Let Λ = (Λ1, · · · , Λd) with Λν ⊂ Z+
, and set PΛ the family of all vector
PΛ = PΛ : PΛ(t) =
· · · ,
with t ∈
Given PΛ ∈ PΛ, we consider a class of multi-parameter oscillatory singular integrals,
I(PΛ,ξ,r) = p.v.
· · ·
where ξ ∈
When n = 1, the integral I(PΛ,ξ,r) for any PΛ ∈ PΛ 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 ∈ PΛ, sup
|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.
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
Author aﬃliation: Department of Mathematics, Yonsei University, Seoul 120-749, Korea.
c 2015 American Mathematical Society