is equal to 1 on the disk of radius 2.

Define an operator $M_T$ associated to a tube $T \in {\mathbb{T}}_{j, \tau}$ by

$$M_T f := \eta_T (\psi_{j, \tau} \hat f)^{\vee}.$$

Morally, $M_T f$ is the part of $f$ which has Fourier support in $\tau$ and physical support in $T$. We also let $M_0 f := (\psi_0 \hat f)^{\vee}$. We denote ${\mathbb{T}}_j = \cup_{\tau} {\mathbb{T}}_{j, \tau}$ and ${\mathbb{T}}= \cup_{j \ge 1} {\mathbb{T}}_j$. If $f$ is a function supported on the unit disk, then $f = M_0 f + \sum_{T \in {\mathbb{T}}} M_T f$, up to a tiny error (see Lemma \[f=MTf\] below for a precise statement).

We call a tube $T \in {\mathbb{T}}_{j, \tau}$ bad if

$$\mu_2 (T) \ge R_j^{-1/2 + 100 \delta}.$$

To get a sense of what this means, notice that the number of tubes $T \in {\mathbb{T}}_{j, \tau}$ is $\sim R_j^{1/2 - \delta}$. If each tube $T \in {\mathbb{T}}_{j, \tau}$ contained the same amount of the measure $\mu_2$, then for each tube we would have $\mu_2(T) \sim R_j^{-1/2 + \delta}$. A tube is bad, if it contains significantly more $\mu_2$ measure than this. A tube is good if it is not bad. Now we define ${\mu_{1, good}}$ 