Revista Matemática Iberoamericana

Uniform Bounds for the Bilinear Hilbert Transforms, II

Xiaochun Li

Full-text: Open access


We continue the investigation initiated in [Grafakos and Li: Uniform bounds for the bilinear Hilbert transforms, I. Ann. of Math. (2) 159 (2004), 889-933] of uniform $L^{p}$ bounds for the family of bilinear Hilbert transforms $$ H_{\alpha,\beta} (f,g)(x) = \text{p.v.} \displaystyle\int_{\mathbb{R}} f(x-\alpha t) g(x-\beta t) \frac{dt}{t} \,. $$ In this work we show that $H_{\alpha,\beta}$ map $L^{p_1}(\mathbb R)\times L^{p_2}(\mathbb R)$ into $L^p(\mathbb R)$ uniformly in the real parameters $\alpha$, $\beta$ satisfying $|\frac{\alpha}{\beta}-1|\ge c > 0$ when $1 < p_1, p_2 < 2$ and $\frac{2}{3} < p= \frac{p_1p_2}{p_1+p_2} < \infty$. As a corollary we obtain $L^p \times L^\infty \to L^p$ uniform bounds in the range $4/3 < p < 4 $ for the $H_{1,\alpha}$'s when $\alpha\in [0,1)$.

Article information

Rev. Mat. Iberoamericana, Volume 22, Number 3 (2006), 1069-1126.

First available in Project Euclid: 22 January 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46B70: Interpolation between normed linear spaces [See also 46M35] 47B38: Operators on function spaces (general)

time-frequency analysis bilinear Hilbert transform uniform bounds


Li, Xiaochun. Uniform Bounds for the Bilinear Hilbert Transforms, II. Rev. Mat. Iberoamericana 22 (2006), no. 3, 1069--1126.

Export citation


  • Calderón, A.: Commutators of singular integral operators. Proc. Nat. Acad. Sci. U.S.A. 53 (1977), 1092-1099.
  • Calderón, A. P. and Zygmund, A.: On singular integrals. Amer. J. Math. 78 (1956), 289-309.
  • Carleson, L.: On convergence and growth of partials sumas of Fourier series. Acta Math. 116 (1966), 135-157.
  • Coifman, R. R. and Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212 (1975), 315-331.
  • Coifman, R. R. and Meyer, Y.: Commutateurs d' intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28 (1978), 177-202.
  • Fefferman, C.: Pointwise convergence of Fourier series. Ann. of Math. (2) 98 (1973), 551-571.
  • Fefferman, C. and Stein, E. M.: Some maximal inequalities. Amer. J. Math. 93 (1971), 107-115.
  • Grafakos, L. and Li, X.: Uniform bounds for the bilinear Hilbert transforms. I. Ann. of Math. (2) 159 (2004), 889-933.
  • Grafakos, L. and Torres, R.: Multilinear Calderón-Zygmund theory. Adv. Math. 165 (2002), no. 1, 124-164.
  • Hunt R. A.: On the convergence of Fourier Series. In 1968 Orthogonal Expansions and their Continuous Analogues (Proc. Conf. Edwardsville, 1967), 235-255. Southern Illinois Univ. Press, Carbondale Ill.
  • Lacey, M. T. and Thiele, C. M.: $L^p$ estimates on the bilinear Hilbert transform for $2<p<\infty$. Ann. of Math. (2) 146 (1997), 693-724.
  • Lacey, M. T. and Thiele, C. M.: On Calderón's conjecture. Ann. of Math. (2) 149 (1999), 475-496.
  • Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.
  • Thiele, C. M.: On the Bilinear Hilbert Transform. Habilitationsschrift, Universität Kiel, 1998.
  • Thiele, C. M.: A uniform estimate. Ann. of Math. (2) 156 (2002), 519-563.