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Oscillation and variation for the Riesz transform associated with Bessel operators

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  18 September 2018

Huoxiong Wu ,
Dongyong Yang  and
Jing Zhang
Huoxiong Wu
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China (huoxwu@xmu.edu.cn; dyyang@xmu.edu.cn)
Dongyong Yang*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China (huoxwu@xmu.edu.cn; dyyang@xmu.edu.cn)
Jing Zhang
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China and School of Mathematics and Statistics, Yili Normal College, Yining Xinjiang 835000, People's Republic of China (zjmath66@126.com)
*
*Corresponding author.
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Abstract

Let λ > 0 and let

be the Bessel operator on ℝ+ := (0,∞). We show that the oscillation operator 𝒪(RΔλ,∗) and variation operator 𝒱ρ(RΔλ,∗) of the Riesz transform RΔλ associated with Δλ are both bounded on Lp(ℝ+, dmλ) for p ∈ (1,∞), from L1(ℝ+, dmλ) to L1,∞(ℝ+, dmλ), and from L∞(ℝ+, dmλ) to BMO(ℝ+, dmλ), where ρ ∈ (2,∞) and dmλ(x) := x2λ dx. As an application, we give the corresponding Lp-estimates for β-jump operators and the number of up-crossings.

Keywords

oscillationvariationBessel operatorRiesz transform

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 1 , February 2019 , pp. 169 - 190
Copyright
Copyright © Royal Society of Edinburgh 2018 

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