Algebras of singular integral operators with kernels controlled by multiple norms

✍ Scribed by Nagel A., Stein E.M

• Publisher: American Mathematical Society
• Year: 2018
• Tongue: English
• Leaves: 156
• Series: Memoires of American Mathematical Society 1230
• Category: Library

✦ Table of Contents

Cover......Page 1
Title page......Page 2
1.1. Background......Page 10
1.2. Some motivating examples......Page 11
1.3. Outline of results......Page 13
2.1. Notation......Page 20
2.2. Global norms......Page 22
2.3. Classes of distributions and multipliers......Page 23
3.1. Dominant terms in ��ⱼ(\t) and ̂��ⱼ(\t)......Page 26
3.2. Marked partitions and the sets ��{��} and ̂��{��}......Page 27
3.3. Characterizing the sets ��{��}^{��} and ̂��{��}^{��}......Page 28
3.4. Estimates of kernels and multipliers on ��{��} and ̂��{��}......Page 31
3.5. A coarser decomposition of \R^{��} associated to ��∈\SS(��)......Page 32
4.1. Fourier transforms of multipliers......Page 34
4.2. Fourier transforms of kernels......Page 38
5.1. Cones associated to a matrix \EEE......Page 42
5.2. Inclusions among classes of kernels associated to different matrices......Page 43
5.3. Coarser decompositions and lower dimensional matrices......Page 44
5.4. Size estimates......Page 46
5.5. Cancellation properties......Page 47
5.6. Dyadic sums with weak cancellation......Page 50
6.1. New matrices \EEE_{��}......Page 54
6.2. Road map for the dyadic decomposition......Page 55
6.3. Partitions of unity......Page 56
6.4. Dyadic decomposition of a multiplier......Page 59
6.5. Dyadic decomposition of a kernel......Page 61
7.1. The rank 1 case: \CZ kernels......Page 66
7.2. Higher rank and integrability at infinity......Page 68
7.3. Higher rank and weak-type estimates near zero......Page 70
8.1. Convolution of scaled bump functions; compatibility of dilations and convolution......Page 72
8.2. Automorphic flag kernels and ��^{��}-boundedness of convolution operators......Page 75
Chapter 9. Composition of operators......Page 78
9.2. Reduction to the case of finite sets......Page 81
9.3. Properties of the convolution [��^{��}{��}]{��}[��^{��}{��}]{��}......Page 82
9.4. A further decomposition of Γ_{\Z}(\EEE_{��})×Γ_{\Z}(\EEE_{��})......Page 85
9.5. Fixed and free indices......Page 88
9.6. A finer decomposition of \R^{��}......Page 90
9.7. The matrix \EEE_{��,��,��}......Page 91
9.8. The class \PP₀(\EEE_{��,��,��})......Page 93
9.9. Estimates for [��]_{��}[��]_{��}......Page 95
9.10. Proof of Theorem 9.2......Page 98
10.2. A general convolution theorem......Page 100
10.3. Convolution of two \CZ kernels......Page 101
11.1. Flag kernels and multipliers......Page 108
11.2. Pairs of opposite flags......Page 109
11.3. Two-step flags......Page 110
11.4. General two-flag kernels......Page 112
11.5. Proof of Lemma 11.6......Page 115
12.1. The ��^{��} boundedness of the operators......Page 118
12.2. The algebra of operators......Page 121
Chapter 13. The role of pseudo-differential operators......Page 124
13.1. The isotropic extended kernels......Page 125
13.2. Proof of Theorem 13.1......Page 127
13.3. The space ��^{��}₁......Page 130
14.1. Optimal inequalities and the basic hypothesis......Page 132
14.2. Partial matrices......Page 133
14.3. Projections......Page 134
14.4. The dimension of Γ(\EEE)......Page 135
14.5. The reduced matrix \EEE^{♭}......Page 137
Chapter 15. Appendix II: Estimates for homogeneous norms......Page 140
Chapter 16. Appendix III: Estimates for geometric sums......Page 144
Bibliography......Page 148
Index of Symbols......Page 150
Back Cover......Page 156