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

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

${\{e^{-t{\mathcal{L}}}\}}_{t>0}$

${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$

$\unicode[STIX]{x1D6E5}$

$V$

$RH_{q}$

$q>n/2$

$n\geqslant 3$

$N=(N_{1},N_{2})\in \mathbb{Z}^{2}$

$N_{1}<N_{2}$

${\{v_{j}\}}_{j\in \mathbb{Z}}$

${\{a_{j}\}}_{j\in \mathbb{Z}}$

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.

We consider the fractional derivative of a general Poisson semigroup. With this fractional derivative, we define the generalized fractional Littlewood–Paley g-function for semigroups acting on Lp-spaces of functions with values in Banach spaces. We give a characterization of the classes of Banach spaces for which the fractional Littlewood–Paley g-function is bounded on Lp-spaces. We show that the class of Banach spaces is independent of the order of derivation and coincides with the classical (Lusin-type/-cotype) case. We also show that the same kind of results exist for the case of the fractional area function and the fractional gλ*-function on ℝn. Finally, we consider the relationship of the almost sure finiteness of the fractional Littlewood–Paley g-function, the area function and the gλ*-function with the Lusin-cotype property of the underlying Banach space. As a byproduct of the techniques developed, one can find some results of independent interest for vector-valued Calderón–Zygmund operators. For example, one can find the following characterization: a Banach space is the unconditional martingale difference if and only if, for some (or, equivalently, for every) p ∈ [1, ∞), dy exists for almost every x ∈ ℝ and every .

We define the higher order Riesz transforms and the Littlewood-Paley $g$ -function associated to the differential operator ${{L}_{\lambda }}f(\theta )\,=\,-{f}''(\theta )-2\lambda \cot \theta {f}'(\theta )+{{\lambda }^{2}}f(\theta )$ . We prove that these operators are Calderón–Zygmund operators in the homogeneous type space $((0,\,\pi ),\,{{(\sin t)}^{2\lambda }}dt)$ . Consequently, ${{L}^{p}}$ weighted, ${{H}^{1}}\,-\,{{L}^{1}}$ and ${{L}^{\infty }}\,-\,BMO$ inequalities are obtained.

Riesz transforms $R_\mu$ related to the Bessel operators

$$ \Delta_\mu=x^{-\mu-1/2}Dx^{2\mu+1}Dx^{-\mu-1/2} $$

are studied in this work. We develop for $R_\mu$ a theory that runs parallel to that for the Euclidean Hilbert transform. It is proved that $R_\mu$ is actually a Calderón–Zygmund singular integral operator. Also, $R_\mu$ is seen to be the boundary value of the appropriate harmonic extension for this context. Finally, we analyse weighted inequalities involving $R_\mu$.

Let Β1, Β2 be a pair of Banach spaces and T be a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent: T is bounded from into for some p (or equivalently for every p) in the range 1 < p < ∞; T is bounded from into BMOB2; T is bounded from BMOB1 into BMOB2; T is bounded from into . Applications to UMD and martingale cotype properties are given. We also prove that the Hardy space defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.