Singular integrals generated by zonal measures

Dmitry Ryabogin; Boris Rubin

doi:10.1090/S0002-9939-01-06242-6

Singular integrals generated by zonal measures

Dmitry Ryabogin^*
, Boris Rubin

^*Corresponding author for this work

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

1 Scopus citations

Abstract

We study L^p-mapping properties of the rough singular integral operator T_νf(x) = ∫₀^∞ dr/r ∫_σn-1 f(x - rθ)dν(θ) depending on a finite Borel measure ν on the unit sphere σ_n-1 in ℝⁿ. It is shown that the conditions sup_|ξ|=1 ∫_σn-1 log (1/|θ · ξ|)d|ν|(θ) < ∞, ν(σ_n-1) = 0 imply the L^p-boundedness of T_ν for all 1 < p < ∞ provided that n > 2 and ν is zonal.

Original language	English
Pages (from-to)	745-751
Number of pages	7
Journal	Proceedings of the American Mathematical Society
Volume	130
Issue number	3
DOIs	https://doi.org/10.1090/S0002-9939-01-06242-6
State	Published - 2002
Externally published	Yes

Keywords

L-boundedness
Singular integrals

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10.1090/S0002-9939-01-06242-6

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abstract = "We study Lp-mapping properties of the rough singular integral operator Tνf(x) = ∫0∞ dr/r ∫σn-1 f(x - rθ)dν(θ) depending on a finite Borel measure ν on the unit sphere σn-1 in ℝn. It is shown that the conditions sup|ξ|=1 ∫σn-1 log (1/|θ · ξ|)d|ν|(θ) < ∞, ν(σn-1) = 0 imply the Lp-boundedness of Tν for all 1 < p < ∞ provided that n > 2 and ν is zonal.",

keywords = "L-boundedness, Singular integrals",

author = "Dmitry Ryabogin and Boris Rubin",

year = "2002",

doi = "10.1090/S0002-9939-01-06242-6",

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T1 - Singular integrals generated by zonal measures

AU - Ryabogin, Dmitry

AU - Rubin, Boris

PY - 2002

Y1 - 2002

N2 - We study Lp-mapping properties of the rough singular integral operator Tνf(x) = ∫0∞ dr/r ∫σn-1 f(x - rθ)dν(θ) depending on a finite Borel measure ν on the unit sphere σn-1 in ℝn. It is shown that the conditions sup|ξ|=1 ∫σn-1 log (1/|θ · ξ|)d|ν|(θ) < ∞, ν(σn-1) = 0 imply the Lp-boundedness of Tν for all 1 < p < ∞ provided that n > 2 and ν is zonal.

AB - We study Lp-mapping properties of the rough singular integral operator Tνf(x) = ∫0∞ dr/r ∫σn-1 f(x - rθ)dν(θ) depending on a finite Borel measure ν on the unit sphere σn-1 in ℝn. It is shown that the conditions sup|ξ|=1 ∫σn-1 log (1/|θ · ξ|)d|ν|(θ) < ∞, ν(σn-1) = 0 imply the Lp-boundedness of Tν for all 1 < p < ∞ provided that n > 2 and ν is zonal.

KW - L-boundedness

KW - Singular integrals

UR - https://www.scopus.com/pages/publications/0035994180

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SP - 745

EP - 751

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 3

ER -

Singular integrals generated by zonal measures

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Keywords

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