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 Schr\"odinger 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 $\alpha \geq 0$ and $\partial _t^\alpha$ denotes the Weyl fractional derivative. The range of values $p$ that works is different when $a\geq 0$ and when $-\frac{(n-2)^2}{4}<a<0$.