supported functions $f, g $ there is a bilinear form $ \Lambda _{r} $ so that $$\lvert  \langle T _{P} f ,g  \rangle\rvert  \lesssim \Lambda _{r} (f,g).$$ The implied constant depends only on $ T$, the degree $ d$, and dimension $ n$ and choice of $ r>1$.

The bound above continues to hold for polynomials $ P $ of two variables, but we suppress the details, as the estimate above can most likely be improved. And, as written is quite easy to prove, yet yields a non-trivial corollary.

\[c:wtd\] For $ 1 < p < \infty $, the operator $ T_P$, where $ P = P (y)$ is of degree $ d$, is bounded on $ L ^{p} (w)$, where $ w $ is a Muckenhoupt weight $ w\in A_p $.

Weak-type and weighted estimates for oscillatory singular integrals have been studied in this and more general contexts by various authors, see for instance [@MR2900003; @MR2910762; @MR2949870; @MR2115460; @MR1782909]. Y. Ding and H. Liu [@MR2900003] were interested in $ L ^{p} (w)$ inequalities for more general operators $ T$. The approach of these authors entails many complications.

The method of proof of Theorem \[t:main1\] is very simple. And, so 