Abstract
Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bényi, Á. Bilinear singular integrals and pseudodifferential operators, PhD Thesis, University of Kansas, Lawrence, (2002).
Bényi, Ä. Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces,J. Math. Anal. Appl. 284, 97–103, (2003).
Bényi, Á., Gröchenig, K., Heil, C., and Okoudjou, K. Modulation spaces and a class of bounded multilinear pseudodifferential operators,J. Operator Theory to appear.
Bényi, Á. and Torres, R. H. Symbolic calculus and the transposes of bilinear pseudodifferential operators,Comm. Partial Differential Equations 28, 1161–1181, (2003).
Bényi, Á. and Torres, R. H. Almost orthogonality and a class of bounded bilinear pseudodifferential operators,Math. Res. Lett. 11, 1–11, (2004).
Bourdaud, G. Lp-estimates for certain nonregular pseudo-differential operators,Comm. Partial Differential Equations 7, 1023–1033, (1982).
Brown, R. M. and Torres, R. H. Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives inL P,p > 2n,J. Fourier Anal. Appl. 9(6), 563–574, (2003).
Chae, D. On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces,Comm. Pure Appl. Math. 55, 654–678, (2002).
Christ, F. M. and Weinstein, M. I. Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation,J. Func. Anal. 100, 87–109, (1991).
Coifman, R. R. and Grafakos, L. Hardy space estimates for multilinear operators, I,Rev. Mat. Iberoamericana 8, 45–67, (1992).
Coifman, R. R., Lions, P. L., Meyer, Y., and Semmes, S. Compensated compactness and Hardy spaces,J. Math. Pures Appl. (9) 72, 247–286, (1993).
Coifman, R. R. and Meyer, Y. On commutators of singular integrals and bilinear singular integrals,Trans. Amer. Math. Soc. 212, 315–331, (1975).
Coifman, R. R. and Meyer, Y. Commutateurs d’intégrales singulièrs et opérateurs multilinéaires,Ann. Inst. Fourier (Grenoble) 28, 177–202, (1978).
Coifman, R.R. and Meyer, Y. Au-delà des opérateurs pseudo-diffeŕentiels,Astérisque 57, Société Math. de France, (1978).
Gilbert, J. and Nahmod, A. Hardy spaces and a Walsh model for bilinear cone operators,Trans. Amer. Math. Soc. 351, 3267–3300, (1999).
Gilbert, J. and Nahmod, A. Boundedness of bilinear operators with nonsmooth symbols,Math. Res. Lett. 7, 767–778, (2000).
Gilbert, J. and Nahmod, A. Bilinear operators with nonsmooth symbols. IJ. Fourier Anal. Appl. 7(5), 435–467, (2001).
Gilbert, J. and Nahmod, A. Lp-boundedness of time-frequency paraproducts, II,J. Fourier Anal. Appl. 8(2), 109–172, (2002).
Grafakos, L. Hardy space estimates for multilinear operators, II,Rev. Mat. Iberoamericana 8, 69–92, (1992).
Grafakos, L. and Torres, R. H. Discrete decompositions for bilinear operators and almost diagonal conditions,Trans. Amer. Math. Soc. 354, 1153–1176, (2002).
Grafakos, L. and Torres, R. H. Multilinear Calderón-Zygmund theory,Adv. Math. 165, 124–164, (2002).
Gröchenig, K.Foundations of Time-Frequency Analysis, Birkhäuser, Boston, (2001).
Hörmander, L. Pseudo-differential operators,Comm. Pure Appl. Math. 18, 501–517, (1965).
Kato, T. and Ponce, G. Commutator estimates and the Euler and Navier-Stokes equations,Comm. Pure Appl. Math. 41, 891–907, (1988).
Kenig, C., Ponce, G., and Vega, L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,Comm. Pure Appl. Math. 46, 527–620, (1993).
Kenig, C. and Stein, E. Multilinear estimates and fractional integration,Math. Res. Lett. 6, 1–15, (1999).
Kohn, J. J. and Nirenberg, L. An algebra of pseudodifferential operators,Comm. Pure Appl. Math. 18, 269–305, (1965).
Lacey, M. and Thiele, C. Lp-bounds for the bilinear Hilbert transform, 2 <p < ∞,Ann. of Math. (2) 146, 693–724, (1997).
Lacey, M. and Thiele, C. Calderón’s conjecture,Ann. of Math. (2) 149, 475–496, (1999).
Meyer, Y. Remarques sur un théorème de J. M. Bony, Prépub. Dept. Math. Univ Paris-Sud, 91405 Orsay, France, (1980).
Muscalu, C., Pipher, J., Tao, T., and Thiele, C. Bi-parameter paraproducts,Acta Math. 193, 269–296, (2004).
Muscalu, C., Tao, T., and Thiele, C. Multilinear operators given by singular multipliers,J. Amer. Math. Soc. 15, 469–496, (2002).
Okoudjou, K. Embeddings of some classical Banach spaces into modulation spaces,Proc. Amer. Math. Soc. 132, 1639–1647, (2004).
Runst, T. and Sickel, W.Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, (1996).
Stein, E. M.Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, (1993).
Wainger, S. Special trigonometric series in k-dimensions,Mem. Amer. Math. Soc. 59, (1965).
Author information
Authors and Affiliations
Additional information
Communicated by Guido Weiss
Rights and permissions
About this article
Cite this article
Bényi, Á., Nahmod, A.R. & Torres, R.H. Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators. J Geom Anal 16, 431–453 (2006). https://doi.org/10.1007/BF02922061
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02922061