Dimension-free estimates for positivity-preserving Riesz transforms related to Schr\"odinger operators with certain potentials

TL;DR

This paper establishes dimension-free bounds for certain positivity-preserving Riesz transforms associated with Schrödinger operators with specific potentials, using semigroup factorization and separate coordinate analysis.

Contribution

It introduces a novel approach to estimate Riesz transforms for Schrödinger operators with coordinate-wise potentials, achieving dimension-free bounds on $L^ abla$ and $L^1$ spaces.

Findings

Dimension-free $L^ abla$ boundedness for Riesz transforms with specific potentials.

Factorization of semigroup into one-dimensional components for analysis.

Extension of results to $L^1$ space under additional assumptions.

Abstract

We study the $L^\infty(\mathbb{R}^d)$ boundedness for Riesz transforms of the form ${V^{a}(-\frac12\Delta+V)^{-a}},$ where $a > 0$ and $V$ is a non-negative potential with power growth acting independently on each coordinate. We factorize the semigroup $e^{-tL}$ into one-dimensional factors, estimate them separately and combine the results to estimate the original semigroup. Similar results with additional assumption $a \leqslant 1$ are obtained on $L^1(\mathbb{R}^d)$.