Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-30T00:27:54.507Z Has data issue: false hasContentIssue false
  • English
  • Français

H1 boundedness of oscillatory singular integrals with degenerate phase functions

Published online by Cambridge University Press:  24 October 2008

Yibiao Pan
Yibiao Pan
Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh PA 15260, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the uniform boundedness of oscillatory singular integral operators with degenerate phase functions on the Hardy space H1. The H1 boundedness was previously known when the phase function is nondegenerate. Here we obtain a sufficient condition for H1 boundedness which allows the phase function vanishing to infinite order.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 353 - 358
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Carbery, A. et al. Operators associated to flat plane curves: Lp estimates via dilation methods. Duke Math. J. 59 (1989), 675–700.CrossRefGoogle Scholar
[2]Carlsson, H. et al. Lp estimates for maximal functions and Hilbert transforms along flat convex curves in R2. Bull. Amer. Math. Soc. 14 (1986), 263–267.CrossRefGoogle Scholar
[3]Chanillo, S. and Cheist, M.. Weak (1,1) bounds for oscillatory singular integrals. Duke Math. J. 55 (1987), 141–155.CrossRefGoogle Scholar
[4]Coifman, R.. A real variable characterization ofHp. Studia Math. 51 (1974), 269–274.CrossRefGoogle Scholar
[5]Fefferman, C. and Stein, E. M.. Hp spaces of several variables. Ada Math. 129 (1972), 137–193.Google Scholar
[6]Hu, Y. and Pan, Y.. Boundedness of oscillatory singular integrals on Hardy spaces. Ark. Mat. 30 (1992), 311–320.CrossRefGoogle Scholar
[7]Nagel, A., Vance, J., Wainger, S. and Weinberg, D.. Hilbert transforms for convex curves. Duke Math. J. 50 (1983), 735–744.CrossRefGoogle Scholar
[8]Nagel, A. and Wainger, S.. Hilbert transforms associated with plane curves. Trans. Amer. Math. Soc. 223 (1976), 235–252.CrossRefGoogle Scholar
[9]Pan, Y.. Uniform estimates for oscillatory integral operators. J. Func. Anal. 100 (1991), 207–220.CrossRefGoogle Scholar
[10]Pan, Y.. Hardy spaces and oscillatory singular integrals. Rev. Mat. Iberoamericana 7 (1991), 55–64.CrossRefGoogle Scholar
[11]Pan, Y.. Boundedness of oscillatory singular integrals on Hardy spaces: II. Indiana U. Math. J. 41 (1992), 279–293.CrossRefGoogle Scholar
[12]Phong, D. H. and Stein, E. M.. Hilbert integrals, singular integrals and Radon transforms I. Ada Math. 157 (1986), 99–157.Google Scholar
[13]Ricci, F. and Stein, E. M.. Harmonic analysis on nilpotent groups and singular integrals. I. J. Func. Anal. 73 (1987), 179–194.CrossRefGoogle Scholar
[14]Sjölin, P.. Convolution with oscillating kernels on Hp spaces. J. London Math. Soc. 23 (1981), 442–454.Google Scholar
[15]Stein, E. M.. Oscillatory integrals in Fourier analysis. In Beijing Lectures in Harmonic Analysis (Princeton University Press, 1986).Google Scholar
[16]Stein, E. M.. Singular Integrals and Differentiability Properties of Functions (Princeton University Press, 1970).Google Scholar