Boundedness of differential transforms for Poisson semigroups generated by Bessel operators

Abstract

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(\mathcal{P}_{a_{j+1}} f(x)-\mathcal{P}_{a_{j}} f(x)\Big),\quad x\in \mathbb R_+, \end{equation*} where $\{{\mathcal{P}}_t{\}}_{t>0}$ is the Poisson semigroup of the Bessel operator ${\Delta }_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ with $\lambda$ being a positive constant, $N=(N_1,N_2)\in {\mathbb{Z}}^2$ 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 $L^p({\mathbb{R}}_{+})$ and in $BMO({\mathbb{R}}_{+})$, of the operators $T_N$ and its maximal operator $ T^*f(x)= sup_N \abs{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.

17 pages