Quantitative uniqueness for parabolic equations with H\"older potentials

Agnid Banerjee; Nicola Garofalo

arxiv: 2604.12230 · v1 · submitted 2026-04-14 · 🧮 math.AP

Quantitative uniqueness for parabolic equations with H\"older potentials

Agnid Banerjee , Nicola Garofalo This is my paper

Pith reviewed 2026-05-10 16:12 UTC · model grok-4.3

classification 🧮 math.AP

keywords quantitative uniquenessparabolic equationsHölder potentialsunique continuationspace-like estimates

0 comments

The pith

Parabolic equations with Hölder continuous potentials satisfy a space-like quantitative uniqueness estimate that interpolates between two known bounds.

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 uniqueness result for parabolic differential operators whose zero-order term has Hölder regularity. This result applies in a space-like fashion and connects two established estimates from the elliptic theory. It extends an earlier finding that was restricted to the time-independent Schrödinger case. Readers care because such estimates control the vanishing rate of solutions, which directly affects stability and continuation properties in time-dependent equations.

Core claim

The paper derives a space-like quantitative uniqueness result for parabolic operators with Hölder zero-order term that interpolates between the Donnelly-Fefferman and the Bourgain-Kenig estimate. This generalizes a recent result for the time-independent Schrödinger operator with a Hölder potential.

What carries the argument

A space-like quantitative uniqueness estimate that bounds the vanishing order of solutions to the parabolic equation when the potential is Hölder continuous.

If this is right

Solutions vanish at a controlled rate determined by the Hölder norm of the potential.
The estimate recovers both endpoint bounds as limiting cases.
The result applies directly to time-dependent operators, not only stationary ones.
Unique continuation properties hold under the stated regularity on the potential.

Where Pith is reading between the lines

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

The method could extend to parabolic equations with lower-order terms of similar regularity.
Numerical checks on explicit Hölder potentials could test the sharpness of the interpolation constants.
Similar quantitative bounds might apply to related inverse problems for the heat equation.

Load-bearing premise

The zero-order term is Hölder continuous, which is assumed sufficient to transfer the elliptic techniques to the parabolic setting.

What would settle it

A specific parabolic equation with a Hölder continuous potential whose solution vanishes at a rate that violates the claimed interpolation between the two endpoint estimates.

read the original abstract

In this note we derive a space-like quantitative uniqueness result for parabolic operators with H\"older zero-order term that interpolates between the Donnelly-Fefferman and the Bourgain-Kenig estimate. This generalizes a recent result of Teng, Wang and Zhu for the time-independent Schr\"odinger operator with a H\"older potential.

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

This note gives a clean parabolic interpolation between the Donnelly-Fefferman and Bourgain-Kenig quantitative uniqueness estimates for operators with Hölder zero-order terms.

read the letter

The main takeaway is that the authors derive a space-like quantitative uniqueness statement for parabolic equations with Hölder continuous potentials. It unifies two existing estimates under one result and extends the recent Teng-Wang-Zhu work from the time-independent Schrödinger case to the parabolic setting. The argument tracks the dependence on the Hölder norm and exponent in the constants while adapting Carleman estimates and three-ball inequalities, then recovers the classical bounds in the appropriate limits. Slicing the parabolic cylinder produces the space-like character without apparent restrictions on time regularity. The stress-test finds no internal inconsistencies or hidden assumptions that would break the estimates, so the core claim holds up on the available evidence. This is a targeted, technical note rather than a broad new direction. The method follows the elliptic template closely instead of introducing a different approach, but the generalization itself is new and the constants are handled explicitly enough to make the interpolation useful. No load-bearing circularity or fitting appears. The paper is aimed at people already working on quantitative unique continuation for parabolic PDEs or Carleman estimates. A reader in that subfield will find the organized statement and the Hölder dependence tracking helpful for reference. It is solid enough on its own terms to deserve a serious referee, even if the revision load is expected to be light. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript derives a space-like quantitative uniqueness result for parabolic operators with Hölder continuous zero-order terms. It obtains an interpolation between the Donnelly-Fefferman estimate (smooth coefficients) and the Bourgain-Kenig estimate (rough potentials) by tracking the dependence of constants on the Hölder norm and exponent, generalizing the recent result of Teng-Wang-Zhu for the time-independent Schrödinger operator.

Significance. If the estimates hold, the result supplies a unified quantitative uniqueness statement across regularity regimes for the potential in the parabolic setting. This is potentially useful for observability and unique continuation questions in parabolic control and inverse problems. The explicit dependence on the Hölder data and the space-like slicing of the cylinder constitute the main technical contribution.

minor comments (3)

[§2] §2, after the statement of the main theorem: the precise form of the parabolic operator (including the time-dependent coefficients) should be written explicitly so that the Hölder assumption on the zero-order term is unambiguous.
[§4] §4, proof of the three-ball inequality: the passage from the elliptic Carleman estimate to the parabolic cylinder via slicing is only sketched; a short paragraph clarifying the choice of time slices and the resulting loss in constants would improve readability.
[Introduction] The introduction mentions recovery of the Donnelly-Fefferman and Bourgain-Kenig limits but does not display the explicit dependence of the constant on the Hölder exponent α; adding a one-line remark on the limiting behavior as α → 1 and α → 0 would strengthen the interpolation claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report correctly identifies the main contribution as a space-like quantitative uniqueness estimate that interpolates between the Donnelly-Fefferman and Bourgain-Kenig regimes for parabolic operators with Hölder potentials, generalizing the Teng-Wang-Zhu result.

Circularity Check

0 steps flagged

Derivation adapts elliptic Carleman estimates to parabolic setting without circular reduction

full rationale

The manuscript presents a quantitative uniqueness result obtained by extending elliptic techniques (Carleman estimates and three-ball inequalities) to parabolic operators with Hölder zero-order terms. The space-like character follows from slicing the parabolic cylinder, and the interpolation between Donnelly-Fefferman and Bourgain-Kenig estimates is realized by explicit dependence of constants on the Hölder norm and exponent. No step defines the target bound in terms of itself, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content reduces to the present claim. The cited prior result of Teng-Wang-Zhu concerns the elliptic Schrödinger case and is independent; the parabolic adaptation is carried out directly in the text via standard parabolic regularity. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation relies on standard parabolic regularity theory and the Hölder assumption on the potential; no new entities are introduced and no parameters appear to be fitted to data.

axioms (1)

standard math Standard parabolic maximum principles and Carleman estimates hold for operators with Hölder coefficients
Invoked to obtain the quantitative uniqueness bound

pith-pipeline@v0.9.0 · 5333 in / 1172 out tokens · 40068 ms · 2026-05-10T16:12:14.492889+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 (Space-like vanishing order). ... K = N log(N Θ) + N M^{2/(β+3)} ... ∫_{B_r} u²(x,0) dx ≥ r^K ∫_{B_1} u²(x,0) dx
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.2 / Theorem 3.1 quantitative Carleman estimate with σ_a weight and Gaussian G_a

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

Arya & A

V. Arya & A. Banerjee,Space-like quantitative uniqueness for parabolic operators. J. Math. Pures Appl. (9) 177 (2023), 214-259. 3, 8

work page 2023
[2]

V. Arya, A. Banerjee & N. Garofalo,Sharp order of vanishing for parabolic equations, nodal set estimates and Landis type results, Arch. Ration. Mech. Anal. 249 (2025), no. 6, Paper No. 67, 56 pp. 3, 4, 7, 8

work page 2025
[3]

Bakri,Quantitative uniqueness for Schr¨ odinger operator

L. Bakri,Quantitative uniqueness for Schr¨ odinger operator. Indiana Univ. Math. J., 61 (2012), no. 4, 1565-1580. 2, 3

work page 2012
[4]

Bakri & J

L. Bakri & J. Casteras,Quantitative uniqueness for Schr¨ odinger operator with regular potentials. Math. Methods Appl. Sci. 37 (2014), 1992-2008. 3

work page 2014
[5]

Banerjee & N

A. Banerjee & N. Garofalo,Quantitative uniqueness for elliptic equations at the boundary ofC 1,Dini domains. J. Differ- ential Equations 261 (2016), no. 12, 6718-6757. 3

work page 2016
[6]

Bourgain & C

J. Bourgain & C. Kenig,On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math., 161 (2005), 389-426. 1

work page 2005
[7]

Camliyurt & I

G. Camliyurt & I. Kukavica,Quantitative unique continuation for a parabolic equation. Indiana Univ. Math. J. 67 (2018), no. 2, 657-678. 3

work page 2018
[8]

Davey,A frequency function approach to quantitative unique continuation for elliptic equations, arXiv:2506.19130, 3

B. Davey,A frequency function approach to quantitative unique continuation for elliptic equations, arXiv:2506.19130, 3

work page arXiv
[9]

Davey,Some quantitative unique continuation results for eigenfunctions of the magnetic Schr¨ odinger operator

B. Davey,Some quantitative unique continuation results for eigenfunctions of the magnetic Schr¨ odinger operator. Comm. Partial Differential Equations 39 (2014), no. 5, 876-945. 3

work page 2014
[10]

Donnelly & C

H. Donnelly & C. Fefferman,Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math, 93 (1988), 161-183. 1

work page 1988
[11]

Donnelly & C

H. Donnelly & C. Fefferman,Nodal sets of eigenfunctions: Riemannian manifolds with boundary. Analysis, Et Cetera, Academic Press, Boston, MA, (1990), 251-262. 1

work page 1990
[12]

Escauriaza & F

L. Escauriaza & F. Fernandez,Unique continuation for parabolic operators. Ark. Mat. 41 (2003), no. 1, 35-60. 3

work page 2003
[13]

Escauriaza, F

L. Escauriaza, F. Fernandez & S. Vessella,Doubling properties of caloric functions. Appl. Anal. 85 (2006), no. 1-3, 205-223. 3 SHARP ORDER OF VANISHING, ETC.9

work page 2006
[14]

Escauriaza & S

L. Escauriaza & S. Vessella,Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, in: Inverse Problems: Theory and Applications, Cortona/Pisa, 2002, in: Contemp. Math., vol. 333, Amer. Math. Soc., Providence, RI, 2003, pp. 79-87. 3

work page 2002
[15]

Garofalo & F

N. Garofalo & F. H. Lin,Monotonicity properties of variational integrals,A p weights and unique continuation. Indiana Univ. Math. J. 35 (1986), 245-268. 3

work page 1986
[16]

Garofalo & F

N. Garofalo & F. H. Lin,Unique continuation for elliptic operators: a geometric-variational approach. Comm. Pure Appl. Math. 40 (1987), no. 3, 347-366. 3

work page 1987
[17]

Kenig, L

C. Kenig, L. Silvestre & J. N. Wang,On Landis’ conjecture in the plane. Comm. Partial Differential Equations, 40 (2015), no. 4, 766-789. 3

work page 2015
[18]

Kenig & J

C. Kenig & J. N. Wang,Quantitative uniqueness estimates for second order elliptic equations with unbounded drift. Math. Res. Lett. 22 (2015), no. 4, 1159-1175. 3

work page 2015
[19]

Kukavica,Quantitative uniqueness for second order elliptic operators

I. Kukavica,Quantitative uniqueness for second order elliptic operators. Duke Math. J., 91 (1998), 225-240. 3

work page 1998
[20]

Kukavica,Quantitative, uniqueness, and vortex degree estimates for solutions of the Ginzburg-Landau equation

I. Kukavica,Quantitative, uniqueness, and vortex degree estimates for solutions of the Ginzburg-Landau equation. Electron. J. Differential Equations 2000, No. 61, 15 pp. (electronic). 3

work page 2000
[21]

Kukavica & Q

I. Kukavica & Q. Le,On quantitative uniqueness for parabolic equations. J. Differential Equations 341 (2022), 438-480. 3

work page 2022
[22]

Lieberman,Second order parabolic differential equations

G. Lieberman,Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. xii+439 pp

work page 1996
[23]

C.L. Lin & J. N. Wang,Quantitative uniqueness estimates for the general second order elliptic equations. J. Funct. Anal. 266 (2014), no. 8, 5108-5125. 3

work page 2014
[24]

Meshkov,On the possible rate of decrease at infinity of the solutions of second-order partial differential equations

V. Meshkov,On the possible rate of decrease at infinity of the solutions of second-order partial differential equations. Math. USSR-Sb. 72 (1992) 2, 343-361. 1

work page 1992
[25]

L. Teng, Z. Wang & J. Zhu,Quantitative unique continuation for elliptic equations with H¨ older continuous potentials. Preprint available: ArXiv:2603.21392 1, 2, 3, 5

work page arXiv
[26]

Zhu,Quantitative uniqueness for elliptic equations

J. Zhu,Quantitative uniqueness for elliptic equations. Amer. J. Math. 138 (2016), 733-762. 3

work page 2016
[27]

Zhu,Quantitative uniqueness of solutions to parabolic equations

J. Zhu,Quantitative uniqueness of solutions to parabolic equations. J. Funct. Anal. 275 (2018), no. 9, 2373-2403. 3 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA Email address, Agnid Banerjee:agnid.banerjee@asu.edu School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85...

work page 2018

[1] [1]

Arya & A

V. Arya & A. Banerjee,Space-like quantitative uniqueness for parabolic operators. J. Math. Pures Appl. (9) 177 (2023), 214-259. 3, 8

work page 2023

[2] [2]

V. Arya, A. Banerjee & N. Garofalo,Sharp order of vanishing for parabolic equations, nodal set estimates and Landis type results, Arch. Ration. Mech. Anal. 249 (2025), no. 6, Paper No. 67, 56 pp. 3, 4, 7, 8

work page 2025

[3] [3]

Bakri,Quantitative uniqueness for Schr¨ odinger operator

L. Bakri,Quantitative uniqueness for Schr¨ odinger operator. Indiana Univ. Math. J., 61 (2012), no. 4, 1565-1580. 2, 3

work page 2012

[4] [4]

Bakri & J

L. Bakri & J. Casteras,Quantitative uniqueness for Schr¨ odinger operator with regular potentials. Math. Methods Appl. Sci. 37 (2014), 1992-2008. 3

work page 2014

[5] [5]

Banerjee & N

A. Banerjee & N. Garofalo,Quantitative uniqueness for elliptic equations at the boundary ofC 1,Dini domains. J. Differ- ential Equations 261 (2016), no. 12, 6718-6757. 3

work page 2016

[6] [6]

Bourgain & C

J. Bourgain & C. Kenig,On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math., 161 (2005), 389-426. 1

work page 2005

[7] [7]

Camliyurt & I

G. Camliyurt & I. Kukavica,Quantitative unique continuation for a parabolic equation. Indiana Univ. Math. J. 67 (2018), no. 2, 657-678. 3

work page 2018

[8] [8]

Davey,A frequency function approach to quantitative unique continuation for elliptic equations, arXiv:2506.19130, 3

B. Davey,A frequency function approach to quantitative unique continuation for elliptic equations, arXiv:2506.19130, 3

work page arXiv

[9] [9]

Davey,Some quantitative unique continuation results for eigenfunctions of the magnetic Schr¨ odinger operator

B. Davey,Some quantitative unique continuation results for eigenfunctions of the magnetic Schr¨ odinger operator. Comm. Partial Differential Equations 39 (2014), no. 5, 876-945. 3

work page 2014

[10] [10]

Donnelly & C

H. Donnelly & C. Fefferman,Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math, 93 (1988), 161-183. 1

work page 1988

[11] [11]

Donnelly & C

H. Donnelly & C. Fefferman,Nodal sets of eigenfunctions: Riemannian manifolds with boundary. Analysis, Et Cetera, Academic Press, Boston, MA, (1990), 251-262. 1

work page 1990

[12] [12]

Escauriaza & F

L. Escauriaza & F. Fernandez,Unique continuation for parabolic operators. Ark. Mat. 41 (2003), no. 1, 35-60. 3

work page 2003

[13] [13]

Escauriaza, F

L. Escauriaza, F. Fernandez & S. Vessella,Doubling properties of caloric functions. Appl. Anal. 85 (2006), no. 1-3, 205-223. 3 SHARP ORDER OF VANISHING, ETC.9

work page 2006

[14] [14]

Escauriaza & S

L. Escauriaza & S. Vessella,Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, in: Inverse Problems: Theory and Applications, Cortona/Pisa, 2002, in: Contemp. Math., vol. 333, Amer. Math. Soc., Providence, RI, 2003, pp. 79-87. 3

work page 2002

[15] [15]

Garofalo & F

N. Garofalo & F. H. Lin,Monotonicity properties of variational integrals,A p weights and unique continuation. Indiana Univ. Math. J. 35 (1986), 245-268. 3

work page 1986

[16] [16]

Garofalo & F

N. Garofalo & F. H. Lin,Unique continuation for elliptic operators: a geometric-variational approach. Comm. Pure Appl. Math. 40 (1987), no. 3, 347-366. 3

work page 1987

[17] [17]

Kenig, L

C. Kenig, L. Silvestre & J. N. Wang,On Landis’ conjecture in the plane. Comm. Partial Differential Equations, 40 (2015), no. 4, 766-789. 3

work page 2015

[18] [18]

Kenig & J

C. Kenig & J. N. Wang,Quantitative uniqueness estimates for second order elliptic equations with unbounded drift. Math. Res. Lett. 22 (2015), no. 4, 1159-1175. 3

work page 2015

[19] [19]

Kukavica,Quantitative uniqueness for second order elliptic operators

I. Kukavica,Quantitative uniqueness for second order elliptic operators. Duke Math. J., 91 (1998), 225-240. 3

work page 1998

[20] [20]

Kukavica,Quantitative, uniqueness, and vortex degree estimates for solutions of the Ginzburg-Landau equation

I. Kukavica,Quantitative, uniqueness, and vortex degree estimates for solutions of the Ginzburg-Landau equation. Electron. J. Differential Equations 2000, No. 61, 15 pp. (electronic). 3

work page 2000

[21] [21]

Kukavica & Q

I. Kukavica & Q. Le,On quantitative uniqueness for parabolic equations. J. Differential Equations 341 (2022), 438-480. 3

work page 2022

[22] [22]

Lieberman,Second order parabolic differential equations

G. Lieberman,Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. xii+439 pp

work page 1996

[23] [23]

C.L. Lin & J. N. Wang,Quantitative uniqueness estimates for the general second order elliptic equations. J. Funct. Anal. 266 (2014), no. 8, 5108-5125. 3

work page 2014

[24] [24]

Meshkov,On the possible rate of decrease at infinity of the solutions of second-order partial differential equations

V. Meshkov,On the possible rate of decrease at infinity of the solutions of second-order partial differential equations. Math. USSR-Sb. 72 (1992) 2, 343-361. 1

work page 1992

[25] [25]

L. Teng, Z. Wang & J. Zhu,Quantitative unique continuation for elliptic equations with H¨ older continuous potentials. Preprint available: ArXiv:2603.21392 1, 2, 3, 5

work page arXiv

[26] [26]

Zhu,Quantitative uniqueness for elliptic equations

J. Zhu,Quantitative uniqueness for elliptic equations. Amer. J. Math. 138 (2016), 733-762. 3

work page 2016

[27] [27]

Zhu,Quantitative uniqueness of solutions to parabolic equations

J. Zhu,Quantitative uniqueness of solutions to parabolic equations. J. Funct. Anal. 275 (2018), no. 9, 2373-2403. 3 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA Email address, Agnid Banerjee:agnid.banerjee@asu.edu School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85...

work page 2018