.$$ for compactly supported functions $ f, g$ with disjoint supports. Moreover, the kernel $ K (y)$ satisfies $$\begin{aligned}
\label{e:size}
\lvert \nabla ^{t} K (x,y)\rvert \leq C_t \lvert  x-y\rvert ^{-n-t}  , \qquad x\neq y \in \mathbb R ^{n}, \end{aligned}$$ for $  t \in \{ 0,1\} $. Key examples are $ K (y) = 1/y$ in dimension one, and the Riesz transform kernels $ y/ \lvert  y\rvert ^{n+1} $, in dimension $ n$.

Such operators are of course non-local, and involve subtle cancellative effects. It is thus something of a surprise that such operators are dominated by sparse operators, which have none of these features. This is a special case of [@150105818; @14094351; @150805639].

\[t:czo\] For each Calderón-Zygmund singular integral operator $ T$ and bounded compactly supported function $ f$, there is a sparse operator $ \Lambda  = \Lambda _{T, f}$ so that $
\lvert  T f \rvert \lesssim \Lambda _{1} f 
$.

An immediate corollary are weighted inequalities that are sharp in the $ A_p$ characteristic. See [@MR3085756; @150105818; @14094351].

We consider polynomials of a fixed degree $ d$, given by $  P  (x,y) = \sum_{\alpha , \beta  \;:\; \lvert  \alpha \rvert + \lvert  \beta \rvert \leq d } \lambda 