Almost Everywhere Convergence of Inverse Spherical Transforms on Noncompact Symmetric Spaces

Meaney, Christopher; Prestini, Elena

doi:10.1006/jfan.1997.3123

Cited by 6 publications

(20 citation statements)

References 21 publications

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“…In the case of noncompact symmetric spaces of rank one, the above Corollary was obtained by Meaney and Prestini in [12] and [13], using the boundedness of the related maximal operator.…”

Section: Resultsmentioning

confidence: 83%

Equiconvergence theorems for Chébli-Trimèche hypergroups

Brandolini¹,

Gigante²

2009

Annali Scuola Normale Superiore - Classe Di Scienze

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We consider a Sturm-Liouville operator of the kindon (0, +∞) and the related eigenfunction expansion. We prove that, under suitable assumptions on A (t), 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 (A (t) dt) functions. Our methods rely on asymptotic expansions for the eigenfunctions and the Harish-Chandra function that we prove under very weak hypotheses.

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“…In the case of noncompact symmetric spaces of rank one, the above Corollary was obtained by Meaney and Prestini in [12] and [13], using the boundedness of the related maximal operator.…”

Section: Resultsmentioning

confidence: 83%

Equiconvergence theorems for Chébli-Trimèche hypergroups

Brandolini¹,

Gigante²

2009

Annali Scuola Normale Superiore - Classe Di Scienze

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“…The study of convergence properties of inverse spherical transforms of radial functions on noncompact symmetric spaces [1], [6], [7] requires estimates on singular integrals with exponential weights, as defined below, due to the exponential growth at infinity of the radial part D(t)dt of the measure, where…”

Section: F(t) = 1 W(t) V (F(r)w(r))(t)mentioning

confidence: 99%

“…In the case V = C, α = 0 and β = p+2q 2 (more precisely w(t) = D(t)), Theorem 1 has been used to obtain sharp results on almost everywhere convergence of inverse pherical transforms on noncompact symmetric spaces [7]. In [2] estimates in the context of Lorentz spaces have been obtained for V = H in the case α = 0 and β = 1.…”

Section: Corollary Letmentioning

confidence: 99%

“…Under suitable conditions on the weight w(t) of exponential type, we prove boundedness of V from L p spaces, defined on [1, +∞) with respect to the measure w 2 (t)dt, to L p + L 2 , 1 < p ≤ 2, with the same density measure. These operators, that arise in questions of harmonic analysis on noncompact symmetric spaces, are bounded from L p to L p , 1 < p < ∞, if and only if p = 2.The study of convergence properties of inverse spherical transforms of radial functions on noncompact symmetric spaces [1], [6], [7] requires estimates on singular integrals with exponential weights, as defined below, due to the exponential growth at infinity of the radial part D(t)dt of the measure, wherewith p and q suitable nonnegative integers that depend upon the geometry of the symmetric space.In what follows we define a class of functions w(t), that include exponentials, and prove boundedness of the operators …”

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confidence: 99%

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Singular integrals with exponential weights

Prestini

1996

Proc. Amer. Math. Soc.

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Abstract. We study the operators V f(t) = 1 w(t) V (f(r)w(r))(t)where V is the Hardy-Littlewood maximal function, the Hilbert transform or Carleson operator. Under suitable conditions on the weight w(t) of exponential type, we prove boundedness of V from L p spaces, defined on [1, +∞) with respect to the measure w 2 (t)dt, to L p + L 2 , 1 < p ≤ 2, with the same density measure. These operators, that arise in questions of harmonic analysis on noncompact symmetric spaces, are bounded from L p to L p , 1 < p < ∞, if and only if p = 2.The study of convergence properties of inverse spherical transforms of radial functions on noncompact symmetric spaces [1], [6], [7] requires estimates on singular integrals with exponential weights, as defined below, due to the exponential growth at infinity of the radial part D(t)dt of the measure, wherewith p and q suitable nonnegative integers that depend upon the geometry of the symmetric space.In what follows we define a class of functions w(t), that include exponentials, and prove boundedness of the operators

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“…A natural analogue of the disc multiplier in the framework of spherical analysis on Riemannian symmetric spaces of rank one was introduced by Meaney and Prestini in the mid-90's and the study was completed in the paper [18] with almost sharp statements about the mapping properties of the maximal operator associated with the disc multiplier. In the present paper we follow in their footsteps and generalize their results to Jacobi analysis, and we establish the missing endpoint results in the setting of Jacobi analysis.…”

mentioning

confidence: 99%