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Full Description
The authors study algebras of singular integral operators on $\mathbb R^n$ and nilpotent Lie groups that arise when considering the composition of Calderon-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 Calderon-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
Introduction
The Classes $\mathcal P(\mathbf E)$ and $\mathcal M(\mathbf E)$
Marked partitions and decompositions of $\mathbb R^N$
Fourier transform duality of kernels and multipliers
Dyadic sums of Schwartz functions
Decomposition of multipliers and kernels
The rank of $\mathbf E$ and integrability at infinity
Convolution operators on homogeneous nilpotent Lie groups
Composition of operators
Convolution of Calderon-Zygmund kernels
Two-flag kernels and multipliers
Extended kernels and operators
The role of pseudo-differential operators
Appendix I: Properties of cones $\Gamma (\mathbf A)$
Appendix II: Estimates for homogeneous norms
Appendix III: Estimates for geometric sums
Bibliography.
