Oscillation and variation for Riesz transform associated with Bessel operators

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