Weighted estimates for powers and Smoothing estimates of Schr\"odinger operators with inverse-square potentials

TL;DR

This paper establishes weighted estimates for fractional powers of Schr"odinger operators with inverse-square potentials and derives smoothing estimates for their propagators, advancing understanding of their functional analysis properties.

Contribution

It introduces new weighted Hardy and square function inequalities for these operators and applies them to obtain smoothing estimates for the propagator.

Findings

Weighted estimates for fractional powers of $\\mathcal{L}_a$

Smoothing estimates for the propagator $e^{it\mathcal{L}_a}$

Use of weighted Hardy inequalities and square function inequalities

Abstract

Let $\mathcal{L}_a$ be a Schr\"odinger operator with inverse square potential $a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates regarding the propagator $e^{it\mathcal{L}_a}$.