Boundedness of Differential Transforms for Heat Semigroups Generated by Schrödinger Operators

Abstract

In this paper we analyze the convergence of the following type of series

$$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$

where ${\{e^{-t{\mathcal{L}}}\}}_{t>0}$ is the heat semigroup of the operator ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$ with $\unicode[STIX]{x1D6E5}$ being the classical laplacian, the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q}$ with $q>n/2$ and $n\geqslant 3$, $N=(N_{1},N_{2})\in \mathbb{Z}^{2}$ 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 $T_{N}^{{\mathcal{L}}}$ and its maximal operator $T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$.

It is also shown that the local size of the maximal differential transform operators (with $V=0$) 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.