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