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.
General
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