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# memo_has_moved_text();Multiple Hilbert transforms associated with polynomials

Joonil Kim

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)
DOI: http://dx.doi.org/10.1090/memo/1119
Published electronically: January 21, 2015
Keywords:Multiple Hilbert transform, Newton polyhedron, face, cone

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Chapters

• Chapter 1. Introduction
• Chapter 2. Definitions of Polyhedra, Their Faces and Cones
• Chapter 3. Main Theorem and Background
• Chapter 4. Combinatorial Lemmas
• Chapter 5. Descending Faces vs. Ascending Cones
• Chapter 6. Preliminary Results of Analysis
• Chapter 7. Proof of Sufficiency
• Chapter 8. Necessity Theorem
• Chapter 9. Preliminary Lemmas for Necessity Proof
• Chapter 10. Proof of Necessity
• Chapter 11. Proofs of Corollary 3.1 and Main Theorem 3.1
• Chapter 12. Appendix

### Abstract

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,

Given $P_\Lambda \in \mathcal {P}_{\Lambda }$, we consider a class of multi-parameter oscillatory singular integrals,

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