Variation operators associated with semigroups generated by Hardy operators involving fractional Laplacians in a half space
We represent by $\{W_{λ, t}^α\}_{t>0}$ the semigroup generated by $-\mathbb L^α_λ$, where $\mathbb L^α_λ$ is a Hardy operator on a half space. The operator $\mathbb L^α_λ$ includes a fractional Laplacian and it is defined by \[\mathbb L^α_λ=(-Δ)^{α/2}_{\mathbb{R}^d_+}+λx_d^{-α}, \quad α\in (0,2], λ\geq 0.\] We prove that, for every $k\in \mathbb N$, the $ρ$-variation operator $\mathcal{V}_ρ\left(\left\{t^k\partial_t^k W_{λ,t}^α\right\}\right)$ 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 $\mathbb{R}^d_+$.
math.AP↗