Oscillation and variation for semigroups associated with Bessel operators

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 ${\mathcal O(P^{[\lambda]}_\ast)}$ and variation operator ${\mathcal V}_\rho(P^{[\lambda]}_\ast)$ of the Poisson semigroup $\{P^{[\lambda]}_t\}_{t>0}$ associated with $\Delta_\lambda$ are both bounded on $L^p(\mathbb R_+, dm_\lambda)$ for $p\in(1, \infty)$, $BMO({{\mathbb R}_+},dm_\lambda)$, from $L^1({{\mathbb R}_+},dm_\lambda)$ to $L^{1,\,\infty}({{\mathbb R}_+},dm_\lambda)$, and from $H^1({{\mathbb R}_+},dm_\lambda)$ to $L^1({{\mathbb R}_+},dm_\lambda)$, where $\rho\in(2, \infty)$ and $dm_\lambda(x):=x^{2\lambda}\,dx$. As an application, an equivalent characterization of $H^1({{\mathbb R}_+},dm_\lambda)$ in terms of ${\mathcal V}_\rho(P^{[\lambda]}_\ast)$ is also established. All these results hold if $\{P^{[\lambda]}_t\}_{t>0}$ is replaced by the heat semigroup $\{W^{[\lambda]}_t\}_{t>0}$. }