Skip to main content
SpringerLink
Log in

Multilinear Pseudo-differential Operators with \(S_{0,0}\) Class Symbols of Limited Smoothness

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We consider the boundedness of the multilinear pseudo-differential operators with symbols in the multilinear Hörmander class \(S_{0,0}\). The aim of this paper is to discuss smoothness conditions for symbols to assure the boundedness between local Hardy spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bényi, Á., Torres, R.: Almost orthogonality and a class of bounded bilinear pseudodifferential operators. Math. Res. Lett. 11, 1–11 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bényi, Á., Bernicot, F., Maldonado, D., Naibo, V., Torres, R.: On the Hörmander classes of bilinear pseudodifferential operators II. Indiana Univ. Math. J. 62, 1733–1764 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boulkhemair, A.: \(L^2\) estimates for pseudodifferential operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci 4(22), 155–183 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Calderón, A.P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nat. Acad. Sci. USA 69, 1185–1187 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Coifman, R.R., Meyer, Y.: Au delà des opérateurs pseudo-différentiels. Astérisque 57, 1–185 (1978)

    MATH  Google Scholar 

  6. Cordero, E., Rodino, L.: Time-Frequency Analysis of Operators, De Gruyter Studies in Mathematics 75. De Gruyter, Berlin (2020)

    Google Scholar 

  7. Cordes, H.O.: On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal. 18, 115–131 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cunanan, J., Kobayashi, M., Sugimoto, M.: Inclusion relations between \(L^p\)-Sobolev and Wiener amalgam spaces. J. Funct. Anal. 268, 239–254 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fefferman, C.: \(L^p\) bounds for pseudo-differential operators. Isr. J. Math. 14, 413–417 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feichtinger, H.G.: Banach convolution algebras of Wiener type. In: Functions, Series, Operators, Proceedings of Conference on Budapest, Colloquia Mathematica Societatis János Bolyai, vol. 38, pp. 509–524 (1980)

  11. Feichtinger, H.G.: Banach spaces of distributions of Wiener’s type and interpolation. In: Butzer, P.L., Sz-Nagy, B., Görlich, E. (eds.) Functional Analysis and Approximation, ISNM 60 International Series of Numerical Mathematics, vol. 60, pp. 153–165. Birkhäuser, Basel (1981)

  12. Feichtinger, H.G.: Generalized amalgams, with applications to Fourier transform. Can. J. Math 42, 395–409 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Feichtinger, H.G.: Modulation spaces on locally compact Abelian groups. Technical report, University of Vienna, Vienna, 1983; also in: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and Their Applications, pp. 99–140. Allied Publishers, New Delhi (2003)

  14. Fournier, J.J.F., Stewart, J.: Amalgams of \(L^p\) and \(\ell ^q\). Bull. Am. Math. Soc. (N.S.) 13, 1–21 (1985)

    Article  MATH  Google Scholar 

  15. Galperin, Y.V., Samarah, S.: Time-frequency analysis on modulation spaces \(M^{m}_{p, q}\), \(0 < p, q \le \infty \). Appl. Comput. Harmon. Anal. 16, 1–18 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46, 27–42 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grafakos, L.: Modern Fourier Analysis. GTM 250, 3rd edn. Springer, New York (2014)

    Book  Google Scholar 

  18. Gröchenig, K.: Foundation of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  19. Guo, W., Wu, H., Yang, Q., Zhao, G.: Characterization of inclusion relations between Wiener amalgam and some classical spaces. J. Funct. Anal. 273, 404–443 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Heil, C.: An introduction to weighted Wiener amalgams. In: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and Their Applications, pp. 183–216. Allied Publishers, New Delhi (2003)

    Google Scholar 

  21. Herbert, J., Naibo, V.: Bilinear pseudodifferential operators with symbols in Besov spaces. J. Pseudo-Differ. Oper. Appl. 5, 231–254 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Herbert, J., Naibo, V.: Besov spaces, symbolic calculus, and boundedness of bilinear pseudodifferential operators. In: Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory, vol. 1, pp. 275–305. Association for Women in Mathematics Series 4, Springer. Cham (2016)

  23. Holland, F.: Harmonic analysis on amalgams of \(L^p\) and \(\ell ^q\). J. Lond. Math. Soc. 2(10), 295–305 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hwang, I.L.: The \(L^2\) boundedness of pseudodifferential operators. Trans. Am. Math. Soc. 302, 55–76 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rational Mech. Anal. 58, 181–205 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kato, T., Miyachi, A., Tomita, N.: Boundedness of bilinear pseudo-differential operators of \(S_{0,0}\)-type on \(L^2 \times L^2\). J. Pseudo-Differ. Oper. Appl. 12, 15 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kato, T., Miyachi, A., Tomita, N.: Boundedness of multilinear pseudo-differential operators of \(S_{0,0}\)-type in \(L^2\)-based amalgam spaces. J. Math. Soc. Jpn. 73, 351–388 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kato, T., Miyachi, A., Tomita, N.: Boundedness of multilinear pseudo-differential operators with symbols in the Hörmander class \(S_{0,0}\). J. Funct. Anal. 282, 109329 (2022)

    Article  MATH  Google Scholar 

  29. Kobayashi, M.: Modulation spaces \(M^{p, q}\) for \(0 < p, q \le \infty \). J. Funct. Spaces Appl. 4, 329–341 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Michalowski, N., Rule, D., Staubach, W.: Multilinear pseudodifferential operators beyond Calderón-Zygmund theory. J. Math. Anal. Appl. 414, 149–165 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Miyachi, A.: Estimates for pseudo-differential operators of class \(S_{0,0}\). Math. Nachr. 133, 135–154 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  32. Miyachi, A., Tomita, N.: Calderón-Vaillancourt-type theorem for bilinear operators. Indiana Univ. Math. J. 62, 1165–1201 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Miyachi, A., Tomita, N.: Bilinear pseudo-differential operators with exotic symbols. Ann. Inst. Fourier (Grenoble) 70(6), 2737–2769 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Muramatu, T.: Estimates for the norm of pseudo-differential operators by means of Besov spaces. In: Cordes, H.O., Gramsch, B., Widom, H. (eds.) Pseudo-Differential Operators. Lecture Notes in Mathematics, vol. 1256, pp. 330–349. Springer, Berlin (1987)

  35. Päivärinta, L., Somersalo, E.: A generalization of the Calderón-Vaillancourt theorem to \(L^p\) and \(h^p\). Math. Nachr. 138, 145–156 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ruzhansky, M., Sugimoto, M., Toft, J., Tomita, N.: Changes of variables in modulation and Wiener amalgam spaces. Math. Nachr. 284, 2078–2092 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shida, N.: Boundedness of bilinear pseudo-differential operators with \(BS^{m}_{0,0}\) symbols on Sobolev spaces

  38. Sugimoto, M.: \(L^p\)-boundedness of pseudo-differential operators satisfying Besov estimates I. J. Math. Soc. Jpn. 40, 105–122 (1988)

    Article  MATH  Google Scholar 

  39. Sugimoto, M.: \(L^{p}\)-boundedness of pseudo-differential operators satisfying Besov estimates II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 , 149–162 (1988)

  40. Tomita, N.: On the \(L^p\)-boundedness of pseudo-differential operators with non-regular symbols. Ark. Mat. 49, 175–197 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Triebel, H.: Modulation spaces on the Euclidean \(n\)-space. Z. Anal. Anwendungen 2, 443–457 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  42. Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)

    Book  MATH  Google Scholar 

  43. Wainger, S.: Special Trigonometric Series in \(k\)-dimensions. Mem. Am. Math. Soc. 59 (1965)

  44. Wang, B., Hudzik, H.: The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Equ. 232, 36–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wiener, N.: On the representation of functions by trigonometric integrals. Math. Z. 24, 575–616 (1926)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wiener, N.: Tauberian theorems. Ann. Math. 33, 1–100 (1932)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author expresses deep thanks to Professor A. Miyachi and Professor N. Tomita. Although Proposition 6.3 in the first draft stated only the case \(p_{j} \ge 1\), Prof. Miyachi gave him ideas to develop it to the whole range \(p_{j} > 0\). Prof. Tomita pointed out to him that Theorem 3.2 holds for more improved symbol classes as stated in Remark 3.3. The author sincerely thanks Professor H. G. Feichtinger for letting the author know the history of Wiener amalgam spaces and leading the author’s misunderstanding to the correct way (Remark 2.2). The author is also grateful for the anonymous referee’s careful reading and constructive suggestions, which lead to improvements of this paper.

Author information

Authors and Affiliations

  1. Division of Pure and Applied Science, Faculty of Science and Technology, Gunma University, Kiryu, Gunma, 376-8515, Japan

    Tomoya Kato

Authors
  1. Tomoya Kato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomoya Kato.

Additional information

Communicated by Elena Cordero.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by JSPS KAKENHI, Grant Number 20K14339.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kato, T. Multilinear Pseudo-differential Operators with \(S_{0,0}\) Class Symbols of Limited Smoothness. J Fourier Anal Appl 29, 40 (2023). https://doi.org/10.1007/s00041-023-10016-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00041-023-10016-4

Keywords

Mathematics Subject Classification

Search

Navigation