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)

 
 

 

Commutators of linear and bilinear Hilbert transforms


Authors: Oscar Blasco and Paco Villarroya
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1997-2004
MSC (2000): Primary 42B20, 42B30; Secondary 45P05, 47H60
DOI: https://doi.org/10.1090/S0002-9939-03-07266-6
Published electronically: December 19, 2003
MathSciNet review: 2053971
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\alpha \in \mathbb{R}$, and let $H_\alpha(f,g)(x)=\frac{1}{\pi} p.v. \int f(x-t)g(x-\alpha t)\frac{dt}{t}$ and $Hf(x)= \frac{1}{\pi} p.v.\int f(x-t)\frac{dt}{t}$ denote the bilinear and linear Hilbert transforms, respectively. It is proved that, for $1<p<\infty$ and $\alpha_1\ne\alpha_2$, $H_{\alpha_1}-H_{\alpha_2}$ maps $L^p\times BMO$ into $L^{p}$ and it maps $BMO \times L^p$ into $L^{p}$ if and only if ${sign}(\alpha_1)={sign}(\alpha_2)$. It is also shown that, for $\alpha\le1$the commutator $[H_{\alpha,f},H]$ is bounded on $L^p$ for $1<p<\infty$ if and only if $f\in BMO$, where $H_{\alpha,f}(g)=H_\alpha(f,g)$.


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

  • 1. Calderón, A. P., Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA, Vol. 74, [1977], pp. 1324-1327. MR 57:6445
  • 2. Calderón, A. P., Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA, Vol. 53, [1965], pp. 1092-1099. MR 31:1575
  • 3. Coifman, R. R. and Meyer, Y., Fourier analysis of multilinear convolutions, Calderón's theorem and analysis of Lipschitz curves. Euclidean harmonic analysis (Proc. Sem. Univ. Maryland, College Park, Md.), Lecture Notes in Math., no. 779 [1979], pp. 104-122. MR 81g:42021
  • 4. Coifman, R. R., Rochberg, R., and Weiss, W., Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 [1976] pp. 611-635. MR 54:843
  • 5. Fefferman, C. and Stein, E. M., $H^{p}$ spaces of several variables. Acta Math. 129, [1972], pp. 137-193. MR 56:6263
  • 6. Grafakos, L. and Li, X., Uniform bounds for the bilinear Hilbert transform I, preprint.
  • 7. Grafakos, L. and Li, X., Uniform bounds for the bilinear Hilbert transform II, preprint.
  • 8. Jones, P., Bilinear singular integrals and maximal functions, Havin and Nikolski (eds.), Linear and Complex Analysis Problem Book 3, Part 1. Springer Lecture Notes in Math., no. 1573, [1994].
  • 9. Lacey, M. and Thiele, C., $L^p$estimates on the bilinear Hilbert transform for $2<p<\infty $. Ann. Math. 146, [1997], pp. 693-724. MR 99b:42014
  • 10. Lacey, M. and Thiele, C. M., Weak bounds for the bilinear Hilbert transform on $L^p$. Documenta Mathematica, extra volume ICM, [1997].
  • 11. Lacey, M. and Thiele, C., On Calderón's conjecture. Ann. Math. 149, no. 2 [1999], pp. 475-496. MR 2000d:42003
  • 12. Riesz, M., Sur les fonctions conjugués. Math. Z. 27, [1927], pp. 218-244.
  • 13. Stein, E. M., Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, [1970]. MR 44:7280
  • 14. Stein, E. M., Harmonic Analysis: real-variable methods, orthogonality and oscillatory integrals. Princeton Univ. Press, Princeton, NJ, [1993]. MR 95c:42002
  • 15. Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton, NJ, [1971]. MR 46:4102
  • 16. Thiele, C., On the bilinear Hilbert transform. Universitat Kiel, Habilitationsschrift [1998].

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42B20, 42B30, 45P05, 47H60

Retrieve articles in all journals with MSC (2000): 42B20, 42B30, 45P05, 47H60


Additional Information

Oscar Blasco
Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain
Email: Oscar.Blasco@uv.es

Paco Villarroya
Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain
Email: Paco.Villarroya@uv.es

DOI: https://doi.org/10.1090/S0002-9939-03-07266-6
Keywords: Bilinear Hilbert transform, commutators
Received by editor(s): February 25, 2002
Received by editor(s) in revised form: March 6, 2003, and March 15, 2003
Published electronically: December 19, 2003
Additional Notes: Both authors were supported by Proyecto PB98-0146 and BFM2002-04013-C02-01.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society