R ^{n}$. Throughout, $ \ell Q = \lvert  Q\rvert ^{1/n} $ is the side length of the cube $ Q$. We fix a dyadic grid $ \mathcal D_t$ throughout the remainder of the argument, and set $ \mathcal D_+ = \{Q \;:\; \ell Q > 2 ^{10}\}$. Define $$I _{Q} f = \int e (P  (y)) \varphi _{ k} (y)  (\mathbf 1_{\tfrac 13Q} f ) (x-y)\; dy, \qquad \ell Q = 2 ^{k+2}.$$ Note that $ I _Q f $ is supported on $ Q$, and that we have suppressed the dependence on $ P $, which we will continue below.

The basic estimate is then this Lemma.

\[l:fixed\] For each cube $ Q$ with $ \lvert  Q\rvert \geq 1 $ and $ 1< r < 2$, there holds $$\label{e:fixed}
 \lvert  \langle I _{Q} f, g \rangle\rvert 
\lesssim 2 ^{- \eta k}  \langle f \rangle _{Q, r} \langle g \rangle _{Q,r} \lvert  Q\rvert ,$$ where $ \eta = \eta (d,n,r) >0 $.

Theorem \[t:s\] follows immediately from this Lemma. The oscillatory nature of the problem exhibits itself in the next Lemma. Write $$\label{e:K}
I ^{\ast} _Q I_Q  \phi (x) = \mathbf 1_{\tfrac 13Q}  (x)  \cdot \int _{\tfrac 13 Q}  K 