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Harmonic analysis on nilpotent groups and singular integrals. I: Oscillatory integrals. (English) Zbl 0622.42010

According to the authors, this paper is the first in a series of papers dealing with the following three interrelated problems:

1. The study of singular integrals which also carry oscillatory factors that are exponentials of imaginary polynomials. 2. The study of convolution operators with singular kernels that are supported on lower- dimensional submanifolds. 3. The extension of the properties of some basic operators to the general setting of arbitrary nilpotent Lie groups.

Connections between these three problems had been observed in several recent papers in particular cases, for example between the second problem and certain twisted convoltution operators. The papers of the series of the two authors are now establishing a more comprehensive theory of those three problems.

The present paper is devoted to the first problem. The basic operator considered here is the one given by

$\left(Tf\right)\left(x\right)=p·v·\phantom{\rule{1.em}{0ex}}{\int }_{{ℝ}^{n}}{e}^{iP\left(x,y\right)}K\left(x-y\right)f\left(y\right)dy,$

where K is a standard Calderón-Zygmund kernel on ${ℝ}^{n}$ and P a general real-valued polynomial. The main result is the boundedness of T on ${L}^{p}\left({ℝ}^{n}\right)$ for $1. Several extensions of this result are discussed.

Reviewer: D.Müller

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 22E30 Analysis on real and complex Lie groups