Boundedness of Differential Transforms for Poisson Semigroups Generated by Bessel Operators

In this paper we analyze the convergence of the following type of series: $$ T_N \,\,f(x)=\sum_{j=N_1}<^>{N_2} v_j\Big({\mathcal P}_{a_{j+1}} \,\,f(x)-{\mathcal P}_{a_{j}} \,\,f(x)\Big),\quad x\in \mathbb R_+, $$ where $\{{\mathcal P}_{t} \}_{t\gt0}$ is the Poisson semigroup associated with 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 $N_1 \lt N_2,$ $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequence 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 TN and its maximal operator $\displaystyle T<^>*\,\,f(x)= \sup_N \left\vert T_N \,\,f(x)\right\vert.$ 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.