Abstract
Let Λ = (Λ1, · · · , Λd) with Λν Z+
n
, and set the family of all vector
polynomials,
= : PΛ(t) =
m ∈Λ1
cm
1 tm,
· · · ,
m ∈Λd
cm
d tm
with t
Rn
.
Given PΛ, we consider a class of multi-parameter oscillatory singular integrals,
I(PΛ,ξ,r) = p.v.
[−rj,rj ]
ei ξ,PΛ(t)
dt1
t1
· · ·
dtn
tn
where ξ
Rd,r

R+.n
When n = 1, the integral I(PΛ,ξ,r) for any is bounded uniformly in ξ
and r. However, when n 2, the uniform boundedness depends on each individual
polynomial PΛ. In this paper, we fix Λ and find a necessary and sufficient condition
on Λ that
for all PΛ, sup
ξ, r
|I(PΛ,ξ,r)| CPΛ ∞.
Received by the editor February 7, 2013 and, in revised form, August 18, 2013.
Article electronically published on January 21, 2015.
DOI: http://dx.doi.org/10.1090/memo/1119
2010 Mathematics Subject Classification. Primary 42B20, 42B25.
Key words and phrases. Multiple Hilbert transform, Newton polyhedron, face, cone, oscilla-
tory singular integral.
This research was supported by the Basic Science Research Program through the National
Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2012R 1A
1A2006774).
Author affiliation: Department of Mathematics, Yonsei University, Seoul 120-749, Korea.
E-mail: jikim7030@yonsei.ac.kr.
c 2015 American Mathematical Society
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