_Q (x,y) \phi  (y) \; dy .$$

\[l:K\] For each cube $ Q \in \mathcal D_+$, and $ x\in \tfrac 13 Q$, we have $$\label{e:K<}
\lvert  K_Q (x,y)\rvert 
\lesssim    \lvert  Q\rvert ^{-1}  \mathbf 1_{Z_ {Q}} (x - y) +  \lvert  Q\rvert ^{-1 - \epsilon } \mathbf 1_{ Q} (x) \mathbf 1_{Q} (y),$$ where $ Z_ {Q} \subset   Q   $ has measure at most $  (\ell Q) ^{- \epsilon } \lvert Q\rvert   $, where $ \epsilon = \epsilon (n,d)>0$.

This Lemma is well known, see for instance [@MR1879821]\*[Lemma 4.1]{}. Here is how we use the Lemma. Using Cauchy-Schwartz, we have $$\begin{aligned}
\label{e:2}
\lVert I_Q f \rVert_2 ^2 &\lesssim    \lvert  Q\rvert ^{-1} \int _{Q} \int _{Z_Q} \lvert  f (x)\rvert  \lvert  f (x-y)\rvert  \; dy dx 
+  \lvert  Q\rvert ^{ - \epsilon } \langle f \rangle _{Q,1} ^{2} \lvert  Q\rvert 
\\&
\lesssim 
\lvert  Q\rvert ^{- \epsilon /n}    \lVert f \mathbf 1_{Q}\rVert   _{2 } ^2 . \end{aligned}$$ We also have the trivial but rarely used $ \lVert I _{Q} f \rVert_ \infty  \lesssim   \lvert  Q\rvert ^{-1}  \lVert f \mathbf 1_{Q}\rVert _1     $. By Riesz Thorin interpolation, there holds with $ \ell Q = 2 ^{k}$, $$\lVert I _{Q} f \rVert _{r'} \lesssim 2 ^{- 