Abstract
We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear Calderón–Zygmund operators.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bényi, Á.: Bilinear singular integrals and pseudodifferential operators. Ph.D. Thesis, University of Kansas (2002)
Bényi, Á.: Bilinear pseudodifferential operators on Lipschitz and Besov spaces. J. Math. Anal. Appl. 284, 97–103 (2003)
Bényi, Á., Maldonado D., Nahmod, A., Torres, R.: Bilinear paraproducts revisited. Math. Nachr. (2008, to appear)
Bényi, Á., Torres, R.H.: Symbolic calculus and the transposes of bilinear pseudodifferential operators. Commun. Partial Differ. Equ. 28, 1161–1181 (2003)
Bényi, Á., Torres, R.H.: Almost orthogonality and a class of bounded bilinear pseudodifferential operators. Math. Res. Lett. 11.1, 1–12 (2004)
Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non-linéaires. Ann. Sci. Ec. Norm. Super. Ser. 4 14(2), 209–246 (1981)
Bourdaud, G.: L p-estimates for certain non-regular pseudo-differential operators. Commun. Partial Differ. Equ. 7(9), 1023–1033 (1982)
Cannone, M.: Ondelettes, Paraproduits et Navier–Stokes. Diderot, Paris (1995). (French) [Wavelets, paraproducts and Navier–Stokes]
Cannone, M.: Harmonic analysis tools for solving the incompressible Navier–Stokes equations. In: Handbook of Mathematical Fluid Dynamics, vol. III, pp. 161–244. North-Holland, Amsterdam (2004)
Ching, C.-H.: Pseudo-differential operators with non-regular symbols. J. Differ. Equ. 11, 436–447 (1972)
Coifman, R.R., Dobyinsky, S., Meyer, Y.: Opérateurs bilinéaires et renormalization. In: Stein, M., Fefferman, C., Fefferman, R., Wainger, S. (eds.) Essays on Fourier Analysis in Honor of Elias. Princeton University Press, Princeton (1995)
Coifman, R.R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)
Coifman, R.R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier 28(3), 177–202 (1978)
Coifman, R.R., Meyer, Y.: Au-delà des Opérateurs Pseudo-Différentiels, 2nd edn. Astèrisque, vol. 57 (1978)
Coifman, R.R., Meyer, Y.: Wavelets: Calderón–Zygmund and Multilinear Operators. Cambridge University Press, Cambridge (1997)
David, G., Journé, J.-L.: A boundedness criterion for generalized Calderón–Zygmund operators. Ann. Math. 120, 371–397 (1984)
Fournier, J.F., Stewart, J.: Amalgams of L p and l q. Bull. Am. Math. Soc. (New Series) 13(1), 1–21 (1985)
Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–169 (1990)
Frazier, M., Jawerth, B., Weiss, G.: Littlewood–Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series in Mathematics, vol. 79 (1991)
García-Cuerva, J., Kazarian, K.: Calderón–Zygmund operators and unconditional bases of weighted Hardy spaces. Stud. Math. 109(3), 255–276 (1994)
Gilbert, J., Nahmod, A.: Bilinear operators with non-smooth symbols, I. J. Fourier Anal. Appl. 5, 435–467 (2001)
Gilbert, J., Nahmod, A.: L p-boundedness of time-frequency paraproducts, II. J. Fourier Anal. Appl. 8, 109–172 (2002)
Grafakos, L., Kalton, N.: Multilinear Calderón–Zygmund operators on Hardy spaces. Collect. Math. 52, 169–179 (2001)
Grafakos, L., Kalton, N.: The Marcinkiewicz multiplier condition for bilinear operators. Stud. Math. 146(2), 115–156 (2001)
Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165, 124–164 (2002)
Grafakos, L., Torres, R.H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J. 51(5), 1261–1276 (2002)
Journè, J.-L.: Calderón–Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón. Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983)
Kenig, C., Stein, E.: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1–15 (1999). Erratum in Math. Res. Lett. 6(3–4), 467 (1999)
Kikuchi, N., Nakai, E., Tomita, N., Yabuta, K., Yoneda, T.: Calderón–Zygmund operators on amalgam spaces and in the discrete case. J. Math. Anal. Appl. 335, 198–212 (2007)
Lacey, M.: Commutators with Riesz potentials in one and several parameters. Hokkaido Math. J. 36(1), 175–191 (2007)
Lacey, M., Metcalfe, J.: Paraproducts in one and several parameters. Forum Math. 19(2), 325–351 (2007)
Lannes, D.: Sharp Estimates for pseudo-differential operators with symbols of limited smoothness and commutators. J. Funct. Anal. 232, 495–539 (2006)
Marschall, J.: Weighted L p-estimates for pseudo-differential operators with non-regular symbols. Z. Anal. Anwend. 10, 493–501 (1991)
Marschall, J.: Non-regular pseudo-differential operators. Z. Anal. Anwend. 15, 109–148 (1996)
Muscalu, C., Tao, T., Thiele, C.: Multilinear operators given by singular multipliers. J. Am. Math. Soc. 15, 469–496 (2002)
Muscalu, C., Pipher, J., Tao, T., Thiele, C.: Bi-parameter paraproducts. Acta Math. 193, 269–296 (2004)
Nagase, M.: The L p-boundedness of pseudo-differential equations with non-regular symbols. Commun. Partial Differ. Equ. 2(10), 1045–1061 (1977)
Nishigaki, S.: Weighted norm inequalities for certain pseudo-differential operators. Tokyo J. Math. 7, 129–140 (1984)
Petermichl, S.: Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbols. C. R. Acad. Sci. Paris Sèr. I Math. 330, 455–460 (2000)
Sato, S.: A note on weighted estimates for certain classes of pseudo-differential operators. Rocky Mt. J. Math. 35(1), 267–284 (2005)
Stefanov, A.: Pseudodifferential operators with rough symbols. Preprint
Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)
Taylor, M.: Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials. Mathematical Surveys and Monographs, vol. 81. AMS, Providence (2000)
Taylor, M.: Pseudodifferential operators and nonlinear PDE. Progress in Mathematics, vol. 100. Birkhäuser, Boston (1991)
Thiele, C.: Wave Packet Analysis. CBMS Regional Conference Series in Mathematics, vol. 105 (2006)
Yabuta, K.: Generalizations of Calderón–Zygmund operators. Stud. Math. 82(1), 17–31 (1985)
Yabuta, K.: Calderón–Zygmund operators and pseudodifferential operators. Commun. Partial Differ. Equ. 10(9), 1005–1022 (1985)
Yabuta, K.: Weighted norm inequalities for pseudodifferential operators. Osaka J. Math. 23(3), 703–723 (1986)
Youssif, A.: Bilinear operators and the Jacobian-determinant on Besov spaces. Indiana Univ. Math. J. 45, 381–396 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hans G. Feichtinger.
Rights and permissions
About this article
Cite this article
Maldonado, D., Naibo, V. Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity. J Fourier Anal Appl 15, 218–261 (2009). https://doi.org/10.1007/s00041-008-9029-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-008-9029-x