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Oscillation and variation for the Riesz transform associated with Bessel operators

  • Huoxiong Wu (a1), Dongyong Yang (a1) and Jing Zhang (a2)


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.


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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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