Variation operators associated with the semigroups generated by Schr\"odinger operators with inverse square potentials

TL;DR

This paper studies weighted inequalities for various operators related to Schr"odinger semigroups with inverse square potentials, revealing different $p$-ranges depending on the potential's sign and magnitude.

Contribution

It establishes weighted $L^p$-inequalities for maximal, variation, oscillation, and jump operators associated with Schr"odinger semigroups with inverse square potentials, including fractional derivatives.

Findings

Different $p$-ranges for inequalities depending on the sign of $a$.

Weighted inequalities hold for maximal, variation, oscillation, and jump operators.

Results extend understanding of Schr"odinger 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 $-\frac{(n-2)^2}{4}<a<0$.