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 \[\Lambda_\kappa=(-\Delta)^{\alpha/2}+\frac{\kappa}{|x|^\alpha} x\cdot \nabla,\] with $\alpha\in (1,2)$, $\alpha<(d+2)/2$ and $\kappa\in \mathbb{R}$. The operator $\Lambda_\alpha$ 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 $V_\rho\left(\{t^\ell\partial_t^\ell T_t^\alpha\}_{t>0}\right)$ with $\rho> 2$, $\ell\in \mathbb{N}$ and $1\vee \frac{d}{\beta}<p<\infty$, 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 $L^{p,\infty}(\mathbb{R}^d)$ for any $1< p<\infty$.