We establish $L^p$-boundedness for a class of product singular
integral operators on spaces $\widetilde{M} = M_1 \times M_2\times
\cdots \times M_n$. Each factor space $M_i$ is a smooth manifold
on which the basic geometry is given by a control, or
Carnot-Caratheodory, metric induced by a collection of vector
fields of finite type. The standard singular integrals on $M_i$
are non-isotropic smoothing operators of order zero. The
boundedness of the product operators is then a consequence of a
natural Littlewood-Paley theory on $\widetilde M$. This in turn is
a consequence of a corresponding theory on each factor space. The
square function for this theory is constructed from the heat
kernel for the sub-Laplacian on each factor.
Publié le : 2004-06-14
Classification:
product singular integrals,
control metrics,
NIS operators,
Littlewood-Paley theory,
42B20,
42B25
@article{1087482026,
author = {Nagel, Alexander and Stein, Elias M.},
title = {On the product theory of singular integrals},
journal = {Rev. Mat. Iberoamericana},
volume = {20},
number = {1},
year = {2004},
pages = { 531-561},
language = {en},
url = {http://dml.mathdoc.fr/item/1087482026}
}
Nagel, Alexander; Stein, Elias M. On the product theory of singular integrals. Rev. Mat. Iberoamericana, Tome 20 (2004) no. 1, pp. 531-561. http://gdmltest.u-ga.fr/item/1087482026/