Oscillation and variation for Riesz transform associated with Bessel operators

Abstract

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