On two methods for quantitative unique continuation results for some nonlocal operators

Abstract

In this article, we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the logarithmic stability of the moment problem. In our second method we rely on the presence of branchcut singularities for certain Fourier multipliers. As an application we present quantitative Runge approximation results for the operator L s ðDÞ ¼ P n j¼1 ðÀ@ 2 xj Þ s þ q with s 2 ½ 1 2 , 1Þ and q 2 L 1 acting on functions on R n : ARTICLE HISTORY

On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols

“…Ishikawa's results in one dimension do not consider the case s = 1 2 except for p = 1 2 . Next we provide an alternative proof of the one-sided antilocality which also holds for the case s = 1 2 (and general p ∈ [0, 1]) and which builds on the ideas in [GFR20]. We present the result for A s 0 (D), but an analogous result is valid for A s 1 (D).…”

“…Using the observations from [GFR20], it is further possible to turn the uniqueness results provided by antilocality into a conditional stability estimate: Lemma 5.4. Let s ∈ (0, 1) and let Ω, U ⊂ R be open, bounded intervals with Ω to the right of U .…”

“…Crucial inputs in the above results on the inverse problems consisted of the unique continuation property of the fractional Laplacian [FF14, Seo14, Rül15, Yu17, GFR19] and the dual Runge approximation properties for fractional Laplacian [DSV14, GSU20, RS20a, Rül19]. Subsequently, some of these properties have been extended to larger classes of nonlocal operators [DSV16,RS20b,GFR20]. Moreover, connections between local and nonlocal Calderón type problems have been established in [CR20].…”

“…As initiated in [GFR20], it is the purpose of this article to transfer some of the results for the specific problem of the fractional Laplacian to more general nonlocal elliptic operators and to extract the decisive nonlocal features entering in this discussion, such as the operators' antilocality. In particular, our study of the operators as in (1) satisfying (A1)-(A3) illustrates the relevance of the notion of antilocality and indicates that geometric conditions may enter for these only directionally antilocal operators.…”