to the definition of $\tilde{B}_{t}$ and to (3.1) we have that

(4.2)

$$C^{-1}\leqslant\int_{\tilde{B}_{t}}d\mu_{t}\leqslant 1,$$

and so

$$\frac{C^{-1}}{|\log|t||^{n}}\leqslant\int_{\tilde{B}_{t}}\tilde{\omega}_{t}^{n}\leqslant\frac{C}{|\log|t||^{n}}.$$

If we then define

$$\tilde{\mu}_{t}=\frac{\tilde{\omega}^{n}_{t}}{\int_{\tilde{B}_{t}}\tilde{\omega}_{t}^{n}},$$

then $\tilde{\mu}_{t}$ is uniformly comparable to $\mu_{t}$ on $\tilde{B}_{t}$ and

(4.3)

$$\mu_{t}(\tilde{B}_{t})\geqslant C^{-1}\mu_{t}(B_{t})\geqslant C^{-1}.$$