Abstract
This article proves the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies earlier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane, a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding Lp-boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part in Part II, our subsequent article [11], using phase-plane analysis.
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References
Carleson, L. (1966). On convergence and growth of partial sums of Fourier series,Acta Math.,116, 135–157.
Coifman, R.R., Lions, P.L., Meyer, Y., and Semmes, S. (1993). Compensated compactness and Hardy spaces,J. Math. Pures Appl.,72, 247–286.
Coifman, R.R. and Meyer, Y. (1978). Au delà des opérateurs pseudo-differentiels,Astérisque,57, Société Mathématique de France.
Coifman, R.R. and Meyer, Y. (1975). On commutators of singular integrals and bilinear singular integrals,Trans. Am. Math. Soc.,212, 315–331.
Coifman, R.R. and Meyer, Y. (1991). Ondelettes et opérateurs III,Opérateurs Multilinéaires, Hermann, Ed., Paris.
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.
Fefferman, C. (1973). Pointwise convergence of Fourier series,Ann. of Math.,98, 551–571.
Folland, G.B. (1989).Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ.
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.
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.
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.).
Grafakos, L. and Torres, R.H. (1999). Multilinear Calderón-Zygmund theory, preprint.
Hogan, J.A. and Lakey, J.D. (1995). Extensions of the Heisenberg group by dilations and frames,Appl. Comp. Harm Anal.,2, 174–199.
Kenig, C.E. and Stein, E.M. (1999). Multilinear estimates and fractional integration,Math. Research Lett,6, 1–15.
Lacey, M. (2000). The bilinear maximal function maps inL p for 2/3<p≤1,Ann. of Math.,151 (2), 35–57.
Lacey, M. and Thiele, C. (1997).L p estimates on the bilinear Hilbert transform, 2<p<∞,Ann. of Math.,146, 683–724.
Lacey, M. and Thiele, C. (1999). On Calderón's conjecture,Annals of Math.,149, 475–496.
Meyer, Y. (1990). Ondelettes et opérateurs. II,Opérateurs de Calderón-Zygymund, Hermann, Ed., Paris.
Muscalu, C., Tao, T., and Thiele, C. (1999). Multilinear operators given by singular multipliers, preprint
Stein, E.M. (1993).Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ.
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Communicated by Y. Meyer
In memory of A.P. Calderón
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Gilbert, J.E., Nahmod, A.R. Bilinear operators with non-smooth symbol, I. The Journal of Fourier Analysis and Applications 7, 435–467 (2001). https://doi.org/10.1007/BF02511220
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DOI: https://doi.org/10.1007/BF02511220