SpringerLink
Log in

Global regularity of Weyl pseudo-differential operators with radial symbols in each phase-space variable

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

We analyse a class of pseudo-differential operators in the Gelfand–Shilov setting whose Weyl symbols are radial in each phase-space variable separately. Namely, the symbols are of the form

$$\begin{aligned} a_{\vartheta }(x,\xi ):= a(2x_1^2+2\xi _1^2,\ldots ,2x_d^2+2\xi _d^2), \end{aligned}$$

where a is a measurable function on \({\mathbb {R}}^d_+:=\{r\in {\mathbb {R}}^d\,|\, r_j>0,\, j=1,\ldots ,d\}\) and has Gelfand–Shilov \(L^p\)-growths. We prove that the action of these pseudo-differential operators on a Gelfand–Shilov ultradistribution f can be given by a series of Hermite functions with coefficients that are explicitly computed in terms of the Laguerre coefficients of a and the Hermite coefficients of f. As a consequence, we give a characterisation of the functions a in terms of the growths of their Laguerre coefficients for which the Weyl quantisation of \(a_{\vartheta }\) are globally Gelfand–Shilov regular.

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

Similar content being viewed by others

References

  1. Abdeljawad, A., Coriasco, S., Toft, J.: Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators. Anal. Appl. 18(4), 523–583 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arias, A. Junior, Ascanelli, A., Cappiello, M.: The Cauchy problem for 3-evolution equations with data in Gelfand–Shilov spaces. J. Evol. Equ. 22(2) (2022), Paper No. 33

  3. Asensio, V., Jornet, D.: Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. RACSAM 113(4), 3477–3512 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barbaroux, J.-M., Hundertmark, D., Ried, T., Vugalter, S.: Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for maxwellian molecules. Arch. Rational Mech. Anal. 225(2), 601–661 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cappiello, M.: Fourier integral operators of infinite order and applications to SG-hyperbolic equations. Tsukuba J. Math. 28, 311–361 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cappiello, M., Pilipović, S., Prangoski, B.: Parametrices and hypoellipticity for pseudodifferential operators on spaces of tempered ultradistributions. J. Pseudo-Differ. Oper. Appl. 5(4), 491–506 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cappiello, M., Toft, J.: Pseudo-differential operators in a Gelfand–Shilov setting. Math. Nachr. 290, 738–755 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantine, G.M., Savits, T.H.: A multivariate Faà di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)

    Article  MATH  Google Scholar 

  9. Cordero, E., Rodino, L.: Time-Frequency Analysis of Operators, De Gruyter (2020)

  10. Duran, A.J.: Gel’fand–Shilov spaces for the Hankel transform. Indag. Math. 3, 137–151 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duran, A.J.: Laguerre expansions of Gel’fand–Shilov spaces. J. Approx. Theory 74, 280–300 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gel’fand, I.M., Shilov, G.E.: Les Distributions, Vol. 2,3,4, Dunod, Paris (1965)

  13. Jakšić, S., Maksimović, S., Pilipović, S., Prangoski, B.: Relations between Hermite and Laguerre expansions of ultradistributions over \({\mathbb{R}}^d\) and \({\mathbb{R}}^d_+\). J. Pseudo-Differ. Oper. Appl. 8, 275–296 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jakšić, S., Pilipović, S., Prangoski, B.: G-type spaces of ultradistributions over \({\mathbb{R}}_+^d\) and the Weyl pseudo-differential operators with radial symbols. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. RACSAM 111, 613–640 (2017)

  15. Komatsu, H.: Ultradistributions, I: Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20(1), 25–105 (1973)

  16. Köthe, G.: Topological Vector Spaces I. Springer, New York (1969)

    MATH  Google Scholar 

  17. Langenbruch, M.: Hermite functions and weighted spaces of generalized functions. Manuscr. Math. 119, 269–285 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lerner, N., Morimoto, Y., Pravda-Starov, K., Xu, C.-J.: Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff. J. Differ. Equ. 256(2), 797–831 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Laptev, A., Schimmer, L., Takhtajan, L.: Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves. Geom. Funct. Anal. 26, 288–305 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nicola, F., Rodino, L.: Global pseudo-differential calculus on Euclidean spaces. Birkhäuser Basel (2010)

  21. Pilipović, S., Prangoski, B.: Complex powers for a class of infinite order hypoelliptic operators. Dissert. Math. 529, 1–58 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pilipović, S., Prangoski, B., Vindas, J.: Infinite order \(\Psi \)DOs: composition with entire functions, new Shubin–Sobolev spaces, and index theorem. Anal. Math. Phys. 11(3) (2021), Article No. 109, 48p

  23. Pilipović, S., Prangoski, B., Vučković, Đ.: Convolution with the kernel \(e^{s\langle x\rangle ^q}\), \(q\ge 1\), \(s>0\) within ultradistribution spaces. Mediterr. J. Math. 18(4) (2021), Article No. 164, 14p

  24. Reiher, K.: Weighted inductive and projective limits of normed Köthe function spaces. Result. Math. 13(1–2), 147–161 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific Publishing Co., Inc, River Edge (1993)

  26. Teofanov, N., Toft, J.: Pseudo-differential calculus in a Bargmann setting. Ann. Acad. Sci. Fenn. Math. 45(1), 227–257 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vučković, Đ, Vindas, J.: Eigenfunction expansions of ultradifferentiable functions and ultradistributions in \({\mathbb{R} }^n\). J. Pseudo-Differ. Oper. Appl. 7(4), 519–531 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wong, M.W.: Weyl Transforms. Springer, New York (1998)

    MATH  Google Scholar 

  29. Zhang, T.-F., Yin, Z.: Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff. J. Differ. Equ. 253(4), 1172–1190 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Forestry, University of Belgrade, Kneza Višeslava 1, Belgrade, Serbia

    Smiljana Jakšić

  2. Department of Mathematics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, Serbia

    Stevan Pilipović

  3. Department of Mathematics, Faculty of Mechanical Engineering, Ss. Cyril and Methodius University in Skopje, Karpos II bb, 1000, Skopje, Macedonia

    Bojan Prangoski

Authors
  1. Smiljana Jakšić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stevan Pilipović
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bojan Prangoski
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors contributed equally in writing the manuscript.

Corresponding author

Correspondence to Smiljana Jakšić.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

S. Jakšić: The work of S. Jakšić was supported by the project 451-03-68/2022-14/200169 financed by the Ministry of Education, Science and Technological Development of the Republic of Serbia.

S. Pilipović: The work of S. Pilipović was supported by the project F10 financed by the Serbian Academy of Sciences and Arts.

B. Prangoski: The work of B. Prangoski was partially supported by the bilateral project “Microlocal analysis and applications” funded by the Macedonian Academy of Sciences and Arts and the Serbian Academy of Sciences and Arts.

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

Jakšić, S., Pilipović, S. & Prangoski, B. Global regularity of Weyl pseudo-differential operators with radial symbols in each phase-space variable. J. Pseudo-Differ. Oper. Appl. 14, 10 (2023). https://doi.org/10.1007/s11868-023-00505-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11868-023-00505-x

Keywords

Mathematics Subject Classification

Search

Navigation