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 \begin{equation*} \mathbb L^{\alpha}_\lambda=(-\Delta)^{\alpha/2}_{\mathbb R^d_+}+\lambda x_d^{-\alpha}, \quad \alpha\in (0,2], \lambda\geq 0. \end{equation*} We prove that, for every k\in \mathbb{N} , the \rho -variation operator \mathcal{V}_{\rho}(\{t^{k}\partial_{t}^{k} W_{\lambda,t}^{\alpha}\}) is bounded on L^{p}(\mathbb{R}^{d}_{+}, w) for each 1 and w\in A_{p}(\mathbb{R}^{d}_{+}) , with A_{p}(\mathbb{R}^{d}_{+}) being the Muckenhoupt p -class of weights on \mathbb{R}^{d}_{+} .
Fields of Study & Subjects
Advanced Harmonic Analysis Research
Nonlinear Partial Differential Equations
Advanced Mathematical Physics Problems
Publication details
- Journal / source
- Zeitschrift für Analysis und ihre Anwendungen
- Volume
- 45
- Issue
- 3
- Pages
- 251–277
- Publication date
- 15 April 2026
- Language
- EN