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On the $ L^\infty\times L^\infty\rightarrow BMO$ mapping property for certain bilinear pseudodifferential operators


Author: Virginia Naibo
Journal: Proc. Amer. Math. Soc. 143 (2015), 5323-5336
MSC (2010): Primary 35S05, 47G30; Secondary 42B20, 42B35
DOI: https://doi.org/10.1090/proc12775
Published electronically: June 10, 2015
MathSciNet review: 3411149
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Abstract: Boundedness from $ L^\infty \times L^\infty $ into $ BMO$ is proved for bilinear pseudodifferential operators with symbols in a range of bilinear Hörmander classes of critical order. This is achieved by means of new continuity results for bilinear operators with symbols in certain classes and a new pointwise inequality relating bilinear operators and maximal functions.


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  • [1] Josefina Álvarez and Jorge Hounie, Estimates for the kernel and continuity properties of pseudo-differential operators, Ark. Mat. 28 (1990), no. 1, 1-22. MR 1049640 (91d:35255), https://doi.org/10.1007/BF02387364
  • [2] Árpad Bényi, Frédéric Bernicot, Diego Maldonado, Virginia Naibo, and Rodolfo H. Torres, On the Hörmander classes of bilinear pseudodifferential operators II, Indiana Univ. Math. J. 62 (2013), no. 6, 1733-1764. MR 3205530, https://doi.org/10.1512/iumj.2013.62.5168
  • [3] Árpád Bényi, Diego Maldonado, Virginia Naibo, and Rodolfo H. Torres, On the Hörmander classes of bilinear pseudodifferential operators, Integral Equations Operator Theory 67 (2010), no. 3, 341-364. MR 2660466 (2011g:47111), https://doi.org/10.1007/s00020-010-1782-y
  • [4] Alberto-P. Calderón and Rémi Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971), 374-378. MR 0284872 (44 #2096)
  • [5] Sagun Chanillo and Alberto Torchinsky, Sharp function and weighted $ L^p$ estimates for a class of pseudodifferential operators, Ark. Mat. 24 (1986), no. 1, 1-25. MR 852824 (87h:47111), https://doi.org/10.1007/BF02384387
  • [6] Charles Fefferman, $ L^{p}$ bounds for pseudo-differential operators, Israel J. Math. 14 (1973), 413-417. MR 0336453 (49 #1227)
  • [7] Loukas Grafakos and Rodolfo H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124-164. MR 1880324 (2002j:42029), https://doi.org/10.1006/aima.2001.2028
  • [8] Lars Hörmander, Pseudo-differential operators and hypoelliptic equations, Singular integrals (Proc. Sympos. Pure Math., Vol. X, Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 138-183. MR 0383152 (52 #4033)
  • [9] Jürgen Marschall, Pseudodifferential operators with nonregular symbols of the class $ S^m_{\rho \delta }$, Comm. Partial Differential Equations 12 (1987), no. 8, 921-965. MR 891745 (88g:47102a), https://doi.org/10.1080/03605308708820514
  • [10] Jürgen Marschall, Correction to: ``Pseudodifferential operators with nonregular symbols of the class $ S^m_{\rho ,\delta }$'', Comm. Partial Differential Equations 13 (1988), no. 1, 129-130. MR 914817 (88g:47102b)
  • [11] Nicholas Michalowski, David Rule, and Wolfgang Staubach, Multilinear pseudodifferential operators beyond Calderón-Zygmund theory, J. Math. Anal. Appl. 414 (2014), no. 1, 149-165. MR 3165300, https://doi.org/10.1016/j.jmaa.2013.12.062
  • [12] Akihiko Miyachi and Naohito Tomita, Calderón-Vaillancourt-type theorem for bilinear operators, Indiana Univ. Math. J. 62 (2013), no. 4, 1165-1201. MR 3179688, https://doi.org/10.1512/iumj.2013.62.5059
  • [13] Virginia Naibo,
    On the bilinear Hörmander classes in the scales of Triebel-Lizorkin and Besov spaces.
    J. Fourier Anal. Appl., 2015. DOI 10.1007/s00041-015-9398-x
  • [14] Salvador Rodríguez-López and Wolfgang Staubach, Estimates for rough Fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators, J. Funct. Anal. 264 (2013), no. 10, 2356-2385. MR 3035059, https://doi.org/10.1016/j.jfa.2013.02.018

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Additional Information

Virginia Naibo
Affiliation: Department of Mathematics, 138 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506
Email: vnaibo@math.ksu.edu

DOI: https://doi.org/10.1090/proc12775
Keywords: Bilinear pseudodifferential operators, H\"ormander classes, critical order, BMO
Received by editor(s): October 5, 2014
Published electronically: June 10, 2015
Additional Notes: This work was partially supported by NSF under grant DMS 1101327.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society

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