Bilinear operators with non-smooth symbol. I. (English) Zbl 0994.42014

A bilinear operator that maps pairs of Schwartz functions to Schwartz distributions and commutes with simultaneous translations can be written in terms of its multiplier \(m(\xi ,\eta)\) as \[ B(f,g)(x)=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }m(\xi ,\eta)\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi ix(\xi +\eta)} d\xi d\eta. \] While boundedness of operators for which \(m\) is smooth away from the origin and satisfies symbol estimates can be proven by more or less standard methods of Calderón-Zygmund theory, multipliers with singularities along other sets require new techniques. The symbol \(m(\xi ,\eta)=\chi _{\{\xi <\eta \}}\) of the bilinear Hilbert transform is prototypical. Here, the authors prove the following extension to ‘cone operators’ of Lacey and Thiele’s theorem on boundedness of the bilinear Hilbert transform [M. Lacey and C. Thiele, Ann. Math. 149, No. 2, 475-496 (1999; Zbl 0934.42012)] : Let \(\Gamma \) be a one-sided cone with vertex at the origin and suppose that, inside \(\Gamma \), \(m\) satisfies the symbol estimates \(|D^{\alpha }m(\xi ,\eta)|\leq C_{\alpha }\text{dist}((\xi ,\eta),\Gamma)^{-|\alpha |}\) (\(|\alpha |\geq 0\)). Then the operator \(B\) defined as above with multiplier \(m(\xi ,\eta)\chi _{\Gamma }(\xi ,\eta)\) is bounded from \(L^{p}\times L^{q}\) to \(L^{r}\) where \(1/p+1/q=1/r<3/2\), provided no edge of \(\Gamma \) lies on the diagonal \(\xi +\eta =0\) or on a coordinate axis. The authors also show that the range of \(B\) lies in the corresponding complex Hardy subspace of \(L^{r}\) when \(r\geq 1\) and \(\Gamma \) strictly lies inside \(\xi +\eta >0\). These facts follow from a boundedness result for generalized paraproduct type operators built from modulated wavelets or wave-packets. The paraproduct bound is proved in a companion paper in the same journal. In the present paper the reduction of the cone operator to a discretized modulated paraproduct is proved. Furthermore, \( L^{p}\)-boundedness is proved for analogues of standard paraproducts having the form \[ (c,f,g)\mapsto \sum_{k,l=-\infty }^{\infty }c_{kl}2^{jk/2}\left\langle f,\psi _{jkl}^{(1)}\right\rangle \left\langle g,\psi _{jkl}^{(3)}\right\rangle \psi _{jkl}^{(3)} \] where \(\psi _{jkl}^{(i)}(x)=2^{jk/2}\psi ^{(i)}(2^{jk}x-la_{i})\). This operator is bounded – with a bound depending on the shifts \(a_{i}\) – from \( l^{\infty }\times L^{p}\times L^{q}\) to \(L^{r}\) provided \(1/r=1/p+1/q<2\). Here one assumes at least two of the \(\psi ^{(i)\text{ }}\) have vanishing moment. The full paraproduct result assumes that the \(\widehat{\psi}^{(i)}\) have disjoint supports. The techniques used are substantially different from those of C. Muscalu, T. Tao and C. Thiele [J. Am. Math. Soc. 15, No. 2, 469-496 (2002; following review)] who independently generalized the \(L^{p}\) result – though not the Hardy space result – to the case \(n\geq 2\).


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42B15 Multipliers for harmonic analysis in several variables
42B25 Maximal functions, Littlewood-Paley theory
47G10 Integral operators
47H60 Multilinear and polynomial operators
Full Text: DOI EuDML


[1] Carleson, L. (1966). On convergence and growth of partial sums of Fourier series,Acta Math.,116, 135–157. · Zbl 0144.06402
[2] Coifman, R.R., Lions, P.L., Meyer, Y., and Semmes, S. (1993). Compensated compactness and Hardy spaces,J. Math. Pures Appl.,72, 247–286. · Zbl 0864.42009
[3] Coifman, R.R. and Meyer, Y. (1978). Au delà des opérateurs pseudo-differentiels,Astérisque,57, Société Mathématique de France.
[4] Coifman, R.R. and Meyer, Y. (1975). On commutators of singular integrals and bilinear singular integrals,Trans. Am. Math. Soc.,212, 315–331. · Zbl 0324.44005
[5] Coifman, R.R. and Meyer, Y. (1991). Ondelettes et opérateurs III,Opérateurs Multilinéaires, Hermann, Ed., Paris.
[6] Daubechies, I., Landau, H.J., and Landau, Z. (1995). Gabor time-frequency lattices and the Wexler-Raz identity,J. Fourier Anal. Appl.,1, 437–478. · Zbl 0888.47018
[7] Fefferman, C. (1973). Pointwise convergence of Fourier series,Ann. of Math.,98, 551–571. · Zbl 0268.42009
[8] Folland, G.B. (1989).Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ. · Zbl 0682.43001
[9] Gilbert, J.E., Han, Y.S., Hogan, J.A., Lakey, J.D., Weiland, D., and Weiss, G. (1998). Calderón reproducing formula and frames with applications to singular integrals, preprint.
[10] Gilbert, J.E. and Nahmod, A.R. (1999). Hardy spaces and a Walsh model for bilinear cone operators,Trans. Am. Math. Soc.,351, 3267–3300. · Zbl 0919.42009
[11] Gilbert, J.E. and Nahmod, A.R. (1999).L p -boundedness of time-frequency paraproducts, II, preprint, to appear in JFAA (Journal of Fourier Anal. Appl.).
[12] Grafakos, L. and Torres, R.H. (1999). Multilinear Calderón-Zygmund theory, preprint.
[13] Hogan, J.A. and Lakey, J.D. (1995). Extensions of the Heisenberg group by dilations and frames,Appl. Comp. Harm Anal.,2, 174–199. · Zbl 0840.43017
[14] Kenig, C.E. and Stein, E.M. (1999). Multilinear estimates and fractional integration,Math. Research Lett,6, 1–15. · Zbl 0952.42005
[15] Lacey, M. (2000). The bilinear maximal function maps inL p for 2/3<p,Ann. of Math.,151 (2), 35–57. · Zbl 0967.47031
[16] Lacey, M. and Thiele, C. (1997).L p estimates on the bilinear Hilbert transform, 2<p<Ann. of Math.,146, 683–724. · Zbl 0914.46034
[17] Lacey, M. and Thiele, C. (1999). On Calderón’s conjecture,Annals of Math.,149, 475–496. · Zbl 0934.42012
[18] Meyer, Y. (1990). Ondelettes et opérateurs. II,Opérateurs de Calderón-Zygymund, Hermann, Ed., Paris.
[19] Muscalu, C., Tao, T., and Thiele, C. (1999). Multilinear operators given by singular multipliers, preprint · Zbl 0994.42015
[20] Stein, E.M. (1993).Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ. · Zbl 0821.42001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.