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Multiple Hilbert transforms associated with polynomials

About this Title

Joonil Kim, Department of Mathematics, Yonsei University, Seoul 121, Korea

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 237, Number 1119
ISBNs: 978-1-4704-1435-1 (print); 978-1-4704-2505-0 (online)
Published electronically: January 21, 2015
Keywords: Multiple Hilbert transform, Newton polyhedron, face, cone, oscillatory singular integral
MSC: Primary 42B20, 42B25

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Table of Contents


  • 1. Introduction
  • 2. Definitions of Polyhedra, Their Faces and Cones
  • 3. Main Theorem and Background
  • 4. Combinatorial Lemmas
  • 5. Descending Faces vs. Ascending Cones
  • 6. Preliminary Results of Analysis
  • 7. Proof of Sufficiency
  • 8. Necessity Theorem
  • 9. Preliminary Lemmas for Necessity Proof
  • 10. Proof of Necessity
  • 11. Proofs of Corollary and Main Theorem
  • 12. Appendix


Let $\Lambda =(\Lambda _1,\cdots ,\Lambda _d)$ with $\,\Lambda _\nu \subset \mathbb {Z}_+^n\,$, and set $\mathcal {P}_{\Lambda }$ the family of all vector polynomials, \[ \mathcal {P}_{\Lambda }=\left \{P_{\Lambda }: P_{\Lambda }(t)=\left (\sum _{\mathfrak {m}\,\in \Lambda _{1}}\,c_{\mathfrak {m}}^{1}\,t^{\mathfrak {m}},\cdots ,\sum _{\mathfrak {m}\,\in \Lambda _{d}}\,c_{\mathfrak {m}}^{d}\,t^{\mathfrak {m}}\right )\ \quad \text {with}\quad t\in \mathbb {R}^n\right \}.\] Given $P_\Lambda \in \mathcal {P}_{\Lambda }$, we consider a class of multi-parameter oscillatory singular integrals, \[ \mathcal {I}(P_\Lambda ,\xi ,r)=\text {p.v.}\int _{\prod [-r_j,r_j]}e^{i\langle \xi ,P_{\Lambda }(t)\rangle } \frac {dt_1}{t_1}\cdots \frac {dt_n}{t_n} \quad \text {where} \quad \xi \in \mathbb {R}^d, r\in \mathbb {R}_+^n.\] When $n=1$, the integral $\mathcal {I}(P_\Lambda ,\xi ,r)$ for any $P_\Lambda \in \mathcal {P}_\Lambda$ is bounded uniformly in $\xi$ and $r$. However, when $n\ge 2$, the uniform boundedness depends on each individual polynomial $P_\Lambda$. In this paper, we fix $\Lambda$ and find a necessary and sufficient condition on $\Lambda$ that \[ \text {for all} \quad P_\Lambda \in \mathcal {P}_\Lambda , \ \ \sup _{\xi , \, r} |\mathcal {I}(P_\Lambda ,\xi ,r)|\le C_{P_\Lambda }<\infty .\]

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