Quantitative Invertibility and Approximation for the Truncated Hilbert and Riesz Transforms

TL;DR

This paper establishes quantitative uniqueness and approximation properties for perturbations of Riesz transforms using PDE techniques, harmonic extensions, and propagation of smallness estimates, advancing the understanding of nonlocal operator approximation.

Contribution

It introduces a PDE-based approach to quantify invertibility and approximation for Riesz transforms, extending recent nonlocal operator results.

Findings

Quantitative propagation of smallness estimates derived

Approximation properties for Riesz transforms quantified

Harmonic extension method applied to nonlocal operators

Abstract

In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realize our operators as harmonic extensions, which makes the problem accessible to PDE tools. In this context we then invoke quantitative propagation of smallness estimates in combination with qualitative Runge approximation results. These results can be viewed as quantifications of the approximation properties which have recently gained prominence in the context of nonlocal operators, c.f. [DSV14], [DSV16].