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Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres. (English) Zbl 1479.42031

This paper is devoted to the proof of a theorem of Mihlin-Hörmander type for a distinguished sub-Laplacian on the unit quaternionic sphere \(\mathbb S\) in \(\mathbb H^n\).
The problem is to obtain sufficient conditions on a bounded multiplier \(m\), defined on \((0, +\infty)\), to imply that the operator \(m(\mathcal L)\) is bounded on \(L^p\) for \(p\) in some interval around \(2\). In this kind of results, multipliers are assumed to be locally in some fractional Sobolev space \(H^s\), \(s > 0\), and it is important to understand how small the smoothness order \(s\) can be in order to imply \(L^p\) boundedness. In the main result of the paper, the authors prove that, under the condition \(s>(4n-1)/2\), \(m(\mathcal L)\) is of weak type \((1, 1)\) and bounded on \(L^p (\mathbb S)\) for all \(1< p<\infty\). The exponent \((4n-1)/2\) (equal to half the topological dimension) is optimal, that is, it cannot be replaced by any smaller number.
The proof hinges on a detailed and particularly clear description of the quaternionic spherical harmonic decomposition, which in turn depend on some already known properties of complex spherical harmonic decomposition.
This is the first multiplier result that applies to a sub-Laplacian on a compact sub-Riemannian manifold of corank greater than 1.

MSC:

42B15 Multipliers for harmonic analysis in several variables
43A85 Harmonic analysis on homogeneous spaces
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] Agrachev, A.; Boscain, U.; Gauthier, J-P; Rossi, F., The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal., 256, 8, 2621-2655 (2009) · Zbl 1165.58012 · doi:10.1016/j.jfa.2009.01.006
[2] Ahrens, J., Spectral Decomposition of a Distinct Sub-Laplacian on the Quaternionic Sphere (2016), Masterarbeit: Christian-Albrechts-Universität zu Kiel, Masterarbeit
[3] Astengo, F.; Cowling, M.; Di Blasio, B., The Cayley transform and uniformly bounded representations, J. Funct. Anal., 213, 2, 241-269 (2004) · Zbl 1054.22006 · doi:10.1016/j.jfa.2003.12.009
[4] Axler, Sheldon; Bourdon, Paul; Ramey, Wade, Harmonic Function Theory (2001), New York, NY: Springer New York, New York, NY · Zbl 0959.31001
[5] Baudoin, F.; Wang, J., The subelliptic heat kernels of the quaternionic Hopf fibration, Potential Anal., 41, 3, 959-982 (2014) · Zbl 1304.35234 · doi:10.1007/s11118-014-9403-z
[6] Bellaïche, André, The tangent space in sub-Riemannian geometry, Sub-Riemannian Geometry, 1-78 (1996), Basel: Birkhäuser Basel, Basel · Zbl 0862.53031
[7] Biquard, O.: Quaternionic contact structures, In: Marchiafava, S., Piccinni, P., Pontecorvo, M. (eds) Quaternionic structures in mathematics and physics (Rome, 1999), Univ. Studi Roma “La Sapienza”, Rome, pp. 23-30 (1999) · Zbl 0993.53017
[8] Casarino, V.; Ciatti, P., \(L^p\) joint eigenfunction bounds on quaternionic spheres, J. Fourier Anal. Appl., 23, 4, 886-918 (2017) · Zbl 1378.43006 · doi:10.1007/s00041-016-9506-6
[9] Casarino, V.; Cowling, Mg; Martini, A.; Sikora, A., Spectral multipliers for the Kohn Laplacian on forms on the sphere in \({\mathbb{C}}^n\), J. Geom. Anal., 27, 4, 3302-3338 (2017) · Zbl 1392.32017 · doi:10.1007/s12220-017-9806-3
[10] Cowling, Michael G.; Martini, Alessio, Sub-Finsler Geometry and Finite Propagation Speed, Trends in Harmonic Analysis, 147-205 (2013), Milano: Springer Milan, Milano · Zbl 1271.58013
[11] Cowling, Mg; Sikora, A., A spectral multiplier theorem for a sublaplacian on \(\text{SU(2)} \), Math. Z., 238, 1, 1-36 (2001) · Zbl 0996.42006 · doi:10.1007/PL00004894
[12] Cowling, Mg; Klima, O.; Sikora, A., Spectral multipliers for the Kohn sublaplacian on the sphere in \(\mathbb{C}^n\), Trans. Amer. Math. Soc., 363, 2, 611-631 (2011) · Zbl 1213.42025 · doi:10.1090/S0002-9947-2010-04920-7
[13] Duong, Xt; Ouhabaz, Em; Sikora, A., Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196, 2, 443-485 (2002) · Zbl 1029.43006 · doi:10.1016/S0022-1236(02)00009-5
[14] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla (1981) · Zbl 0051.30303
[15] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. II. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla (1981) · Zbl 0051.30303
[16] Fefferman, C., Phong, D.H.: Subelliptic eigenvalue problems. In: Beckner, W., Calderón, A.P., Fefferman, R., Jones P.W. (eds.) Conference on Harmonic Analysis in Honor of Antoni Zygmund, vol. II (Chicago, Ill., 1981). Wadsworth Math. Ser., Wadsworth, Belmont, CA, pp. 590-606 (1983) · Zbl 0503.35071
[17] Folland, Gb, The tangential Cauchy-Riemann complex on spheres, Trans. Amer. Math. Soc., 171, 83-133 (1972) · Zbl 0249.35013 · doi:10.1090/S0002-9947-1972-0309156-X
[18] Hall, Bc, Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics (2003), New York: Springer-Verlag, New York · Zbl 1026.22001
[19] Hebisch, W.: Functional calculus for slowly decaying kernels, preprint (1995)
[20] Hebisch, W., Multiplier theorem on generalized Heisenberg groups, Colloq. Math., 65, 2, 231-239 (1993) · Zbl 0841.43009 · doi:10.4064/cm-65-2-231-239
[21] Johnson, Kd; Wallach, Nr, Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc., 229, 137-173 (1977) · Zbl 0349.43010 · doi:10.1090/S0002-9947-1977-0447483-0
[22] Martini, A., Müller, D., Nicolussi Golo, S.: Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type, preprint (2018). arXiv:1812.02671 · Zbl 1514.35114
[23] Martini, A., Spectral multipliers on Heisenberg-Reiter and related groups, Ann. Mat. Pura Appl., 194, 4, 1135-1155 (2015) · Zbl 1333.43002 · doi:10.1007/s10231-014-0414-6
[24] Martini, A., Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres, Math. Z., 286, 1539-1574 (2017) · Zbl 1375.42006 · doi:10.1007/s00209-016-1813-8
[25] Martini, A.; Müller, D., Spectral multiplier theorems of Euclidean type on new classes of \(2\)-step stratified groups, Proc. Lond. Math. Soc. (3), 109, 5, 1229-1263 (2014) · Zbl 1310.43003 · doi:10.1112/plms/pdu033
[26] Martini, A.; Müller, D., Spectral multipliers on \(2\)-step groups: topological versus homogeneous dimension, Geom. Funct. Anal., 26, 2, 680-702 (2016) · Zbl 1366.43002 · doi:10.1007/s00039-016-0365-8
[27] Melrose, Richard, Propagation for the Wave Group of a Positive Subelliptic Second-Order Differential Operator, Hyperbolic Equations and Related Topics, 181-192 (1986) · Zbl 0696.35064
[28] Montgomery, Richard, Dido meets Heisenberg, Mathematical Surveys and Monographs, 3-21 (2006), Providence, Rhode Island: American Mathematical Society, Providence, Rhode Island
[29] Müller, D.; Stein, Em, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. (9), 73, 4, 413-440 (1994) · Zbl 0838.43011
[30] Nagel, A.; Rudin, W., Moebius-invariant function spaces on balls and spheres, Duke Math. J., 43, 4, 841-865 (1976) · Zbl 0343.32017 · doi:10.1215/S0012-7094-76-04365-9
[31] Pajas, P.; Raçzka, R., Degenerate representations of the symplectic groups. I. The compact group, J. Math. Phys., 9, 1188-1201 (1968) · Zbl 0165.04201 · doi:10.1063/1.1664699
[32] Seeger, A.; Sogge, Cd, On the boundedness of functions of (pseudo-) differential operators on compact manifolds, Duke Math. J., 59, 3, 709-736 (1989) · Zbl 0698.35169 · doi:10.1215/S0012-7094-89-05932-2
[33] Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton (1971) · Zbl 0232.42007
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