Variation operators associated with semigroups generated by Hardy operators involving fractional Laplacians in a half space

We represent by $\{W_{\lambda ,t}^{\alpha }{\}}_{t>0}$ the semigroup generated by $-{\mathbb{L}}_{\lambda }^{\alpha }$, where ${\mathbb{L}}_{\lambda }^{\alpha }$ is a Hardy operator on a half space. The operator ${\mathbb{L}}_{\lambda }^{\alpha }$ includes a fractional Laplacian and it is defined by $${\mathbb{L}}_{\lambda }^{\alpha }=(-\mathrm{\Delta }{)}_{{\mathbb{R}}_{+}^d}^{\alpha \mathrm{/}2}+\lambda x_d^{-\alpha },\alpha \in (0,2],\lambda \ge 0.$$ We prove that, for every $k\in \mathbb{N}$, the $\rho$-variation operator ${\mathcal{V}}_{\rho }\left(\left\{t^k{\mathrm{\partial }}_t^k{W}_{\lambda ,t}^{\alpha }\right\}\right)$ is bounded on $L^p({\mathbb{R}}_{+}^d,w)$ for each $1<p<\mathrm{\infty }$ and $w\in A_p({\mathbb{R}}_{+}^d)$, being $A_p({\mathbb{R}}_{+}^d)$ the Muckenhoupt $p$-class of weights on ${\mathbb{R}}_{+}^d$.