On the transfer of stability from the local to the fractional anisotropic Calder\'on problem with exterior measurements

Angkana R\"uland; Hendrik Baers

arxiv: 2604.10749 · v1 · submitted 2026-04-12 · 🧮 math.AP

On the transfer of stability from the local to the fractional anisotropic Calder\'on problem with exterior measurements

Hendrik Baers , Angkana R\"uland This is my paper

Pith reviewed 2026-05-10 15:42 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional Calderón problemstability estimatesexterior measurementsunique continuationRunge approximationanisotropicisotropicinverse problems

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The pith

Stability estimates transfer from the classical to the fractional Calderón problem with exterior measurements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a quantitative transfer of uniqueness and stability from the classical anisotropic Calderón problem to its fractional counterpart when measurements come from exterior data. This transfer is achieved through quantitative unique continuation and Runge approximation estimates that handle the specific difficulties of unbounded domains and the mismatch between open-set exterior data and co-dimension-one boundary data. A sympathetic reader would care because the result yields the first stability estimates for the principal part of the isotropic fractional Calderón problem without invoking Liouville transforms, thereby connecting local and nonlocal inverse problems in a setting where only exterior observations are available.

Core claim

By establishing quantitative unique continuation and Runge approximation estimates valid for the unbounded geometry and the mismatch between exterior measurements on open sets and boundary data on co-dimension one manifolds, the authors deduce the first stability estimates for the principal part of the isotropic fractional Calderón problem with exterior data in the absence of Liouville transforms.

What carries the argument

Quantitative unique continuation and Runge approximation estimates that transfer stability from the local to the fractional setting despite the unbounded geometry and the dimensional mismatch between exterior open-set data and co-dimension-one boundary data.

If this is right

The first stability estimates hold for the principal part of the isotropic fractional Calderón problem using only exterior data.
The same transfer technique applies to the anisotropic fractional Calderón problem under the stated geometric conditions.
Exterior open-set measurements suffice to recover stability information previously available only from boundary data.
The approach bypasses the need for Liouville transforms in obtaining quantitative control.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar quantitative transfers might extend to other nonlocal inverse problems where data domains have mismatched dimensions.
Numerical reconstruction algorithms for fractional models could be designed to rely solely on exterior observations.
The handling of unbounded domains suggests the method could apply to problems on unbounded manifolds or with variable coefficients at infinity.

Load-bearing premise

The quantitative unique continuation and Runge approximation estimates remain valid under the unbounded geometry and the mismatch between exterior open-set measurements and co-dimension-one boundary data.

What would settle it

A concrete counterexample in which the fractional problem loses stability for the principal part while the local problem remains stable, specifically when the domain is unbounded or measurements are restricted to an exterior open set, would disprove the transfer.

Figures

Figures reproduced from arXiv: 2604.10749 by Angkana R\"uland, Hendrik Baers.

**Figure 1.** Figure 1: Schematic visualization of the propagation of smallness argument in the proof of Proposition 3.9. We seek to prove smallness of ∥w˜∥L2((Ω1\Ω+δ)×(h,L),x 1−2s n+1 ) on (Ω1 \ Ω+δ) × (h, L) (blue part, in the figure: Γ := Ω1 \ Ω+δ). In particular, Steps 1.1 and 1.3 of the proof are depicted here. By assumption, we have smallness of the data on the boundary W × {0} (orange part). Applying the quantitative boun… view at source ↗

**Figure 2.** Figure 2: Schematic visualization of Step 2 in the proof of Proposition 3.9. Similarly to Step 1, we propagate the smallness of the boundary data (orange part) into the interior (green half-ball) via the boundary-bulk unique continuation, Proposition 3.6, and then to ∂Ω1 × {1} via the three-balls-inequality (gray balls), Proposition 3.7. Then we cover (Ω1 \Ω+δ)×(h, 1) with (violet) balls such that for each ball the… view at source ↗

read the original abstract

We study the quantitative transfer of uniqueness from the classical to the fractional Calder\'on problem with exterior data. This allows us to deduce the first stability estimates for the principal part of the isotropic fractional Calder\'on problem with exterior data in the absence of Liouville transforms. Our argument relies on careful quantitative unique continuation and Runge approximation estimates. Due to the unbounded geometry and the mismatch of the dimensionalities of the measurement domains (exterior data on an open set vs boundary data on a co-dimension one manifold) novel challenges arise compared to the setting of source-to-solution measurements on closed manifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper transfers stability from the local Calderón problem to the fractional isotropic case with exterior data, claiming the first such estimates, but the Runge approximation constants in unbounded domains with dimensional mismatch are the part that needs checking.

read the letter

This paper deduces stability estimates for the principal part of the isotropic fractional Calderón problem using exterior measurements by moving them over from the local anisotropic version. That transfer is the concrete new piece: no Liouville transforms, and the first quantitative stability in this exterior-data setting. They do it with quantitative unique continuation plus Runge approximation, and they flag the extra work required by unbounded geometry and the open-set versus hypersurface measurement mismatch. That acknowledgment is useful and keeps the argument honest. The strategy itself follows known lines in the field, so the value sits in carrying those tools across without losing the quantitative character. The soft spot is exactly the one the stress-test note raises. Runge approximation has to produce approximants whose constants do not blow up with distance to infinity or with the codimension gap between full-dimensional exterior sets and codimension-one boundary data. If those constants deteriorate, the transferred stability estimates stop being uniform and the main claim weakens. The abstract treats this as a novel challenge rather than a solved detail, so the paper stands or falls on whether the bounds are written down with explicit dependence. Readers who already know the local Calderón stability results and the standard quantitative unique-continuation machinery will see the most. It is a targeted advance inside inverse-problem theory for nonlocal operators, not a broad methodological shift. I would send it to peer review. The claim is precise enough and the method is grounded enough that referees can check the constants and either confirm or tighten the estimates.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes a quantitative transfer of stability from the local Calderón problem to the fractional anisotropic Calderón problem with exterior measurements. It deduces the first stability estimates for the principal part of the isotropic fractional Calderón problem with exterior data without Liouville transforms. The argument relies on quantitative unique continuation combined with Runge approximation estimates, explicitly addressing novel challenges from the unbounded geometry and the mismatch between full-dimensional exterior open-set data and co-dimension-one boundary measurements.

Significance. If the estimates hold with uniform constants, the result is significant for inverse problems involving nonlocal operators: it extends stability results beyond closed manifolds and settings requiring Liouville transforms, providing the first quantitative bounds in this exterior-data unbounded-domain regime. The paper's use of standard quantitative tools while carefully handling the geometric novelties is a clear strength and advances the field.

major comments (1)

[§4] §4 (Runge approximation): The quantitative Runge estimates must explicitly verify that the approximation constants remain finite and independent of the distance to infinity and the codimension gap between the exterior open set and the boundary hypersurface. If the error deteriorates under the unbounded geometry, the stability transfer to the principal-part estimate in §5 loses its quantitative character; the local-chart argument needs global control to support the main claim.

minor comments (2)

[Abstract] Abstract: The title refers to the anisotropic problem while the abstract focuses on the isotropic case; clarify the precise scope of the stability result.
[Introduction] Introduction: Add explicit comparison of the new stability constants with those from prior works on bounded domains to better highlight the improvement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major comment on the Runge approximation estimates is well-taken, and we address it point by point below. We will revise the manuscript to make the required verifications explicit.

read point-by-point responses

Referee: [§4] §4 (Runge approximation): The quantitative Runge estimates must explicitly verify that the approximation constants remain finite and independent of the distance to infinity and the codimension gap between the exterior open set and the boundary hypersurface. If the error deteriorates under the unbounded geometry, the stability transfer to the principal-part estimate in §5 loses its quantitative character; the local-chart argument needs global control to support the main claim.

Authors: We agree that explicit verification of the independence of the constants is necessary to preserve the quantitative character of the stability transfer. In §4 the Runge approximation is constructed by combining the quantitative unique continuation principle (which holds uniformly by the local-to-fractional transfer) with a global approximation procedure that uses a countable covering of the unbounded domain by local charts together with a partition of unity whose derivatives are controlled by a fixed constant independent of location. The cut-off functions are chosen with supports whose diameters are bounded uniformly (depending only on the fixed exterior open set), so that the approximation error constants depend solely on the ellipticity constants, the dimension, and the geometry of the fixed sets; they do not grow with distance to infinity. The codimension gap is handled by performing the approximation in the full-dimensional exterior open set and recovering the boundary trace via the trace theorem, whose operator norm is likewise uniform on the fixed hypersurface. We will revise §4 by adding a short paragraph after the statement of the Runge estimate that records these uniform bounds and explains the global control of the local-chart argument via the partition of unity. This clarification will be referenced in §5 to confirm that the stability estimates remain quantitative. We therefore mark this revision as 'yes'. revision: yes

Circularity Check

0 steps flagged

No circularity: stability transfer rests on independent quantitative estimates

full rationale

The paper derives stability estimates for the fractional Calderón problem by transferring uniqueness from the local setting via quantitative unique continuation and Runge approximation. These tools are invoked to address the unbounded geometry and exterior-versus-boundary measurement mismatch, without any equation or definition that reduces the target stability to a fitted parameter, self-referential ansatz, or prior self-citation whose content is itself the result being proved. The argument is therefore self-contained and relies on external estimates rather than constructing its own inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The argument presupposes standard quantitative unique continuation and Runge approximation results from prior literature.

pith-pipeline@v0.9.0 · 5395 in / 1119 out tokens · 37676 ms · 2026-05-10T15:42:17.157488+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Transfer of stability from the classical to the fractional anisotropic Calder´ on problem.arXiv preprint arXiv:2510.02242,

[BR25] Hendrik Baers and Angkana R¨ uland. Transfer of stability from the classical to the fractional anisotropic Calder´ on problem.arXiv preprint arXiv:2510.02242,

work page arXiv
[2]

Boundary reconstruction for the anisotropic fractional Calder´ on problem.arXiv preprint arXiv:2502.12287,

[CR25] Xiaopeng Cheng and Angkana R¨ uland. Boundary reconstruction for the anisotropic fractional Calder´ on problem.arXiv preprint arXiv:2502.12287,

work page arXiv
[3]

Fractional anisotropic Calder´ on problem with external data.arXiv preprint arXiv:2502.00710,

[FGK+25] Ali Feizmohammadi, Tuhin Ghosh, Katya Krupchyk, Angkana R¨ uland, Johannes Sj¨ ostrand, and Gunther Uhlmann. Fractional anisotropic Calder´ on problem with external data.arXiv preprint arXiv:2502.00710,

work page arXiv
[4]

Calder´ on problem for fractional Schr¨ odinger operators on closed Riemannian manifolds.arXiv preprint arXiv:2407.16866,

[FKU24] Ali Feizmohammadi, Katya Krupchyk, and Gunther Uhlmann. Calder´ on problem for fractional Schr¨ odinger operators on closed Riemannian manifolds.arXiv preprint arXiv:2407.16866,

work page arXiv
[5]

The Calder´ on problem for variable coefficients nonlocal elliptic operators.Communications in Partial Differential Equations, 42(12):1923–1961,

[GLX17] Tuhin Ghosh, Yi-Hsuan Lin, and Jingni Xiao. The Calder´ on problem for variable coefficients nonlocal elliptic operators.Communications in Partial Differential Equations, 42(12):1923–1961,

work page 1923
[6]

The Calder´ on problem for nonlocal operators.arXiv preprint arXiv:2110.09265,

[GU21] Tuhin Ghosh and Gunther Uhlmann. The Calder´ on problem for nonlocal operators.arXiv preprint arXiv:2110.09265,

work page arXiv
[7]

The Calder´ on problem with partial data on manifolds and applications

[KS13] Carlos Kenig and Mikko Salo. The Calder´ on problem with partial data on manifolds and applications. Anal. PDE, 6(8):2003–2048,

work page 2003
[8]

The Calder´ on problem for nonlocal parabolic operators: A new reduction from the nonlocal to the local.arXiv preprint arXiv:2308.09654,

[LLU23] Ching-Lung Lin, Yi-Hsuan Lin, and Gunther Uhlmann. The Calder´ on problem for nonlocal parabolic operators: A new reduction from the nonlocal to the local.arXiv preprint arXiv:2308.09654,

work page arXiv
[9]

The Calder´ on problem for the Schr¨ odinger equation in transversally anisotropic geometries with partial data.arXiv preprint arXiv:2408.08298,

[LNZ24] Yi-Hsuan Lin, Gen Nakamura, and Philipp Zimmermann. The Calder´ on problem for the Schr¨ odinger equation in transversally anisotropic geometries with partial data.arXiv preprint arXiv:2408.08298,

work page arXiv
[10]

Quantitative Runge approximation and inverse problems.Inter- national Mathematics Research Notices, 2019(20):6216–6234, 01

[RS18b] Angkana R¨ uland and Mikko Salo. Quantitative Runge approximation and inverse problems.Inter- national Mathematics Research Notices, 2019(20):6216–6234, 01

work page 2019
[11]

Revisiting the anisotropic fractional Calder´ on problem.International Mathematics Research Notices, 2025(5):rnaf036, 03

[R¨ u25] Angkana R¨ uland. Revisiting the anisotropic fractional Calder´ on problem.International Mathematics Research Notices, 2025(5):rnaf036, 03

work page 2025
[12]

Extension problem and Harnack’s inequality for some fractional operators.Comm

[ST10] Pablo Ra´ ul Stinga and Jos´ e Luis Torrea. Extension problem and Harnack’s inequality for some fractional operators.Comm. Partial Differential Equations, 35(11):2092–2122,

work page 2092

[1] [1]

Transfer of stability from the classical to the fractional anisotropic Calder´ on problem.arXiv preprint arXiv:2510.02242,

[BR25] Hendrik Baers and Angkana R¨ uland. Transfer of stability from the classical to the fractional anisotropic Calder´ on problem.arXiv preprint arXiv:2510.02242,

work page arXiv

[2] [2]

Boundary reconstruction for the anisotropic fractional Calder´ on problem.arXiv preprint arXiv:2502.12287,

[CR25] Xiaopeng Cheng and Angkana R¨ uland. Boundary reconstruction for the anisotropic fractional Calder´ on problem.arXiv preprint arXiv:2502.12287,

work page arXiv

[3] [3]

Fractional anisotropic Calder´ on problem with external data.arXiv preprint arXiv:2502.00710,

[FGK+25] Ali Feizmohammadi, Tuhin Ghosh, Katya Krupchyk, Angkana R¨ uland, Johannes Sj¨ ostrand, and Gunther Uhlmann. Fractional anisotropic Calder´ on problem with external data.arXiv preprint arXiv:2502.00710,

work page arXiv

[4] [4]

Calder´ on problem for fractional Schr¨ odinger operators on closed Riemannian manifolds.arXiv preprint arXiv:2407.16866,

[FKU24] Ali Feizmohammadi, Katya Krupchyk, and Gunther Uhlmann. Calder´ on problem for fractional Schr¨ odinger operators on closed Riemannian manifolds.arXiv preprint arXiv:2407.16866,

work page arXiv

[5] [5]

The Calder´ on problem for variable coefficients nonlocal elliptic operators.Communications in Partial Differential Equations, 42(12):1923–1961,

[GLX17] Tuhin Ghosh, Yi-Hsuan Lin, and Jingni Xiao. The Calder´ on problem for variable coefficients nonlocal elliptic operators.Communications in Partial Differential Equations, 42(12):1923–1961,

work page 1923

[6] [6]

The Calder´ on problem for nonlocal operators.arXiv preprint arXiv:2110.09265,

[GU21] Tuhin Ghosh and Gunther Uhlmann. The Calder´ on problem for nonlocal operators.arXiv preprint arXiv:2110.09265,

work page arXiv

[7] [7]

The Calder´ on problem with partial data on manifolds and applications

[KS13] Carlos Kenig and Mikko Salo. The Calder´ on problem with partial data on manifolds and applications. Anal. PDE, 6(8):2003–2048,

work page 2003

[8] [8]

The Calder´ on problem for nonlocal parabolic operators: A new reduction from the nonlocal to the local.arXiv preprint arXiv:2308.09654,

[LLU23] Ching-Lung Lin, Yi-Hsuan Lin, and Gunther Uhlmann. The Calder´ on problem for nonlocal parabolic operators: A new reduction from the nonlocal to the local.arXiv preprint arXiv:2308.09654,

work page arXiv

[9] [9]

The Calder´ on problem for the Schr¨ odinger equation in transversally anisotropic geometries with partial data.arXiv preprint arXiv:2408.08298,

[LNZ24] Yi-Hsuan Lin, Gen Nakamura, and Philipp Zimmermann. The Calder´ on problem for the Schr¨ odinger equation in transversally anisotropic geometries with partial data.arXiv preprint arXiv:2408.08298,

work page arXiv

[10] [10]

Quantitative Runge approximation and inverse problems.Inter- national Mathematics Research Notices, 2019(20):6216–6234, 01

[RS18b] Angkana R¨ uland and Mikko Salo. Quantitative Runge approximation and inverse problems.Inter- national Mathematics Research Notices, 2019(20):6216–6234, 01

work page 2019

[11] [11]

Revisiting the anisotropic fractional Calder´ on problem.International Mathematics Research Notices, 2025(5):rnaf036, 03

[R¨ u25] Angkana R¨ uland. Revisiting the anisotropic fractional Calder´ on problem.International Mathematics Research Notices, 2025(5):rnaf036, 03

work page 2025

[12] [12]

Extension problem and Harnack’s inequality for some fractional operators.Comm

[ST10] Pablo Ra´ ul Stinga and Jos´ e Luis Torrea. Extension problem and Harnack’s inequality for some fractional operators.Comm. Partial Differential Equations, 35(11):2092–2122,

work page 2092