Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms

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].

A Remark on the Arcsine Distribution and the Hilbert transform

“…One simple question one could ask is the following: let f ∈ C ∞ c (−1, 1), how big does the Hilbert transform have to be on, say, the interval (2, 3)? A sharp result was given by Alaifari, Pierce and the second author in [1] (see also Rüland [15]) and reads…”

Local Density of Solutions to Fractional Equations

“…The approximation results with solutions of nonlocal operators have been first introduced in [DSV17] for the case of the fractional Laplacian, and since then widely studied under different perspectives, including harmonic analysis, see [RS18,GSU16,Rül17,RS17a,RS17b]. The approximation result for the one dimensional case of a fractional derivative of Caputo type has been considered in [Buc17,CDV18], and operators involving classical time derivatives and additional classical derivatives in space have been studied in [DSV18].…”

Quantitative approximation properties for the fractional heat equation

“…These quantitative arguments were partly motivated by stability results in inverse problems for nonlocal operators, c.f. [RS17], and can also be considered as a continuation of the investigation started in [Rül17]. In the context of the model problem (1) our main result can be formulated as the following proposition:…”

“…Similarly as in [Rül17], our approach to the question on the cost of control relies on (i) a propagation of smallness result, (ii) quantitative unique continuation properties of the adjoint equation (8),…”