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, $1<\rho \le a_{j+1}\mathrm{/}{a}_j,\text{for all }j\in \mathbb{Z}\mathrm{.}$ 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 $L^p(\mathbb{R},\omega )(\omega \in A_p^{-}$), $1<p<\mathrm{\infty }$. 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 $L^1(\mathbb{R},\omega )(\omega \in A_1^{-})$, $L^{\mathrm{\infty }}(\mathbb{R})$ and $BMO(\mathbb{R})$ are also obtained. It is also shown that the local size of $T^{\ast }f$, 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.