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Algebras of Singular Integral Operators with Kernels Controlled by Multiple Norms

About this Title

Alexander Nagel, University of Wisconsin-Madison, Madison, Wisconsin 53706, Fulvio Ricci, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Elias M. Stein, Princeton University, Princeton, New Jersey 08544 and Stephen Wainger, University of Wisconsin-Madison, Madison, Wisconsin 53706

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 256, Number 1230
ISBNs: 978-1-4704-3438-0 (print); 978-1-4704-4923-0 (online)
DOI: https://doi.org/10.1090/memo/1230
Published electronically: September 24, 2018
Keywords: Singular integral operators, Multiplier operators, Multi-parameter theory, Pseudo-differential operators, Homogeneous nilpotent Lie groups, Flag kernels
MSC: Primary 42B20, 35S05

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Table of Contents

Chapters

  • 1. Introduction
  • 2. The Classes $\mathcal P(\mathbf E)$ and $\mathcal M({\mathbf E})$

Abstract

We 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 $1<p<\infty$. While the usual class of Calderón-Zygmund operators is invariant under a one-parameter family of dilations, the operators we study fall outside this class, and reflect a multi-parameter structure.