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Multiple Hilbert transforms associated with polynomials


About this Title

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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Table of Contents


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 with , and set the family of all vector polynomials,

Given , we consider a class of multi-parameter oscillatory singular integrals,

When , the integral for any is bounded uniformly in and . However, when , the uniform boundedness depends on each individual polynomial . In this paper, we fix and find a necessary and sufficient condition on that

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