Abstract:

In this paper we analyze the convergence of the following type of series $T_N f(x)=\sum\limits_{j=N_1}^{N_2} v_j\big(e^{-a_{j+1}(-\Delta)^\alpha} f(x)-e^{-a_{j}(-\Delta)^\alpha} f(x)\big), \quad x\in \mathbb{R} ^n,$ where $\{e^{-t(-\Delta)^\alpha} \}_{t>0}$ is the heat semigroup of the fractional Laplacian $(-\Delta)^\alpha,$ $N=(N_1, N_2)\in {\Bbb Z}^2$ with $N_1<N_2,$ $\{v_j\}_{j\in {\Bbb Z}}$ is a bounded real sequences and $\{a_j\}_{j\in {\Bbb Z}}$ 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.

Key words: Differential transforms, Heat semigroup, Fractional Laplacian, Maximal operator, Lacunary sequence