(\lambda ) = e ^{i \lambda } $ for $ \lambda \in \mathbb R $. If the kernel $ K$ of $ T$ is supported on $ 2B= \{y \;:\; \lvert  y\rvert \leq 2\}$, then we have $$\lvert  e (P (y)) K (y) - K (y)\rvert \lesssim  \mathbf 1_{2B} (y) \lvert  y \rvert ^{-n+1},$$ so that $ \lvert  T _{P} f - T f \rvert \lesssim M f $. Both $ T$ and $ M$ admit pointwise domination by sparse forms, hence also by bilinear forms. (This is the main result of [@150105818].)

Thus, we can proceed under the assumption that the kernel $ K $ is not supported on $ B$. We can then write $$K  =  \sum_{j =1} ^{\infty  } \varphi _j$$ where $ \varphi _j$ is supported on $ 2 ^{j-1} B \setminus 2 ^{j-2}B$, with $ \lVert  \nabla ^{s} \varphi _ j\rVert _{\infty } \lesssim 2 ^{-nj -sj}$, for $ s=0,1$.

We use shifted dyadic grids, $ \mathcal D _{t}$, for $ 1\leq t \leq 3 ^{n}$. These grids have the property that $$\{ \tfrac 13  Q  \;:\; Q\in \mathcal D_t,\  \ell Q= 2 ^{k}, 1\leq t \leq 3 ^{n}\}$$ form a partition of $ \mathbb 