Ricci, Fulvio; Stein, Elias M. Harmonic analysis on nilpotent groups and singular integrals. I: Oscillatory integrals. (English) Zbl 0622.42010 J. Funct. Anal. 73, 179-194 (1987). 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 \[ (Tf)(x)=p.v.\quad \int_{{\mathbb{R}}^ n}e^{iP(x,y)}K(x-y)f(y)dy, \] where K is a standard Calderón-Zygmund kernel on \({\mathbb{R}}^ n\) and P a general real-valued polynomial. The main result is the boundedness of T on \(L^ p({\mathbb{R}}^ n)\) for \(1<p<\infty\). Several extensions of this result are discussed. Reviewer: D.Müller Cited in 9 ReviewsCited in 145 Documents MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 22E30 Analysis on real and complex Lie groups Keywords:singular integrals; oscillatory factors; convolution operators; Calderón-Zygmund kernel × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Coifman, R.; Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51, 241-250 (1979) · Zbl 0291.44007 [2] David, G.; Journé, J. L., A boundedness criterion for generalized Calderón-Zygmund operators, Ann. Math., 120, 371-397 (1984) · Zbl 0567.47025 [3] Geller, D.; Stein, E. M., Estimates for singular convolution operators on the Heisenberg group, Math. Ann., 267, 1-15 (1984) · Zbl 0537.43005 [4] Mauceri, G.; Picardello, M. A.; Ricci, F., Twisted convolution, Hardy spaces, and Hörmander multipliers, Suppl. Rend. Circ. Mat. Palermo, 1, 191-202 (1981) · Zbl 0472.43006 [5] Müller, D., Singular kernels supported by homogeneous sub-manifolds, J. Rein. Angew. Math., 356, 90-118 (1985) · Zbl 0551.43005 [6] Phong, D. H.; Stein, E. M., Singular integrals related to the Radon transform and boundary value problems, (Proc. Nat. Acad. U.S.A., 80 (1983)), 7697-7710 · Zbl 0567.42010 [7] Phong, D. H.; Stein, E. M., Hilbert integrals, singular integrals and Radon transforms I, Acta. Math., 157, 99-157 (1986) · Zbl 0622.42011 [8] Ricci, F.; Stein, E. M., Oscillatory singular integrals and harmonic analysis on nilpotent groups, (Proc. Nat. Acad. Sci. U.S.A., 83 (1986)), 1-3 · Zbl 0583.43010 [9] Sjölin, P., Convolution with oscillating kernels, Indiana Univ. Math. J., 30, 47-56 (1981) · Zbl 0419.47020 [10] Stein, E. M., Oscillatory integrals in Fourier analysis, Ann. Math. Stud., 112 (1986) · Zbl 0618.42006 [11] Stein, E. M.; Waingner, S., Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., 84, 1239-1295 (1978) · Zbl 0393.42010 [12] Strichartz, R. S., Singular integrals supported on sub-manifolds, Studia Math., 74, 137-151 (1982) · Zbl 0501.43007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.