and yet deliver striking applications.

Proof of Theorem \[t:main1\]
============================

Our conclusion is invariant under dilations of the operator. Hence, we can proceed under the assumption that $ \lVert P\rVert = \sum_{\alpha } \lvert  \lambda _{\alpha }\rvert  =1$. We can also assume that the polynomial $ P$ has no linear term, as it can be absorbed into the function $ f$. Under these assumptions we prove

\[t:s\] Let $ P$ be a polynomial without linear terms, and $ \lVert P\rVert=1$. Then, for bounded compactly supported functions $ f, g$ and $ 1< r < \infty $, there is a sparse form $ \Lambda_1 $ and a $ \eta >0$ so that $$\label{e:s}
\lvert  \langle T_P f, g \rangle\rvert \lesssim \Lambda _1 (f,g) 
+ \sum_{ Q  \in \mathcal D\;:\; \lvert  Q\rvert \geq 1 }  
\langle  f  \rangle _{Q,r} \langle g \rangle _{Q,r} \lvert  Q\rvert  ^{1- \eta }$$

It is easy to see that this implies Theorem \[t:main1\], since the second term on the right is restricted to dyadic cubes of volume at least one, and there is a gain of $ \lvert  Q\rvert ^{- \eta } $. Moreover, we will see that this Theorem implies the weighted result.

Let $ e 