Positivity 13 (2009), 339–366
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020339-28, published online October 28, 2008
A bilinear oscillatory integral along parabolas
Dashan Fan and Xiaochun Li
Abstract. We establish an L
norm estimate for a bilinear
oscillatory integral operator along parabolas incorporating oscillatory factors
Mathematics Subject Classiﬁcation (2000). Primary 42B20, 42B25, Secondary
Keywords. Bilinear operator, oscillatory integral.
It is well-known that the Hilbert transform along curves:
f(x − ν(t))
is bounded on L
)for1<p<∞, where ν(t) is an appropriate curve in R
Among various curves, one simple model case is the parabola (t, t
) in the two
dimensional plane. This work was initiated by Fabes and Riviere  in order to
study the regularity of parabolic diﬀerential equations. A nice survey  on this
type of operators was written by Stein and Wainger. A lot of work on the Hilbert
transform along curves had been done in the last thirty years by many people.
Readers can ﬁnd some of them in [4,5,10,19]. The general results were established
in  for the singular Radon transforms and their maximal analogues over smooth
submanifolds of R
with some curvature conditions.
The cancellation condition of p.v.
plays an important role for obtaining L
boundedness of the Hilbert transform. However, this condition is not necessary if
there is an oscillatory factor e
(β>0) in the kernel (see [26,12,15]). Due to
the high oscillation of the factor e
estimates can be obtained for corre-
sponding operators with the kernel e
/|t|.In, Zielinski studied the following
The ﬁrst author was partially supported by NSF grant of China grant 10671079. The second
author was supported by NSF grant DMS-0456976.