Algebras of singular integral operators with kernels controlled by multiple norms / Alexander Nagel [and three others].

Format:

Book

Author/Creator:

Nagel, Alexander, 1945- author.

Contributor:

Ricci, Fulvio, 1948-

Stein, Elias M., 1931-2018.

Wainger, Stephen, 1936-

Series:

Memoirs of the American Mathematical Society ; Number 1230.

Memoirs of the American Mathematical Society ; Number 1230

Language:

English

Subjects (All):

Integral operators.

Singular integrals.

Algebra.

Kernel functions.

Physical Description:

1 online resource (vii, 141 pages).

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2018]

Summary:

The authors study algebras of singular integral operators on \mathbb R^n and nilpotent Lie groups that arise when considering the composition of Calderón-Zygmund operators with different homogeneities, such as operators occuring in sub-elliptic problems and those arising in elliptic problems. These algebras are characterized in a number of different but equivalent ways: in terms of kernel estimates and cancellation conditions, in terms of estimates of the symbol, and in terms of decompositions into dyadic sums of dilates of bump functions. The resulting operators are pseudo-local and bounded on L^p for 1 \lt p \lt \infty . While the usual class of Calderón-Zygmund operators is invariant under a one-parameter family of dilations, the operators studied here fall outside this class, and reflect a multi-parameter structure.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Background

1.2. Some motivating examples

1.3. Outline of results

Chapter 2. The Classes \PP(\EEE) and \MM(\EEE)

2.1. Notation

2.2. Global norms

2.3. Classes of distributions and multipliers

Chapter 3. Marked partitions and decompositions of \R^{ }

3.1. Dominant terms in ⱼ(\t) and ̂ ⱼ(\t)

3.2. Marked partitions and the sets _{ } and ̂ _{ }

3.3. Characterizing the sets _{ }^{ } and ̂ _{ }^{ }

3.4. Estimates of kernels and multipliers on _{ } and ̂ _{ }

3.5. A coarser decomposition of \R^{ } associated to ∈\SS( )

Chapter 4. Fourier transform duality of kernels and multipliers

4.1. Fourier transforms of multipliers

4.2. Fourier transforms of kernels

Chapter 5. Dyadic sums of Schwartz functions

5.1. Cones associated to a matrix \EEE

5.2. Inclusions among classes of kernels associated to different matrices

5.3. Coarser decompositions and lower dimensional matrices

5.4. Size estimates

5.5. Cancellation properties

5.6. Dyadic sums with weak cancellation

Chapter 6. Decomposition of multipliers and kernels

6.1. New matrices \EEE_{ }

6.2. Road map for the dyadic decomposition

6.3. Partitions of unity

6.4. Dyadic decomposition of a multiplier

6.5. Dyadic decomposition of a kernel

Chapter 7. The rank of \EEE and integrability at infinity

7.1. The rank 1 case: \CZ kernels

7.2. Higher rank and integrability at infinity

7.3. Higher rank and weak-type estimates near zero

Chapter 8. Convolution operators on homogeneous nilpotent Lie groups

8.1. Convolution of scaled bump functions

compatibility of dilations and convolution

8.2. Automorphic flag kernels and ^{ }-boundedness of convolution operators

Chapter 9. Composition of operators

9.1. A preliminary decomposition.

9.2. Reduction to the case of finite sets

9.3. Properties of the convolution [ ^{ }_{ }]_{ }*[ ^{ }_{ }]_{ }

9.4. A further decomposition of Γ_{\Z}(\EEE_{ })×Γ_{\Z}(\EEE_{ })

9.5. Fixed and free indices

9.6. A finer decomposition of \R^{ }

9.7. The matrix \EEE_{ , , }

9.8. The class \PP₀(\EEE_{ , , })

9.9. Estimates for [ ]_{ }*[ ]_{ }

9.10. Proof of Theorem 9.2

Chapter 10. Convolution of Calderón-Zygmund kernels

10.1. \CZ kernels

10.2. A general convolution theorem

10.3. Convolution of two \CZ kernels

Chapter 11. Two-flag kernels and multipliers

11.1. Flag kernels and multipliers

11.2. Pairs of opposite flags

11.3. Two-step flags

11.4. General two-flag kernels

11.5. Proof of Lemma 11.6

Chapter 12. Extended kernels and operators

12.1. The ^{ } boundedness of the operators

12.2. The algebra of operators

Chapter 13. The role of pseudo-differential operators

13.1. The isotropic extended kernels

13.2. Proof of Theorem 13.1

13.3. The space ^{ }₁

Chapter 14. Appendix I: Properties of cones Γ(\AAA)

14.1. Optimal inequalities and the basic hypothesis

14.2. Partial matrices

14.3. Projections

14.4. The dimension of Γ(\EEE)

14.5. The reduced matrix \EEE^{♭}

Chapter 15. Appendix II: Estimates for homogeneous norms

Chapter 16. Appendix III: Estimates for geometric sums

Bibliography

Index of Symbols

Back Cover.

Notes:

Includes bibliographical references and index of symbols.

Description based on print version record.

ISBN:

1-4704-4923-4