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

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