Almost Everywhere Convergence of Inverse Spherical Transforms on Noncompact Symmetric Spaces

C. Meaney; E. Prestini

doi:10.1006/jfan.1997.3123

Almost Everywhere Convergence of Inverse Spherical Transforms on Noncompact Symmetric Spaces

C. Meaney^*, E. Prestini

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

LetGbe a connected noncompact semisimple Lie group with finite center and real rank one. Fix a maximal subgroupK. We considerKbi-invariant functionsfonGand their spherical transformf(λ)=∫_Gf(g)φ _λ(g)dg,whereφ_λdenote the elementary spherical functions onG/Kandλ≥0. We consider the maximal operatorsS_*f(t)=Sup_R>1∫^R ₁f(λ)_λ(a(t))c(λ) ^-2dλand prove thatS* maps boundedly^KL^K_s(G)→L_s(G)+L ₂(G) for 2n/(n+1)<s≤2 wheren=dim(G/K). The result is sharp and it implies a.e. convergence properties of the inverse spherical transforms.

Original language	English
Pages (from-to)	277-304
Number of pages	28
Journal	Journal of Functional Analysis
Volume	149
Issue number	2
DOIs	https://doi.org/10.1006/jfan.1997.3123
Publication status	Published - 1 Oct 1997

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abstract = "LetGbe a connected noncompact semisimple Lie group with finite center and real rank one. Fix a maximal subgroupK. We considerKbi-invariant functionsfonGand their spherical transformf(λ)=∫Gf(g)φ λ(g)dg,whereφλdenote the elementary spherical functions onG/Kandλ≥0. We consider the maximal operatorsS*f(t)=SupR>1∫R 1f(λ)λ(a(t))c(λ) -2dλand prove thatS* maps boundedlyKLKs(G)→Ls(G)+L 2(G) for 2n/(n+1)",

author = "C. Meaney and E. Prestini",

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AU - Prestini, E.

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N2 - LetGbe a connected noncompact semisimple Lie group with finite center and real rank one. Fix a maximal subgroupK. We considerKbi-invariant functionsfonGand their spherical transformf(λ)=∫Gf(g)φ λ(g)dg,whereφλdenote the elementary spherical functions onG/Kandλ≥0. We consider the maximal operatorsS*f(t)=SupR>1∫R 1f(λ)λ(a(t))c(λ) -2dλand prove thatS* maps boundedlyKLKs(G)→Ls(G)+L 2(G) for 2n/(n+1)

AB - LetGbe a connected noncompact semisimple Lie group with finite center and real rank one. Fix a maximal subgroupK. We considerKbi-invariant functionsfonGand their spherical transformf(λ)=∫Gf(g)φ λ(g)dg,whereφλdenote the elementary spherical functions onG/Kandλ≥0. We consider the maximal operatorsS*f(t)=SupR>1∫R 1f(λ)λ(a(t))c(λ) -2dλand prove thatS* maps boundedlyKLKs(G)→Ls(G)+L 2(G) for 2n/(n+1)

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VL - 149

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JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -

Almost Everywhere Convergence of Inverse Spherical Transforms on Noncompact Symmetric Spaces

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