Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators

Abstract

In this paper we consider fractional Kolmogorov operators defined, in ${\mathbb{R}}^d$, by \[Λ_κ=(-Δ)^{α/2}+\fracκ{|x|^α} x\cdot \nabla,\] with $\hat{I}\pm \in (1,2)$, $\hat{I}\pm <(d+2)/2$ and $\hat{I}º\in \mathbb{R}$. The operator $\hat{I}{›}_{\hat{I}}\pm$ generates a holomorphic semigroup $\{T_t^{\hat{I}}\pm {\}}_{t>0}$ in $L^2({\mathbb{R}}^d)$ provided that $\hat{I}º<\hat{I}{º}_c$ where $\hat{I}{º}_c$ is a critical coupling constant. We establish $L^p$-boundedness properties for the variation operators $V_{\ddot{I}}\left(\{t^{\ell }{\partial }_t^{\ell }{T}_t^{\hat{I}}\pm {\}}_{t>0}\right)$ with $\ddot{I}>2$, $\ell \in \mathbb{N}$ and $1\vee {\frac{d}{\hat{I}}}^2<p<\infty$, where ${\hat{I}}^2$ depends on $\hat{I}º$. We also study the behavior of these variation operators in the endpoint $L^{1\vee {\frac{d}{\hat{I}}}^2}({\mathbb{R}}^d)$ and we prove that $V_2(\{T_t^{\hat{I}}\pm {\}}_{t>0})$ is not bounded from $L^p({\mathbb{R}}^d)$ to $L^{p,\infty }({\mathbb{R}}^d)$ for any $1<p<\infty$.