Equiconvergence theorems for Chébli-Trimèche hypergroups

Brandolini, Luca; Gigante, Giacomo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 2, pp. 211-265.

We consider a Sturm-Liouville operator of the kind $\frac{d^2}{d{t}^2}+\frac{A^{\text{'}}\left(t\right)}{A\left(t\right)}\frac{d}{dt}$ on $\left(0,+\infty \right)$ and the related eigenfunction expansion. We prove that, under suitable assumptions on $A\left(t\right)$, the partial sums of the Fourier integral associated to such expansion behave like the partial sums of the classical Fourier-Bessel transform. This implies an almost everywhere convergence result for $L^p\left(A\left(t\right)dt\right)$ functions. Our methods rely on asymptotic expansions for the eigenfunctions and the Harish-Chandra function that we prove under very weak hypotheses.

Classification : 43A62, 43A32, 34L10

[1] W. R. Bloom and H. Heyer, “Harmonic Analysis of Probability Measures on Hypergroups”, de Gruyter Studies in Mathematics, Vol. 20, 1994. | MR

[2] W. R. Bloom and Z. Xu, The Hardy-Littlewood maximal function for Chébli-Trimèche hypergroups, Contemp. Math. 183 (1995), 45–70. | MR | Zbl

[3] M. Bôcher, On regular singular points of linear differential equations of the second order whose coefficients are not necessarily analytic, Trans. Amer. Math. Soc. 1 (1990), 40–52. | JFM

[4] O. Bracco, “Fonction Maximale Associée à des Opérateurs de Sturm-Liouville Singuliers”, Thèse, Université Louis Pasteur, Strasbourg, 1999. | MR

[5] H. Chebli, Sur un Théorème de Paley-Wiener Associé à la Décomposition Spectrale d’un Opérateur de Sturm-Liouville sur $]0,+\infty [$, J. Funct. Anal. 17 (1974), 447–461. | MR | Zbl

[6] L. Colzani, A. Crespi, G. Travaglini and M. Vignati, Equiconvergence theorems for Fourier-Bessel expansions with applications to the harmonic analysis of radial functions in Euclidean and non-Euclidean spaces, Trans. Amer. Math. Soc. 338 (1993), 43–55. | MR | Zbl

[7] A. Fitouhi and M. M. Hamza, A uniform expansion for the eigenfunction of a singular second-order differential operator, SIAM J. Math Anal. 21 (1990), 1619–1632. | MR | Zbl

[8] M. Flensted-Jensen and T. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Mat. 11 (1973), 245–262. | MR | Zbl

[9] L. Gallardo and K. Trimèche, Renewal theorems for Singular Differential Operators, J. Theoret. Probab. 15 (2002), 161–205. | MR | Zbl

[10] N. N. Lebedev, “Special Functions and their Applications”, Dover Publications, 1972. | MR

[11] F. W. J. Olver, “Asymptotic and Special function”, AKP Classics, 1997. | MR

[12] C. Meaney and E. Prestini, Almost everywhere convergence of inverse spherical transforms on noncompact symmetric spaces, J. Funct. Anal. 149 (1997), 277–304. | MR | Zbl

[13] C. Meaney and E. Prestini, On almost-everywhere convergence of inverse spherical transforms, Pacific J. Math. 170 (1995), 203–215. | MR | Zbl

[14] R. J. Stanton and P. A. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), 251–276. | MR | Zbl

[15] G. N. Watson, “A treatise on the Theory of Bessel Functions”, 2nd edition, Cambridge, University Press, 1996. | MR

[16] H. Zeuner, One dimensional hypergroup, Adv. Math. 76 (1989), 1–18. | MR | Zbl