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Operational calculus and integral transforms for groups with finite propagation speed

Gordon Blower and Ian Doust ORCID logo

Abstract

Let A be the generator of a strongly continuous cosine family (cos(tA))t on a complex Banach space E. The paper develops an operational calculus for integral transforms and functions of A using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on E. Examples include the Mellin transform and the Mehler–Fock transform. The paper uses functional calculus for the cosine family cos(tΔ) which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm–Liouville hypergroups with Laplace representation as well as analogues to the Kunze–Stein phenomenon in the hypergroup convolution setting.

MSC 2010: 47A60; 47D09

Funding statement: This research was partially supported by a Scheme 2 Grant from the London Mathematical Society.

Acknowledgements

Gordon Blower thanks the University of New South Wales for hospitality.

References

[1] E. Berkson and T. A. Gillespie, Stečkin’s theorem, transference and spectral decompositions, J. Funct. Anal. 70 (1987), 140–170. 10.1016/0022-1236(87)90128-5Search in Google Scholar

[2] W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, 1994. Search in Google Scholar

[3] G. Blower, Multipliers for semigroups, Proc. Edin. Math. Soc. (2) 39 (1996), 241–252. 10.1017/S0013091500022975Search in Google Scholar

[4] K. Boyadzhiev and R. deLaubenfels, Spectral theorem for unbounded strongly continuous groups on a Hilbert space, Proc. Amer. Math. Soc. 120 (1994), 127–136. 10.1090/S0002-9939-1994-1186983-0Search in Google Scholar

[5] I. Chavel, Riemannian Geometry – A Modern Introduction, Cambridge Tracts in Math. 108, Cambridge University Press, Cambridge, 1993. Search in Google Scholar

[6] I. Chavel, Isoperimetric Inequalities, Cambridge Tracts in Math. 145, Cambridge University Press, Cambridge, 2001. Search in Google Scholar

[7] H. Chebli, Sur un théorème de Paley–Wiener associé à la décomposition spectrale d’une opérateur de Sturm–Liouville sur ]0,[, J. Funct. Anal. 17 (1974), 447–461. Search in Google Scholar

[8] H. Chebli, Théorème de Paley–Wiener associé à un opérateur différentiel singulier sur (0,), J. Math. Pures Appl. (9) 58 (1979), 1–19. Search in Google Scholar

[9] J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15–53. 10.4310/jdg/1214436699Search in Google Scholar

[10] P. R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973), 401–414. 10.1016/0022-1236(73)90003-7Search in Google Scholar

[11] R. R. Coifman and G. Weiss, Transference Methods in Analysis, CBMS Regional Conf. Ser. Math. 31, American Mathematical Society, Providence, 1976. Search in Google Scholar

[12] M. G. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded H functional calculus, J. Aust. Math. Soc. Ser. A 60 (1996), 51–89. Search in Google Scholar

[13] A. Erdélyi, Higher Transcendental Functions, Volume I, McGraw–Hill, New York, 1953. Search in Google Scholar

[14] A. Erdélyi, Tables of Integral Transforms, Volume I, McGraw–Hill, New York, 1954. Search in Google Scholar

[15] A. Erdélyi, Tables of Integral Transforms, Volume II, McGraw–Hill, New York, 1954. Search in Google Scholar

[16] G. Gigante, Transference for hypergroups, Collect. Math. 52 (2001), 127–155. Search in Google Scholar

[17] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Monogr., Oxford University Press, Oxford, 1985. Search in Google Scholar

[18] M. Haase, A transference principle for general groups and functional calculus on UMD spaces, Math. Ann. 345 (2009), 245–265. 10.1007/s00208-009-0347-3Search in Google Scholar

[19] E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Ontario, 1969. Search in Google Scholar

[20] R. I. Jewett, Spaces with an abstract convolution of measures, Adv. Math. 18 (1975), 1–101. 10.1016/0001-8708(75)90002-XSearch in Google Scholar

[21] P. D. Lax and R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal. 46 (1982), 280–350. 10.1016/0022-1236(82)90050-7Search in Google Scholar

[22] A. McIntosh, Operators which have an H functional calculus, Miniconference on Operator Theory and Partial Differential Equations (North Ryde 1986), Proc. Centre Math. Anal. Austral. Nat. Univ. 14, Australian National University, Canberra (1986), 210–231. Search in Google Scholar

[23] F. G. Mehler, Über eine mit den Kugel- und Cylinderfunctionen verwandte Funktion und ihre Anwendung in der Theorie der Electricitätsvertheilung, Math. Ann. 18 (1881), 161–194. 10.1007/BF01445847Search in Google Scholar

[24] I. N. Sneddon, The Use of Integral Transforms, McGraw–Hill, New York, 1972. Search in Google Scholar

[25] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood–Paley Theory, Ann. of Math. Stud. 63, Princeton University Press, Princeton, 1970. Search in Google Scholar

[26] R. S. Strichartz, Analysis of the Laplacian on a complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48–79. 10.1016/0022-1236(83)90090-3Search in Google Scholar

[27] M. E. Taylor, Lp-estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), 773–793. Search in Google Scholar

[28] M. Taylor, Functions of the Laplace operator on manifolds with lower Ricci and injectivity bounds, Comm. Partial Differential Equations 34 (2009), 1114–1126. 10.1080/03605300902892485Search in Google Scholar

[29] C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar. 32 (1978), 75–96. 10.1007/BF01902205Search in Google Scholar

[30] K. Trimèche, Transformation intégrale de Weyl et théorème de Paley–Wiener associés à un opérateur différentielle singulier sur (0,), J. Math. Pures Appl. (9) 60 (1981), 51–98. Search in Google Scholar

[31] M. Uiterdijk, A functional calculus for analytic generators of C0-groups, Integral Equations Operator Theory 36 (2000), 349–369. Search in Google Scholar

[32] M. Voit, Positive characters on commutative hypergroups and some applications, Math. Z. 198 (1988), 405–421. 10.1007/BF01184674Search in Google Scholar

[33] E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th ed., Cambridge University Press, London, 1927. Search in Google Scholar

[34] H. Zeuner, One-dimensional hypergroups, Adv. Math. 76 (1989), 1–18. 10.1016/0001-8708(89)90041-8Search in Google Scholar

Received: 2015-8-31
Revised: 2016-8-30
Accepted: 2017-3-28
Published Online: 2017-5-25
Published in Print: 2017-10-1

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