{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T09:42:32Z","timestamp":1758706952172,"version":"3.37.3"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T00:00:00Z","timestamp":1684281600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T00:00:00Z","timestamp":1684281600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100007689","name":"Universidade de Aveiro","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100007689","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Fourier Anal Appl"],"published-print":{"date-parts":[[2023,6]]},"abstract":"Abstract<\/jats:title>In this paper, we use the representation theory of the group $$\\textrm{Spin}(m)$$<\/jats:tex-math>\n \n Spin<\/mml:mtext>\n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to develop aspects of the global symbolic calculus of pseudo-differential operators on $$\\textrm{Spin}(3)$$<\/jats:tex-math>\n \n Spin<\/mml:mtext>\n (<\/mml:mo>\n 3<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\textrm{Spin}(4)$$<\/jats:tex-math>\n \n Spin<\/mml:mtext>\n (<\/mml:mo>\n 4<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in the sense of Ruzhansky\u2013Turunen\u2013Wirth. A detailed study of $$\\textrm{Spin}(3)$$<\/jats:tex-math>\n \n Spin<\/mml:mtext>\n (<\/mml:mo>\n 3<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\textrm{Spin}(4)$$<\/jats:tex-math>\n \n Spin<\/mml:mtext>\n (<\/mml:mo>\n 4<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-representations is made including recurrence relations and natural differential operators acting on matrix coefficients. We establish the calculus of left-invariant differential operators and of difference operators on the group $$\\textrm{Spin}(4)$$<\/jats:tex-math>\n \n Spin<\/mml:mtext>\n (<\/mml:mo>\n 4<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and apply this to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some that are locally neither invertible nor hypoelliptic. The paper presents a particular case study for higher dimensional spin groups.<\/jats:p>","DOI":"10.1007\/s00041-023-10015-5","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T23:02:57Z","timestamp":1684364577000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Global Operator Calculus on Spin Groups"],"prefix":"10.1007","volume":"29","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7667-4595","authenticated-orcid":false,"given":"P.","family":"Cerejeiras","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1816-8293","authenticated-orcid":false,"given":"M.","family":"Ferreira","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9066-1819","authenticated-orcid":false,"given":"U.","family":"K\u00e4hler","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3950-4236","authenticated-orcid":false,"given":"J.","family":"Wirth","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,17]]},"reference":[{"issue":"2","key":"10015_CR1","doi-asserted-by":"publisher","first-page":"425","DOI":"10.2307\/2371218","volume":"57","author":"R Brauer","year":"1935","unstructured":"Brauer, R., Weyl, H.: Spinors in $$n$$-dimensions. Am. J. Math. 57(2), 425\u2013449 (1935)","journal-title":"Am. J. Math."},{"key":"10015_CR2","unstructured":"Cardona, D., Ruzhansky, M.: Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups. Preprint (2020). arXiv:2008.09651"},{"issue":"3","key":"10015_CR3","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/j.acha.2011.01.005","volume":"31","author":"P Cerejeiras","year":"2011","unstructured":"Cerejeiras, P., Ferreira, M., K\u00e4hler, U., Teschke, G.: Inversion of the noisy Radon transform on $${\\rm SO} (3)$$ by Gabor frames and sparse recovery algorithms. Appl. Comput. Harmon. Anal. 31(3), 325\u2013345 (2011)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"10015_CR4","doi-asserted-by":"publisher","unstructured":"Connolly, D.: Pseudo-differential Operators on Homogeneous Spaces. PhD Thesis, Imperial College London (2013). https:\/\/doi.org\/10.25560\/23926","DOI":"10.25560\/23926"},{"key":"10015_CR5","doi-asserted-by":"crossref","unstructured":"Delanghe, R., Sommen, F., Souc\u0306ek, V.: Clifford Algebras and Spinor-Valued Functions. A Function Theory for the Dirac Operator. Mathematics and its Applications 53, Kluwer Academic Publishers, Dordrecht etc. (1992)","DOI":"10.1007\/978-94-011-2922-0_2"},{"key":"10015_CR6","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/BF02825255","volume":"41","author":"D Dickinson","year":"1997","unstructured":"Dickinson, D., Gramchev, T., Yoshino, M.: First order pseudodifferential operators on the torus: normal forms, diophantine phenomena and global hypoellipticity. Ann. Univ. Ferrara Sez. VII (N.S.) 41, 51\u201364 (1997)","journal-title":"Ann. Univ. Ferrara Sez. VII (N.S.)"},{"key":"10015_CR7","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1155\/S0161171279000168","volume":"2","author":"J Dieudonn\u00e9","year":"1979","unstructured":"Dieudonn\u00e9, J.: Gelfand pairs and spherical functions. Int. J. Math. Math. Sci. 2, 153\u2013162 (1979)","journal-title":"Int. J. Math. Math. Sci."},{"issue":"11","key":"10015_CR8","doi-asserted-by":"publisher","first-page":"3404","DOI":"10.1016\/j.jfa.2015.03.015","volume":"268","author":"V Fischer","year":"2015","unstructured":"Fischer, V.: Intrinsic pseudo-differential calculi on any compact Lie group. J. Funct. Anal. 268(11), 3404\u20133477 (2015)","journal-title":"J. Funct. Anal."},{"key":"10015_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-29558-9","volume-title":"Quantization on Nilpotent Groups. Progress in Mathematics","author":"V Fischer","year":"2016","unstructured":"Fischer, V., Ruzhansky, M.: Quantization on Nilpotent Groups. Progress in Mathematics, vol. 314. Birkh\u00e4user (2016)"},{"key":"10015_CR10","doi-asserted-by":"publisher","DOI":"10.1201\/b19172","volume-title":"A Course in Abstract Harmonic Analysis","author":"GB Folland","year":"2016","unstructured":"Folland, G.B.: A Course in Abstract Harmonic Analysis, vol. 29, 2nd edn. CRC Press, Boca Raton, FL (2016)","edition":"2"},{"key":"10015_CR11","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511611582","volume-title":"Clifford Algebras and Dirac Operators in Harmonic Analysis","author":"J Gilbert","year":"1991","unstructured":"Gilbert, J., Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)"},{"key":"10015_CR12","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/0040-9383(73)90011-6","volume":"12","author":"S Greenfield","year":"1973","unstructured":"Greenfield, S., Wallach, N.R.: Globally hypoelliptic vectorfields. Topology 12, 247\u2013253 (1973)","journal-title":"Topology"},{"key":"10015_CR13","unstructured":"Hamm, T.: Harmonic Analysis on the 3-Sphere. M.Sc. Thesis, University of Stuttgart (2016)"},{"key":"10015_CR14","volume-title":"Introduction to the Theory of Numbers","author":"GH Hardy","year":"1954","unstructured":"Hardy, G.H., Wright, E.M.: Introduction to the Theory of Numbers. Oxford University Press, Oxford (1954)"},{"key":"10015_CR15","doi-asserted-by":"crossref","unstructured":"H\u00f6rmander, L.: Pseudo-differential operators and hypoelliptic equations. In: A.P. Calder\u00f2n (ed.) Singular Integrals, Proc. Symp. Pure Math. 10, pp. 138\u2013183. Amer. Math. Soc., Providence (1967)","DOI":"10.1090\/pspum\/010\/0383152"},{"key":"10015_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2020.102853","volume":"160","author":"A Kirilov","year":"2020","unstructured":"Kirilov, A., de Moraes, W.A.A., Ruzhansky, M.: Partial Fourier series on compact Lie groups. Bull. Sci. Math. 160, 102853 (2020)","journal-title":"Bull. Sci. Math."},{"issue":"2","key":"10015_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2020.108806","volume":"280","author":"A Kirilov","year":"2021","unstructured":"Kirilov, A., de Moraes, W.A.A., Ruzhansky, M.: Global hypoellipticity and global solvability for vector fields on compact Lie groups. J. Funct. Anal. 280(2), 108806 (2021)","journal-title":"J. Funct. Anal."},{"key":"10015_CR18","doi-asserted-by":"publisher","first-page":"1539","DOI":"10.4171\/dm\/604","volume":"22","author":"M M\u0103ntoiu","year":"2017","unstructured":"M\u0103ntoiu, M., Ruzhansky, M.: Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups. Doc. Math. 22, 1539\u20131592 (2017)","journal-title":"Doc. Math."},{"key":"10015_CR19","doi-asserted-by":"publisher","first-page":"3","DOI":"10.12942\/lrr-2013-3","volume":"16","author":"A Perez","year":"2013","unstructured":"Perez, A.: The spin-foam approach to quantum gravity. Living Rev. Relativ. 16, 3 (2013). https:\/\/doi.org\/10.12942\/lrr-2013-3","journal-title":"Living Rev. Relativ."},{"key":"10015_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-02895-4","volume-title":"Hardy Inequalities on Homogeneous Groups. Progress in Mathematics","author":"M Ruzhansky","year":"2019","unstructured":"Ruzhansky, M., Suragan, D.: Hardy Inequalities on Homogeneous Groups. Progress in Mathematics, vol. 327. Birkh\u00e4user, Basel (2019)"},{"key":"10015_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7643-8514-9","volume-title":"Pseudo-Differential Operators and Symmetries","author":"M Ruzhansky","year":"2010","unstructured":"Ruzhansky, M., Turunen, V.: Pseudo-Differential Operators and Symmetries. Birkh\u00e4user, Basel (2010)"},{"key":"10015_CR22","doi-asserted-by":"publisher","first-page":"2439","DOI":"10.1093\/imrn\/rns122","volume":"11","author":"M Ruzhansky","year":"2013","unstructured":"Ruzhansky, M., Turunen, V.: Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere. Int. Math. Res. Not. IMRN 11, 2439\u20132496 (2013)","journal-title":"Int. Math. Res. Not. IMRN"},{"issue":"3","key":"10015_CR23","doi-asserted-by":"publisher","first-page":"476","DOI":"10.1007\/s00041-014-9322-9","volume":"20","author":"M Ruzhansky","year":"2014","unstructured":"Ruzhansky, M., Turunen, V., Wirth, J.: H\u00f6rmander class of pseudo-differential operators on compact Lie groups and global hypoellipticity. J. Fourier Anal. Appl. 20(3), 476\u2013499 (2014)","journal-title":"J. Fourier Anal. Appl."},{"issue":"1","key":"10015_CR24","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1016\/j.jfa.2014.04.009","volume":"267","author":"M Ruzhansky","year":"2014","unstructured":"Ruzhansky, M., Wirth, J.: Global functional calculus for operators on compact Lie groups. J. Funct. Anal. 267(1), 144\u2013172 (2014)","journal-title":"J. Funct. Anal."},{"key":"10015_CR25","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1007\/s00209-015-1440-9","volume":"280","author":"M Ruzhansky","year":"2015","unstructured":"Ruzhansky, M., Wirth, J.: $$L_p$$-Fourier multipliers on compact Lie groups. Math. Z. 280, 621\u2013642 (2015)","journal-title":"Math. Z."},{"key":"10015_CR26","volume-title":"Pseudo-Differential Operators","author":"M Taylor","year":"1981","unstructured":"Taylor, M.: Pseudo-Differential Operators. Princeton University Press, Princeton, NJ (1981)"},{"issue":"1","key":"10015_CR27","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1007\/BF03042223","volume":"11","author":"P Van Lancker","year":"2001","unstructured":"Van Lancker, P., Sommen, F., Constales, D.: Models for irreducible representations of Spin(m). Adv. Appl. Clifford Algebras 11(1), 271\u2013289 (2001)","journal-title":"Adv. Appl. Clifford Algebras"},{"key":"10015_CR28","first-page":"613","volume":"22","author":"NJ Vilenkin","year":"1968","unstructured":"Vilenkin, N.J.: Special functions and the theory of group representations. Am. Math. Soc. 22, 613 (1968)","journal-title":"Am. Math. Soc."},{"issue":"1","key":"10015_CR29","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1016\/0022-1236(86)90058-3","volume":"68","author":"S Zelditch","year":"1986","unstructured":"Zelditch, S.: Pseudo-differential analysis on hyperbolic surfaces. J. Funct. Anal. 68(1), 72\u2013105 (1986)","journal-title":"J. Funct. Anal."}],"container-title":["Journal of Fourier Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00041-023-10015-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00041-023-10015-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00041-023-10015-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,27]],"date-time":"2023-06-27T00:04:08Z","timestamp":1687824248000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00041-023-10015-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,17]]},"references-count":29,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["10015"],"URL":"https:\/\/doi.org\/10.1007\/s00041-023-10015-5","relation":{},"ISSN":["1069-5869","1531-5851"],"issn-type":[{"type":"print","value":"1069-5869"},{"type":"electronic","value":"1531-5851"}],"subject":[],"published":{"date-parts":[[2023,5,17]]},"assertion":[{"value":"24 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 February 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 April 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 May 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"32"}}