Algebras of singular integral operators with kernels controlled by multiple norms

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