Let \(\lambda \in {(0,\infty )}\) and \(\Delta _{\lambda }\) be the Bessel operator on \(\mathbb {R}_{+}:=(0, \infty )\) defined by \(\begin{aligned} \Delta _{\lambda }:=-x^{-2\lambda }\frac{d}{dx}x^{2\lambda }\frac{d}{dx}. \end{aligned}\) The authors give a decomposition of functions with bounded support in the space \(\textrm{BMO}(\mathbb {R}_{+},dm_{\lambda })\) with \(dm_{\lambda }(x):=x^{2\lambda }dx\) , that is, for any \(f\in {\textrm{BMO}(\mathbb {R}_{+},dm_{\lambda })}\) with bounded support, there exist \(g\in {L^{\infty }(\mathbb {R}_{+},dm_{\lambda })}\) and an \(m_{\lambda }\) -Carleson measure \(\mu \) on \(\mathbb {R}_{+}\times \mathbb {R}_{+}\) such that \(\begin{aligned} f(y)=g(y)+S_{\mu ,P^{[\lambda ]}}(y),\quad a.\,e.\,\,\,y\in \mathbb {R}_{+}, \end{aligned}\) where \(\begin{aligned} S_{\mu ,P^{[\lambda ]}}(y):=\iint _{\mathbb {R}_{+}\times \mathbb {R}_{+}}P^{[\lambda ]}_{t}(y,x)d\mu (x,t),\quad y\in \mathbb {R}_{+} \end{aligned}\) and \(P^{[\lambda ]}_{t}(y,x)\) is the Poisson kernel associated with \(\Delta _{\lambda }\) . Conversely, when \(\mu \) is an \(m_{\lambda }\) -Carleson measure on \(\mathbb {R}_{+}\times \mathbb {R}_{+}\) , the balayage \(S_{\mu ,P^{[\lambda ]}}\) is in \(\textrm{BMO}(\mathbb {R}_{+},dm_{\lambda })\)
