_{\alpha, \beta  } x ^{\alpha } y ^{\beta }$, where we use the usual multi-index notation. The polynomial modulated Calderón-Zygmund operators are $$\label{e:TK}
T_P f (x) = \int e ^{i P  (x,y)}K (y) f (x-y) \; dy .$$ The $ L ^{p}$ result below is a special case of the results of Ricci and Stein [@MR822187; @MR890662], and the weak-type result is due to Chanillo and Christ [@MR883667].

\[t:fixedPoly\] For $ 1 < p < \infty $, the operator $ T_P$ is bounded on $ L ^{p}$, that is $$\lVert T_P  \;:\;  L^{p} \mapsto L ^{p}\rVert  \lesssim  1,$$ where the implied constant depends on the degree of $ P$, and in particular is independent of $ \lambda $. Moreover, $ T_P$ maps $ L ^{1}$ to weak $ L ^{1}$, with the same bound.

The dependence on the polynomial being felt only through the degree of $ P$ is important to the application of these bounds to the setting of nilpotent groups, like the Heisenberg group, see [@MR890662]. This dependence continues to hold in the Theorems below.

\[t:main1\] For each $ 1< r < 2$ Calderón-Zygmund operator $ T$, polynomial $ P = P(y)$ of degree $ d$ and bounded 