Oscillation and variation for the Riesz transform associated with Bessel operators

Abstract

Let λ > 0 and letbe 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.