Abstract
In this paper we prove L p-boundedness properties of spectral multipliers associated with multidimensional Bessel operators. In order to do this we estimate the L p-norm of the imaginary powers of Bessel operators. We also prove that the Hankel multipliers of Laplace transform type on (0,∞)n are principal value integral operators of weak type (1,1).
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Betancor, J.J., Rodríguez-Mesa, L.: Weighted inequalities for Hankel convolution operators. Ill. J. Math. 44(2), 230–245 (2000)
Betancor, J.J., Stempak, K.: Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform. Tohoku Math. J. 53(1), 109–129 (2001)
Betancor, J.J., Martínez, T., Rodríguez-Mesa, L.: Laplace transform type multipliers for Hankel transforms. Can. Math. Bull. 51, 487–496 (2008)
Cowling, M.: Harmonic analysis on semigroups. Ann. Math. 117, 267–283 (1983)
Cowling, M., Meda, S.: Harmonic analysis and ultracontractivity. Trans. Am. Math. Soc. 340(2), 733–752 (1993)
Dinger, U.: Weak type (1,1) estimates of the maximal function for the Laguerre semigroup in finite dimensions. Rev. Mat. Iberoam. 8(1), 93–118 (1992)
García-Cuerva, J., Mauceri, G., Sjögren, P., Torrea, J.L.: Spectral multipliers for the Ornstein-Uhlenbeck semigroup. J. Anal. Math. 78, 281–305 (1999)
Garrigós, G., Seeger, A.: Characterizations of Hankel multipliers. Math. Ann. 342, 31–68 (2008)
Gasper, G., Trebels, W.: Multiplier criteria of Hormander type for Fourier series and applications to Jacobi series and Hankel transforms. Math. Ann. 242(3), 225–240 (1979)
Gosselin, J., Stempak, K.: A weak type estimate for Fourier-Bessel multipliers. Proc. Am. Math. Soc. 106(3), 655–662 (1989)
Guy, D.L.: Hankel transformations and weighted p-norms. Trans. Am. Math. Soc. 95, 137–189 (1960)
Kapelko, R.: A multiplier theorem for Hankel transforms. Rev. Mat. Complut. 11(2), 281–288 (1998)
Lebedev, N.N.: Special Functions and Their Applications. Dover, New York (1972)
Meda, S.: A general multiplier theorem. Proc. Am. Math. Soc. 110(3), 639–647 (1990)
Muckenhoupt, B.: Hardy’s inequality with weights. Stud. Math. 44, 31–38 (1972)
Nowak, A., Sjögren, P.: Weak type (1,1) estimates for maximal operators associated with various multi-dimensional systems of Laguerre functions. Indiana Univ. Math. J. 56, 417–436 (2007)
Nowak, A., Stempak, K.: Weighted estimates for the Hankel transform transplantation operator. Tohoku Math. J. 58, 277–301 (2006)
Sikora, A., Wright, J.: Imaginary powers of Laplace operators. Proc. Am. Math. Soc. 129, 1745–1754 (2001)
Stein, E.M.: Topics in harmonic analysis related to the Littlewood-Paley theory. Ann. of Math. Studies, vol. 63. Princeton Univ. Press, Princeton (1970)
Stempak, K., Torrea, J.L.: Poisson integrals and Riesz transform for Hermite function expansions with weights. J. Funct. Anal. 202, 443–472 (2003)
Stempak, K., Trebels, W.: Hankel multipliers and transplantation operators. Stud. Math. 126(1), 51–66 (1997)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1966)
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Communicated by Hans G. Feichtinger.
J.J. Betancor was partially supported by MTM2007/65609. A.J. Castro was supported by a grant for Master studies of “la Caixa”. J. Curbelo was supported by a grant JAE-Predoc of the CSIC (Spain).
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Betancor, J.J., Castro, A.J. & Curbelo, J. Spectral Multipliers for Multidimensional Bessel Operators. J Fourier Anal Appl 17, 932–975 (2011). https://doi.org/10.1007/s00041-010-9162-1
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DOI: https://doi.org/10.1007/s00041-010-9162-1