Boundedness of differential transforms for Poisson semigroups generated by Bessel operators

In this paper we analyze the convergence of the following type of series \begin{equation*} T_N f(x)=\sum_{j=N_1}^{N_2} v_j\Big(\mathcal{P}_{a_{j+1}} f(x)-\mathcal{P}_{a_{j}} f(x)\Big),\quad x\in \mathbb R_+, \end{equation*} where $\{\mathcal{P}_t \}_{t>0}$ is the Poisson semigroup of the Bessel operator $\displaystyle \Delta_\lambda:=-{d^2\over dx^2}-{2\lambda\over x}{d\over dx}$ with $\lambda$ being a positive constant, $N=(N_1, N_2)\in \mathbb Z^2$ with $N_1<N_2,$ $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequences and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence. {Our analysis will consist in the boundedness, in $L^p(\mathbb{R}_+)$ and in $BMO(\mathbb{R}_+)$, of the operators $T_N$ and its maximal operator $T^*f(x)= sup_N \abs{T_N f(x)}.$} It is also shown that the local size of the maximal differential transform operators is the same with the order of a singular integral for functions $f$ having local support.