Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Oscillatory singular integrals
on $L^{p}$ and Hardy spaces

Author: Yibiao Pan
Journal: Proc. Amer. Math. Soc. 124 (1996), 2821-2825
MSC (1991): Primary 42B20
MathSciNet review: 1328369
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider boundedness properties of oscillatory singular integrals on $L^{p}$ and Hardy spaces. By constructing a phase function, we prove that $H^{1}$ boundedness may fail while $L^{p}$ boundedness holds for all $p \in (1, \infty )$. This shows that the $L^{p}$ theory and $H^{1}$ theory for such operators are fundamentally different.

References [Enhancements On Off] (What's this?)

  • 1. Carlsson, H., et al., $L^{p}$ estimates for maximal functions and Hilbert transforms along flat convex curves in ${\mathbf {R}^{\mathbf {2}}}$, Bull. Amer. Math. Soc. 14 (1986), 263--267. MR 87f:42044
  • 2. Chanillo, S., and Christ, M., Weak (1,1) bounds for oscillatory singular integrals, Duke Math. Jour. 55 (1987), 141--155. MR 88h:42015
  • 3. Hu, Y., and Pan, Y., Boundedness of oscillatory singular integrals on Hardy spaces, Ark. Mat. 30 (1993), 311--320. CMP 94:16
  • 4. Nagel, A., Vance, J., Wainger, S., and Weinberg, D., Hilbert transforms for convex curves, Duke Math. Jour. 50 (1983), 735--744. MR 85a:42025
  • 5. Nagel, A., and Wainger, S., Hilbert transforms associated with plane curves, Trans. Amer. Math. Soc. 223 (1976), 235--252. MR 54:10994
  • 6. Pan, Y., Uniform estimates for oscillatory integral operators, Jour. Func. Anal. 100 (1991), 207--220. MR 93f:42034
  • 7. ------, Boundedness of Oscillatory singular integrals on Hardy spaces: II, Indiana Univ. Math. Jour 41 (1992), 279--293. CMP 92:11
  • 8. ------, $H^{1}$ boundedness of oscillatory singular integrals with degenerate phase functions, Math. Proc. Camb. Phil. Soc. 116 (1994), 353--358. MR 95c:42019
  • 9. Phong, D.H., and Stein, E.M., Hilbert integrals, singular integrals and Radon transforms I, Acta Math. 157 (1986), 99--157. MR 88i:42028a
  • 10. Ricci, F., and Stein, E.M., Harmonic analysis on nilpotent groups and singular integrals, I, Jour. Func. Anal. 73 (1987), 179--194. MR 88g:42023
  • 11. Stein, E.M., Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, 1986. MR 88g:42022
  • 12. ------, Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, NJ, 1993. MR 95c:42002
  • 13. Stein, E.M., and Wainger, S., Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), 1239--1295. MR 80k:42023
  • 14. Zygmund, A., Trigonometric series, Cambridge Univ. Press, Cambridge, 1959. MR 21:6498

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42B20

Retrieve articles in all journals with MSC (1991): 42B20

Additional Information

Yibiao Pan
Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Received by editor(s): November 15, 1994
Received by editor(s) in revised form: March 25, 1995
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society