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

Abstract

In this paper we analyze the convergence of the following type of series $$T_N^{\alpha }f(x,t)=\sum _{j=N_1}^{N_2}{v}_j(P_{a_{j+1}}^{\alpha }f(x,t)-P_{a_j}^{\alpha }f(x,t)),(x,t)\in {\mathbb{R}}^{n+1},N=(N_1,N_2)\in {\mathbb{Z}}^2,\alpha >0,$$ 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 $T_N^{\alpha }$ and its maximal operator $T^{\ast }f(x)={\sup }_{N\in {\mathbb{Z}}^2}|T_N^{\alpha }f(x)|.$ 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.

22 pages

References in corpus (1)

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