Boundedness of differential transforms for heat semigroups generated by fractional Laplacian

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(e^{-a_{j+1}(-\Delta)^\alpha} f(x)-e^{-a_{j}(-\Delta)^\alpha} f(x)\Big),\quad x\in \mathbb R^n, \end{equation*} where $\{e^{-t(-\Delta)^\alpha} \}_{t>0}$ is the heat semigroup of the fractional Laplacian $(-\Delta)^\alpha,$ $N=(N_1, N_2)\in \mathbb Z^2$ 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 $L^p(\mathbb{R}^n)$ and in $BMO(\mathbb{R}^n)$, of the operators $T_N$ and its maximal operator $\displaystyle T^*f(x)= \sup_N |T_N f(x)|.$ 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.