Quantitative Invertibility and Approximation for the Truncated Hilbert and Riesz Transforms

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].