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

Abstract: 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 $\mathbb{R}^d_+$.