Boundedness of differential transforms for fractional Poisson type operators generated by parabolic operators

In this paper we analyze the convergence of the following type of series $$T_N^\alpha f(x,t)=\sum_{j=N_1}^{N_2} v_j(P_{a_{j+1}}^\alpha f(x,t)-P_{a_j}^\alpha f(x,t)),\quad (x,t)\in \mathbb R^{n+1}, \ N=(N_1, N_2)\in \mathbb Z^2,\ \alpha>0,$$ where $\{P_{\tau}^\alpha \}_{\tau>0}$ is the fractional Poisson-type operators generated by the parabolic operator $L=\partial_t-\Delta$ with $\Delta$ being the classical Laplacian, $\{v_j\}_{j\in \mathbb Z}$ a bounded real sequences and $\{a_j\}_{j\in \mathbb Z}$ an increasing real sequence. Our analysis will consist {of} the boundedness, in $L^p(\mathbb{R}^n)$ and in $BMO(\mathbb{R}^n)$, of the operators $T^{\alpha}_N$ and its maximal operator $T^*f(x)= \sup_{N\in \mathbb Z^2} |T^{\alpha}_N f(x)|.$ 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. 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.