Skip to main content
×
×
Home

$L^{p}$ -BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS

  • Julio Delgado (a1) and Michael Ruzhansky (a1)
Abstract

Given a compact Lie group $G$ , in this paper we establish $L^{p}$ -bounds for pseudo-differential operators in $L^{p}(G)$ . The criteria here are given in terms of the concept of matrix symbols defined on the noncommutative analogue of the phase space $G\times \widehat{G}$ , where $\widehat{G}$ is the unitary dual of $G$ . We obtain two different types of $L^{p}$ bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using $\mathscr{S}_{\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF}}^{m}(G)$ classes which are a suitable extension of the well-known $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF})$ ones on the Euclidean space. The results herein extend classical $L^{p}$ bounds established by C. Fefferman on $\mathbb{R}^{n}$ . While Fefferman’s results have immediate consequences on general manifolds for $\unicode[STIX]{x1D70C}>\max \{\unicode[STIX]{x1D6FF},1-\unicode[STIX]{x1D6FF}\}$ , our results do not require the condition $\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$ . Moreover, one of our results also does not require $\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$ . Examples are given for the case of $\text{SU}(2)\cong \mathbb{S}^{3}$ and vector fields/sub-Laplacian operators when operators in the classes $\mathscr{S}_{0,0}^{m}$ and $\mathscr{S}_{\frac{1}{2},0}^{m}$ naturally appear, and where conditions $\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$ fail, respectively.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      $L^{p}$ -BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      $L^{p}$ -BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      $L^{p}$ -BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS
      Available formats
      ×
Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
Hide All
1. Akylzhanov R. and Ruzhansky M., Net spaces on lattices, Hardy–Littlewood type inequalities, and their converses, preprint, 2015, arXiv:1510.01251v1.
2. Beals R., L p and Holder estimates for pseudodifferential operators: necessary conditions, in Harmonic Analysis in Euclidean Spaces (Proc. Symp. Pure Math., Williams Coll., Williamstown, MA, 1978), pp. 153157 (American Mathematical Society, Providence, RI, 1979).
3. Beals R., L p and Holder estimates for pseudodifferential operators: sufficient conditions, Ann. Inst. Fourier. 29(3) (1979), 239260.
4. Dasgupta A. and Ruzhansky M., Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces, Bull. Sci. Math. 138 (2014), 756782.
5. Dasgupta A. and Ruzhansky M., The Gohberg lemma, compactness, and essential spectrum of operators on compact Lie groups, J. Anal. Math. 128 (2016), 179190.
6. Delgado J., Estimations L p pour une classe d’opérateurs pseudo-différentiels dans le cadre du calcul de Weyl–Hörmander, J. Anal. Math. 100 (2006), 337374.
7. Fefferman C., L p bounds for pseudo-differential operators, Israel J. Math. 14 (1973), 413417.
8. Fefferman C. and Stein. E. M., H p -spaces of several variables, Acta Math. 129 (1972), 137193.
9. Fischer V., Intrinsic pseudo-differential calculi on any compact Lie group, J. Funct. Anal. 268 (2015), 34043477.
10. Fischer V., Hörmander condition for Fourier multipliers on compact Lie groups, preprint, 2016, arXiv:1610.06348.
11. Fischer V. and Ruzhansky M., Fourier multipliers on graded Lie groups, preprint, 2014, arXiv:1411.6950.
12. Fischer V. and Ruzhansky M., Quantization on nilpotent Lie groups, Progress in Mathematics, Volume 314 (Birkhäuser, Basel, 2016).
13. Garetto C. and Ruzhansky M., Wave equation for sums of squares on compact Lie groups, J. Differential Equations 258(12) (2015), 43244347.
14. Grigor’yan A., Estimates of heat kernels on Riemannian manifolds, in Spectral theory and geometry. ICMS Instructional Conference, Edinburgh, 1998,(ed. Davies B. and Safarov Yu.), London Mathematical Society Lecture Note Series, Volume 273, pp. 140225 (Cambridge University Press, Cambridge, 1999).
15. Grigor’yan A., Hu J. and Lau K.-S., Heat kernels on metric measure spaces, in Geometry and Analysis on Fractals, Springer Proceedings in Mathematics and Statistics, Volume 88, pp. 147207 (Springer, Heidelberg, 2014).
16. Hirschman I. I., Multiplier transformations I, Duke Math. J. 26 (1956), 222242.
17. Hörmander L., Pseudo-differential operators and hypoelliptic equations, in Proc. Symposium on Singular Integrals, pp. 138183 (American Mathematical Society, vol. 10, Providence, RI, 1967).
18. Hounie J., On the L 2 continuity of pseudo-differential operators, Communications in Partial Differential Equations 11(7) (1986), 765778.
19. Li Ch. and Wang R., On the L p -boundedness of several classes of pseudo-differential operators, Chin. Ann. Math. Ser. B 5(2) (1984), 193213.
20. Molahajloo S. and Wong M. W., Pseudodifferential operators on S1 , in New Developments in Pseudo-differential operators, Operator Theory: Advances and Applications, Volume 189, pp. 297306 (Birkhäuser, Basel, 2009).
21. Nursultanov E., Ruzhansky M. and Tikhonov S., Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XVI(5) (2016), 9811017.
22. Rothschild L. and Stein E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137(3–4) (1976), 247320.
23. Ruzhansky M. and Turunen V., Pseudo-differential Operators and Symmetries. Background Analysis and Advanced Topics, Pseudo-Differential Operators. Theory and Applications, Volume 2 (Birkhäuser Verlag, Basel, 2010).
24. Ruzhansky M. and Turunen V., Sharp Gårding inequality on compact Lie groups, J. Funct. Anal. 260(10) (2011), 28812901.
25. Ruzhansky M. and Turunen V., Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces, Int. Math. Res. Not. IMRN 2013(11) (2013), 24392496.
26. Ruzhansky M. and Wirth J., On multipliers on compact Lie groups, Funct. Anal. Appl. 47(1) (2013), 8791.
27. Ruzhansky M. and Wirth J., Global functional calculus for operators on compact Lie groups, J. Funct. Anal. 267 (2014), 144172.
28. Ruzhansky M. and Wirth J., L p Fourier multipliers on compact Lie groups, Math. Z. 280 (2015), 621642.
29. Ruzhansky M., Turunen V. and Wirth J., Hörmander class of pseudo-differential operators on compact Lie groups and global hypoellipticity, J. Fourier Anal. Appl. 20 (2014), 476499.
30. Shubin M. A., Pseudodifferential Operators and Spectral Theory, second edition (Springer, Berlin, 2001). Translated from the 1978 Russian original by Stig I. Andersson.
31. Stein E. M., Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482492.
32. Stein E. M. and Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32 (Princeton University Press, Princeton, NJ, 1971).
33. Strichartz R., Analysis of the laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 4879.
34. Taylor M. E., Partial Differential Equations III. NonLinear Equations (Springer, New York, 1996).
35. Wainger S., Special trigonometric series in k-dimensions, Mem. Amer. Math. Soc. No. 59 (1965), 102.
36. Zygmund A., Trigonometric Series I and II, second edition (Cambridge University Press, Cambridge, New York, Melbourne, 1977).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 1
Total number of PDF views: 92 *
Loading metrics...

Abstract views

Total abstract views: 151 *
Loading metrics...

* Views captured on Cambridge Core between 3rd April 2017 - 18th January 2018. This data will be updated every 24 hours.