Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 2973–3017 | Cite as

\(L^p\)-Boundedness and \(L^p\)-Nuclearity of Multilinear Pseudo-differential Operators on \({\mathbb {Z}}^n\) and the Torus \({\mathbb {T}}^n\)

  • Duván CardonaEmail author
  • Vishvesh Kumar


In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on \(L^p\)-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the s-nuclearity, \(0<s \le 1,\) of periodic and discrete pseudo-differential operators. To accomplish this, we classify those s-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on \(\sigma \)-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato–Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.


Multilinear pseudo-differential operator Discrete operator Periodic operator Nuclearity Boundedness Fourier integral operators Multilinear analysis 

Mathematics Subject Classification

Primary 58J40 Secondary 47B10 47G30 35S30 



The authors would like to thank the anonymous referees for their valuable suggestions which help us to improve the presentation of this article. Vishvesh Kumar thanks the Council of Scientific and Industrial Research, India, for its senior research fellowship. Duván Cardona was partially supported by the Department of Mathematics, Pontificia Universidad Javeriana.


  1. 1.
    Agranovich, M.S.: Spectral properties of elliptic pseudo-differential operators on a closed curve. Funct. Anal. Appl. 13, 279–281 (1971)zbMATHGoogle Scholar
  2. 2.
    Aoki, S.: On the boundedness and the nuclearity of pseudo-differential operators. Commun. Partial Differ. Equ. 6(8), 849–881 (1981)zbMATHGoogle Scholar
  3. 3.
    Bényi, Á., Bernicot, F., Maldonado, D., Naibo, V., Torres, R.: On the Hörmander classes of bilinear pseudo-differential operators II. Indiana Univ. Math. J. 62, 1733–1764 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bényi, Á., Maldonado, D., Naibo, V., Torres, R.: On the Hörmander classes of bilinear pseudodifferential operators. Integral Equ. Oper. Theory 67, 341–364 (2010)zbMATHGoogle Scholar
  5. 5.
    Botchway L., Kibiti G., Ruzhansky M.: Difference equations and pseudo-differential operators on \({\mathbb{Z}}^{n}\). arXiv:1705.07564
  6. 6.
    Cardona, D.: Weak type (1, 1) bounds for a class of periodic pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 5(4), 507–515 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cardona, D.: On the boundedness of periodic pseudo-differential operators. Monatsh. für Math. 185(2), 189–206 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cardona, D.: Pseudo-differential operators on \({\mathbb{Z}}^n\) with applications to discrete fractional integral operators. Bull. Iran. Math. Soc. (to appear)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cardona, D., Kumar, V.: Multilinear analysis for discrete and periodic pseudo-differential operators in \(L^p\) spaces. Rev. Integr. Temas Mat. 36(2), 151–164 (2018)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cardona, D., Messiouene, R., Senoussaoui, A.: \(L^p\)-bounds for Fourier integral operators on the torus. arXiv:1807.09892
  11. 11.
    Catana, V.: \(L^p\)-Boundedness of multilinear pseudo-differential operators on \({\mathbb{Z}}^n\) and \({\mathbb{T}}^n\). Math. Model. Nat. Phenom. 9(5), 17–38 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Coifman, R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Coifman, R., Meyer, Y.: Ondelettes et operateurs III. Operateurs multilineaires. Hermann, Paris (1991)zbMATHGoogle Scholar
  14. 14.
    Delgado, J.: \(L^p\) bounds for pseudo-differential operators on the torus. Oper. Theory Adv. Appl. 231, 103–116 (2012)zbMATHGoogle Scholar
  15. 15.
    Delgado, J.: A trace formula for nuclear operators on \(L^p\). In: Schulze, B.W., Wong, M.W. (eds.) Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications, vol. 205, pp. 181–193. Birkhäuser, Basel (2010)Google Scholar
  16. 16.
    Delgado, J., Wong, M.W.: \(L^p\)-Nuclear pseudo-differential operators on \({\mathbb{Z}}\) and \({\mathbb{S}}^1\). Proc. Am. Math. Soc. 141(11), 3935–3944 (2013)zbMATHGoogle Scholar
  17. 17.
    Delgado, J.: The trace of nuclear operators on \(L^p(\mu )\) for \(\sigma \)-finite Borel measures on second countable spaces. Integral Equations Operator Theory 68(1), 61–74 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Delgado, J.: On the \(r\)-nuclearity of some integral operators on Lebesgue spaces. Tohoku Math. J. 67(2), 125–135 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Delgado, J., Ruzhansky, M.: \(L^p\)-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups. J. Math. Pures Appl. (9) 102(1), 153–172 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Delgado, J., Ruzhansky, M.: Schatten classes on compact manifolds: Kernel conditions. J. Funct. Anal. 267(3), 772–798 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Delgado, J., Ruzhansky, M.: Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds. C. R. Acad. Sci. Paris. Ser. I. 352, 779–784 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Delgado, J., Ruzhansky, M.: Fourier multipliers, symbols and nuclearity on compact manifolds. J. Anal. Math. 135(2), 757–800 (2018)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Delgado, J., Ruzhansky, M.: Schatten-von Neumann classes of integral operators. arXiv:1709.06446
  24. 24.
    Delgado, J., Ruzhansky, M.: The bounded approximation property of variable Lebesgue spaces and nuclearity. Math. Scand. 122, 299–319 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Delgado, J., Ruzhansky, M.: Schatten classes and traces on compact groups. Math. Res. Lett. 24, 979–1003 (2017)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Delgado, J., Ruzhansky, M., Wang, B.: Approximation property and nuclearity on mixed-norm \(L^p\), modulation and Wiener amalgam spaces. J. Lond. Math. Soc. 94, 391–408 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Delgado, J., Ruzhansky, M., Wang, B.: Grothendieck-Lidskii trace formula for mixed-norm \(L^p\) and variable Lebesgue spaces. J. Spectr. Theory 6(4), 781–791 (2016)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Delgado, J., Ruzhansky, M., Tokmagambetov, N.: Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary (9). J. Math. Pures Appl. 107(6), 758–783 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Fujita, M., Tomita, N.: Weighted norm inequalities for multilinear Fourier multipliers. Trans. Am. Math. Soc. 364(12), 6335–6353 (2012)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Ghaemi, M.B., Jamalpour Birgani, M., Wong, M.W.: Characterizations of nuclear pseudo-differential operators on \({\mathbb{S}}^1\) with applications to adjoints and products. J. Pseudo-Differ. Oper. Appl. 8(2), 191–201 (2017)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ghaemi, M.B., Jamalpour Birgani, M., Wong, M.W.: Characterization, adjoints and products of nuclear pseudo-differential operators on compact and Hausdorff groups. U.P.B. Sci. Bull. Ser. A 79(4), 207–220 (2017)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Grafakos, L., Miyachi, A., Tomita, N.: On multilinear Fourier multipliers of limited smoothness. Can. J. Math. 65, 299–330 (2013)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Grafakos, L., Si, Z.: The Hörmander multiplier theorem for multilinear operators. J. Reine Angew. Math. 668, 133–147 (2012)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Grafakos, L.: Multilinear operators In: Harmonic Analysis and Partial Differential Equations. Research Institute of Mathematical Sciences, Kyoto (2012)Google Scholar
  35. 35.
    Grafakos, L., Torres, R.: Discrete decompositions for bilinear operators and almost diagonal conditions. Trans. Am. Math. Soc. 354, 1153–1176 (2012)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Grafakos, L., Torres, R.: Multilinear Calderón- Zygmund theory. Adv. Math. 165, 124–164 (2002)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Hörmander, L.: The Analysis of the Linear Partial Differential Operators, vol. III. Springer, IV (1985)Google Scholar
  38. 38.
    Jamalpour Birgani, M.: Characterizations of Nuclear Pseudo-differential Operators on \({\mathbb{Z}}\) with some Applications. Math. Model. Nat. Phenom. 13, 13–30 (2018)MathSciNetGoogle Scholar
  39. 39.
    Kenig, C., Stein, E.: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1–15 (1999)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Mclean, W.M.: Local and Global description of periodic pseudo-differential operators. Math. Nachr. 150, 151–161 (1991)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Michalowski, N., Rule, D., Staubach, W.: Multilinear pseudodifferential operators beyond Calderón-Zygmund operators. J. Math. Anal. Appl. 414, 149–165 (2014)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Miyachi, A., Tomita, N.: Minimal smoothness conditions for bilinear Fourier multipliers. Rev. Mat. Iberoam. 29, 495–530 (2013)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Miyachi, A., Tomita, N.: Calderón-Vaillancourt type theorem for bilinear operators. Indiana Univ. Math. J. 62, 1165–1201 (2013)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Miyachi, A., Tomita, N.: Bilinear pseudo-differential operators with exotic symbols. Ann. Inst. Fourier (Grenoble). arXiv:1801.06744 (to appear)
  45. 45.
    Molahajloo, S.: A characterization of compact pseudo-differential operators on \({\mathbb{S}}^1\). Oper. Theory Adv. Appl. Birkhüser/Springer Basel AG, Basel 213, 25–29 (2011)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Molahajloo, S., Wong, M.W.: Pseudo-differential operators on \({\mathbb{S}}^1\). In: Rodino, L., M.W. Wong (eds.) New Developments in Pseudo-differential Operators, pp. 297–306 (2008)Google Scholar
  47. 47.
    Molahajloo, S., Wong, M.W.: Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on \({\mathbb{S}}^1\). J. Pseudo-Differ. Oper. Appl. 1, 183–205 (2010)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Muscalu, C., Tao, T., Thiele, C.: Multilinear operators given by singular multipliers. J. Am. Math. Soc. 15, 469–496 (2002)zbMATHGoogle Scholar
  49. 49.
    Muscalu, C., Schlag, W.: Classical and multilinear harmonic analysis, vol. II. Cambridge Studies in Advanced Mathematics, vol. 138. Cambridge University Press, Cambridge (2013)Google Scholar
  50. 50.
    Rabinovich, V.S.: Exponential estimates of solutions of pseudo-differential equations on the lattice \((\mu {\mathbb{Z}})^n\): applications to the lattice Schrödinger and Dirac operators. J. Pseudo-Differ. Oper. Appl. 1(2), 233–253 (2010)MathSciNetGoogle Scholar
  51. 51.
    Rabinovich, V.S.: Wiener algebra of operators on the lattice \((\mu {\mathbb{Z}})^n\) depending on the small parameter \(\mu >0\). Complex Var. Elliptic Equ. 58(6), 751–766 (2013)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Rabinovich, V.S., Roch, S.: The essential spectrum of Schrödinger operators on lattices. J. Phys. A 39(26), 8377–8394 (2006)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Rabinovich, V.S., Roch, S.: Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics. J. Phys. A 42(38), 385–207 (2009)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Rempala, J.A.: On a proof of the boundedness and nuclearity of pseudodifferential operators in \({\mathbb{R}}^n\). Ann. Pol. Math. 52, 59–65 (1990)zbMATHGoogle Scholar
  55. 55.
    Rodriguez, C.A.: \(L^p-\)estimates for pseudo-differential operators on \({\mathbb{Z}}^n\). J. Pseudo-Differ. Oper. Appl. 1, 183–205 (2011)Google Scholar
  56. 56.
    Ruzhansky, M., Turunen, V.: Quantization of pseudo-differential operators on the torus. J. Fourier Anal. Appl. 16, 943–982 (2010)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Ruzhansky, M., Turunen, V.: Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics. Birkhaüser-Verlag, Basel (2010)zbMATHGoogle Scholar
  58. 58.
    Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, NJ (1971)zbMATHGoogle Scholar
  59. 59.
    Tomita, N.: A Hörmander type multiplier theorem for multilinear operators. J. Funct. Anal. 259, 2028–2044 (2010)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Turunen, V., Vainikko, G.: On symbol analysis of periodic pseudodifferential operators. Z. Anal. Anwendungen. 17, 9–22 (1998)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsPontificia Universidad JaverianaBogotáColombia
  2. 2.Department of Mathematics, Analysis Logic and Discrete MathematicsGhent UniversityGhentBelgium
  3. 3.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia

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