Oscillation and variation for semigroups associated with Bessel operators

Let $\lambda >0$ and $\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$betheBesseloperatoron$\mathbb R_+:=(0,\infty)$. We show that the oscillation operator $\mathcal{O}(P_{\ast }^{[\lambda ]})$ and variation operator ${\mathcal{V}}_{\rho }(P_{\ast }^{[\lambda ]})$ of the Poisson semigroup $\{P_t^{[\lambda ]}{\}}_{t>0}$ associated with ${\mathrm{\Delta }}_{\lambda }$ are both bounded on $L^p({\mathbb{R}}_{+},d{m}_{\lambda })$ for $p\in (1,\mathrm{\infty })$, $BMO({{\mathbb R}+},dm\lambda)$,from$L^1({{\mathbb R}+},dm\lambda)$to$L^{1,,\infty}({{\mathbb R}+},dm\lambda)$,andfrom$H^1({{\mathbb R}+},dm\lambda)$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 $H^1({\mathbb{R}}_{+},d{m}_{\lambda })$ in terms of ${\mathcal{V}}_{\rho }(P_{\ast }^{[\lambda ]})$ is also established. All these results hold if $\{P_t^{[\lambda ]}{\}}_{t>0}$ is replaced by the heat semigroup $\{W_t^{[\lambda ]}{\}}_{t>0}$. }