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.
Bernstein inequalities via the heat semigroup