Multiple-hilbert Transforms Associated With Polynomials

To view the abstract go to http://www.ams.org/books/memo/1119. Cover 1 Title page 2 Chapter 1. Introduction 8 Chapter 2. Definitions of Polyhedra, Their Faces and Cones 12 2.1. Polyhedron 12 2.2. Faces of Polyhedron 14 2.3. A Cone (Dual Face) of Face 15 2.4. Generalized Newton Polyhedron 16 2.5. Basic Decompositions According to Faces and Cones 17 Chapter 3. Main Theorem and Background 20 3.1. Even Sets 20 3.2. Statement of Main Results 21 3.3. Background 24 Chapter 4. Combinatorial Lemmas 30 4.1. Elementary properties of faces and dual faces 30 4.2. Low Dimensional Polyhedron in Rn 31 4.3. Properties of Boundary and Interior of Face 33 4.4. Representations of Faces 34 4.5. Representations of Cones (Dual Faces) 36 4.6. Cone Decompositions 41 4.7. Projective Cone; Boundary Deleted Neighborhood 43 4.8. Representations of Unbounded Faces 47 4.9. Essential Faces 48 4.10. Invariance Property under Isomorphism 50 Chapter 5. Descending Faces vs. Ascending Cones 52 5.1. Construction of Descending Faces and Ascending Cones 53 5.2. Size Control Number 58 Chapter 6. Preliminary Results of Analysis 62 6.1. Preliminary Inequalities 62 6.2. Basic L^{p} estimates 63 Chapter 7. Proof of Sufficiency 66 7.1. Statement of Sufficiency Theorem 66 7.2. Main Estimate for Low Rank Case 67 7.3. Proof of Theorem 7.1 69 Chapter 8. Necessity Theorem 78 8.1. Transitivity Rule for Dual Faces 78 8.2. Lemma for Necessity 82 8.3. Necessity Theorem 84 Chapter 9. Preliminary Lemmas for Necessity Proof 88 9.1. Exclusiveness of Odd Sets 88 9.2. Scheme of Necessity Proof 96 Chapter 10. Proof of Necessity 100 10.1. Proof of Lemma 9.2 100 10.2. Proof of Lemma 9.3 104 Chapter 11. Proofs of Corollary 3.1 and Main Theorem 3.1 110 11.1. Proof of Corollary 3.1 110 11.2. Proof of Sufficient Part of Main Theorem 3.1 112 11.3. Proof of Necessity Part of Main Theorem 3.1 115 Chapter 12. Appendix 120 Bibliography 126 Back Cover 132 This volume includes definitions of polyhedra, their faces and cones; combinatorial lemmas; descending faces versus ascending cones; preliminary results of analysis; proof of sufficiency; necessity theorem' preliminary lemmas for necessity proof; proof of necessity; proofs of corollary 3.1 and main theorem 3.1.