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 $-{\mathbb{L}}_{\lambda }^{\alpha }$, 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 \[\mathbb L^α_λ=(-Δ)^{α/2}_{\mathbb{R}^d_+}+λx_d^{-α}, \quad α\in (0,2], λ\geq 0.\] We prove that, for every $k\in \mathbb{N}$, the $\rho$-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 ${\mathbb{R}}_{+}^d$.

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