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 $\mathbb{ R}^d_+$.
Categories: math.AP