  We prove that the generalized Carleson operator $C_d$ with polynomial phase
function is of strong type $(p,r)$, $1<r<p<\infty$; this yields a positive
answer in the $1<p<2$ case to a conjecture of Stein which asserts that for
$1<p<\infty$ we have that $C_d$ is of strong type $(p,p)$. A key ingredient in
this proof is the further extension of the {\it relational} time-frequency
perspective (introduced in \cite{q}) to the general polynomial phase.
