Boundedness of differential transforms for one-sided fractional Poisson-type operator sequence

In this paper, we analyze the convergence speed of a series related with $\mathcal{P}_\tau^\alpha f$ by discussing the behavior of the family of operators \begin{equation*} T_N^\alpha f(t) = \sum_{j=N_1}^{N_2} v_j(\mathcal{P}_{a_{j+1}}^\alpha f(t)-\mathcal{P}_{a_j}^\alpha f(t)),\quad ~N=(N_1,N_2)\in \mathbb{Z}^2\quad \hbox{with} \quad N_1<N_2, \end{equation*} where $\{v_j\}_{j\in \mathbb Z}$ is a bounded number sequence, and $\{a_j\}_{j\in \mathbb{Z}}$ is a $\rho$-lacunary sequence of positive numbers, that is, $1<\rho \leq a_{j+1}/a_j, \text{for all}\ j\in \mathbb{Z}.$ We shall show the boundedness of the maximal operator \begin{equation*}T^*f(t)=\sup_N |T_N^\alpha f(t)|, \quad t\in\mathbb{R}, \end{equation*} in the one-sided weighted Lebesgue spaces $L^p(\mathbb{R},\omega)(\omega \in A_p^-$), $1< p < \infty$. As a consequence we infer the existence of the limit, in norm and almost everywhere, of the family $T_N^\alpha f$ for functions in $L^p(\mathbb{R},\omega)$. Results for $L^1(\mathbb{R},\omega)(\omega \in A_1^-)$, $L^\infty(\mathbb{R})$ and $BMO(\mathbb{R})$ are also obtained. It is also shown that the local size of $T^*f$, for functions $f$ having local support, is the same with the order of a singular integral. Moreover, if $\{v_j\}_{j\in \mathbb Z}\in \ell^p(\mathbb Z)$, we get an intermediate size between the local size of singular integrals and Hardy-Littlewood maximal operator.