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.

在本文中，我们分析以下类型级数的收敛性$$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_ {1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{ \mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$ where ${\{e^{-t{\mathcal {L}}}\}}_{t>0}$是算子${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$与$\unicode[STIX]的热半群{x1D6E5}$是经典的拉普拉斯算子，属于反向 Hölder 类$RH_{q}$的非负势$V$，其中$q>n/2$和$n\geqslant 3$，$N=(N_{1 },N_{2})\in \mathbb{Z}^{2}$与 $N_{1}<N_{2}$ , ${\{v_{j}\}}_{j\in \mathbb{Z}}$是有界实数序列， ${\{a_{j} \}}_{j\in \mathbb{Z}}$是一个递增的实数序列。

我们的分析将包括在$L^{p}(\mathbb{R}^{n})$和$BMO(\mathbb{R}^{n})$ 中的有界性，运算符$T_{ N}^{{\mathcal{L}}}$及其极大算子$T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f (x)$。

还表明，最大微分变换算子（$V=0$）的局部大小与具有局部支持的函数$f$的奇异积分的阶数相同。此外，如果${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$，我们得到了本地奇异积分的大小和 Hardy-Littlewood 极大算子。