Applications of an inverse abel transform for jacobi analysis

Weak-l1 estimates and the kunze-Stein phenomenon

Research output: Contribution to journalArticle

Abstract

For the Jacobi hypergroup (R+,Δ,), the weak-L1 estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution to the Euclidean convolution. More generally, let T be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood- Paley g-function for the Jacobi hypergroup, which are defined by using. Then we shall give a standard shape of Tf for f ∈ L1(Δ), from which its weak-L1 estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.

Original languageEnglish
Pages (from-to)77-112
Number of pages36
JournalTokyo Journal of Mathematics
Volume41
Issue number1
DOIs
Publication statusPublished - 2018 Jun 1

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Jacobi
Hypergroup
Maximal Operator
Transform
Convolution
Estimate
kernel
G-function
Euclidean
Siméon Denis Poisson
Alternatives
Form

Keywords

  • Jacobi analysis
  • Kunze-Stein phenomenon
  • Maximal function
  • Weak-l1 estimate

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Applications of an inverse abel transform for jacobi analysis: Weak-l1 estimates and the kunze-Stein phenomenon",
abstract = "For the Jacobi hypergroup (R+,Δ,), the weak-L1 estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution to the Euclidean convolution. More generally, let T be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood- Paley g-function for the Jacobi hypergroup, which are defined by using. Then we shall give a standard shape of Tf for f ∈ L1(Δ), from which its weak-L1 estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.",
keywords = "Jacobi analysis, Kunze-Stein phenomenon, Maximal function, Weak-l1 estimate",
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T2 - Weak-l1 estimates and the kunze-Stein phenomenon

AU - Kawazoe, Takeshi

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N2 - For the Jacobi hypergroup (R+,Δ,), the weak-L1 estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution to the Euclidean convolution. More generally, let T be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood- Paley g-function for the Jacobi hypergroup, which are defined by using. Then we shall give a standard shape of Tf for f ∈ L1(Δ), from which its weak-L1 estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.

AB - For the Jacobi hypergroup (R+,Δ,), the weak-L1 estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution to the Euclidean convolution. More generally, let T be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood- Paley g-function for the Jacobi hypergroup, which are defined by using. Then we shall give a standard shape of Tf for f ∈ L1(Δ), from which its weak-L1 estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.

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