Variation operators associated with the semigroups generated by Schrödinger operators with inverse square potentials

Abstract

By $\{T_t^a{\}}_{t>0}$ we denote the semigroup of operators generated by the Friedrichs extension of the Schrödinger operator with the inverse square potential $L_a=-\Delta +\frac{a}{|x{|}^2}$ defined in the space of smooth functions with compact support in ${\mathbb{R}}^n\setminus \{0\}$. In this paper we establish weighted $L^p$-inequalities for the maximal, variation, oscillation and jump operators associated with $\{t^{\alpha }{\partial }_t^{\alpha }{T}_t^a{\}}_{t>0}$, where $\alpha \ge 0$ and ${\partial }_t^{\alpha }$ denotes the Weyl fractional derivative. The range of values $p$ that works is different when $a\ge 0$ and when $-\frac{(n-2{)}^2}{4}<a<0$.