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 Rn + . 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)| CP Λ ∞. 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 v
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