Oscillation and variation for Riesz transform associated with Bessel operators

Let $λ>0$ and $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ be the Bessel operator on $\mathbb R_+:=(0,\infty)$. We show that the oscillation operator $\mathcal{O}(R_{Δ_λ,\ast})$ and variation operator $\mathcal{V}_ρ(R_{Δ_λ,\ast})$ of the Riesz transform $R_{Δ_λ}$ associated with $Δ_λ$ are both bounded on $L^p(\mathbb R_+, dm_λ)$ for $p\in(1,\,\infty)$, from $L^1(\mathbb{R}_{+},dm_λ)$ to $L^{1,\,\infty}(\mathbb{R}_{+},dm_λ)$, and from $L^{\infty}(\mathbb{R}_{+},dm_λ)$ to $BMO(\mathbb{R}_{+},dm_λ)$, where $ρ\in (2,\infty)$ and $dm_λ(x):=x^{2λ}dx$. As an application, we give the corresponding $L^p$-estimates for $β$-jump operators and the number of up-crossing.

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