Oscillation and variation for semigroups associated with Bessel operators

Abstract

Let $\lambda >0$ and ${△}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator on ${\mathbb{R}}_{+}:=(0,\infty )$. We show that the oscillation operator $\mathcal{O}(P_{\ast }^{[\lambda ]})$ and variation operator ${\mathcal{V}}_{\rho }(P_{\ast }^{[\lambda ]})$ of the Poisson semigroup $\{P_t^{[\lambda ]}{\}}_{t>0}$ associated with ${\Delta }_{\lambda }$ are both bounded on $L^p({\mathbb{R}}_{+},d{m}_{\lambda })$ for $p\in (1,\infty )$, $BMO({\mathbb{R}}_{+},d{m}_{\lambda })$, from $L^1({\mathbb{R}}_{+},d{m}_{\lambda })$ to $L^{1,\infty }({\mathbb{R}}_{+},d{m}_{\lambda })$, and from $H^1({\mathbb{R}}_{+},d{m}_{\lambda })$ to $L^1({\mathbb{R}}_{+},d{m}_{\lambda })$, where $\rho \in (2,\infty )$ and $d{m}_{\lambda }(x):=x^{2\lambda }dx$. 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 $\{W_t^{[\lambda ]}{\}}_{t>0}$. }

20 pages