Research proof index

There is a measurable exceptional set $G' \subseteq G$ with $2\mu(G') \le \mu(G)$ such that, for every measurable $f$ bounded by $\mathbf{1}_F$, the integral over $G \setminus G'$ of the finitary oscillatory singular integral, summed only over the scales from $\sigma_1(x)$ to $\sigma_2(x)$, is at most $C(a,q)\mu(G)^{1-1/q}\mu(F)^{1/q}$.

For $x\in G\setminus G'$, summing the localized Carleson contribution over all tiles is exactly the same as summing the corresponding oscillatory kernel integral over the integer scales from $\sigma_1(x)$ to $\sigma_2(x)$. This is the identity that converts the discrete tile model back into the finitary operator.