\eta  k}  \lvert  Q\rvert ^{-1+ 2/ r'}    \lVert f \mathbf 1_{Q}\rVert _r, \qquad 1 < r  \leq 2,\  r'= \tfrac r {r-1}.$$ Above, $ \eta = \eta (\epsilon , r)$ But, this immediately implies . Namely, $$\begin{aligned}
\lvert  \langle I_Q f, g \rangle \rvert & \lesssim  
\lVert I _{Q} f \rVert _{r'}  \lVert g \mathbf 1_{Q}\rVert _{r} 
 \\&\lesssim 2 ^{- \eta  k}  \lvert  Q\rvert ^{-1+ 2/ r'}    \lVert f \mathbf 1_{Q}\rVert _r  \lVert g \mathbf 1_{Q}\rVert _{r}  
\\&=  2 ^{- \eta  k} \langle f \rangle _{Q,r} \langle g \rangle _{Q,r} \lvert  Q\rvert.  \end{aligned}$$ (Alternatively, one can just use bilinear interpolation.)

We now give the weighted result.

The qualitative result that $ T_P $ is bounded on $ L ^{p} (w)$ for $ w \in A_p$, $ 1< p < \infty $ is as follows. Given $ w \in A_p$, recall that the dual weight is $ \sigma = w ^{1- p'}$. Then, it is equivalent to show that $$\lvert  \langle T _{P}  (f \sigma ),   g w    \rangle\rvert \lesssim 
C _{[w] _{A_p}}\lVert f\rVert _{L ^{p} (\sigma) } \lVert g\rVert _{L ^{p'} (w)}.$$ Using the sparse domination from , we see that we need to prove the corresponding bound for the 