Fractional Vector-Valued Littlewood-Paley-Stein Theory for Semigroups

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 $L^p$-spaces of functions with values in Banach spaces. We give a characterization of the classes of Banach spaces for which the fractional Litlewood-Paley $g$-function is bounded on $L^p$-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. It is also shown that the same kind of results exist for the case of the fractional area function and the fractional $g^*_\lambda$-function on $\mathbb{R}^n$. At last, we consider the relationship of the almost sure finiteness of the fractional Littlewood-Paley $g$-function, area function and $g^*_\lambda$-function with the Lusin cotype property of the underlying Banach space. As a byproduct of the techniques developed, one can get some results of independent interest for vector-valued Calder\'on--Zygmund operators. For example, one can get the following characterization, a Banach space $\mathbb{B}$ is UMD if and only if for some (or, equivalently, for every) $p\in [1,\infty)$, $\displaystyle \lim_{\epsilon \rightarrow 0} \int_{|x-y|> \epsilon} \frac{f(y)}{x-y}dy$ exists \textup{a.e.} $x\in \mathbb{R}$ for every $f\in L^p_\mathbb{B}(\mathbb{R}).$