Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators

In this paper we consider fractional Kolmogorov operators defined, in ${\mathbb{R}}^d$, by $${\mathrm{\Lambda }}_{\kappa }=(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}+\frac{\kappa }{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}x\cdot \mathrm{\nabla },$$ with $\alpha \in (1,2)$, $\alpha <(d+2)\mathrm{/}2$ and $\kappa \in \mathbb{R}$. The operator ${\mathrm{\Lambda }}_{\alpha }$ generates a holomorphic semigroup $\{T_t^{\alpha }{\}}_{t>0}$ in $L^2({\mathbb{R}}^d)$ provided that $\kappa <{\kappa }_c$ where ${\kappa }_c$ is a critical coupling constant. We establish $L^p$-boundedness properties for the variation operators $V_{\rho }(\{t^{\mathrm{\ell }}{\mathrm{\partial }}_t^{\mathrm{\ell }}{T}_t^{\alpha }{\}}_{t>0})$ with $\rho >2$, $\mathrm{\ell }\in \mathbb{N}$ and $1\vee \frac{d}{\beta }<p<\mathrm{\infty }$, where $\beta$ depends on $\kappa$. We also study the behavior of these variation operators in the endpoint $L^{1\vee \frac{d}{\beta }}({\mathbb{R}}^d)$ and we prove that $V_2(\{T_t^{\alpha }{\}}_{t>0})$ is not bounded from $L^p({\mathbb{R}}^d)$ to $L^{p,\mathrm{\infty }}({\mathbb{R}}^d)$ for any $1<p<\mathrm{\infty }$.