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## 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}, ${\{v\_{j}\}}\_{j\in \mathbb{Z}}$ 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}\}}

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Author

CHAO, ZHANG; TORREA, JOSÃ L.

Publisher

Canadian Mathematical Society; Test accounts (ISSN 0008-414X)

Published

2020

Language

EN

Field

Mathematics (Physical Sciences)