Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting

Mario Rastrelli; Vladimir Georgiev

arxiv: 2504.19732 · v1 · submitted 2025-04-28 · 🧮 math.AP

Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the L^p setting

Vladimir Georgiev , Mario Rastrelli This is my paper

Pith reviewed 2026-05-22 18:50 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional Sobolev spacessingular perturbationLaplace operatorL^p spacesStrichartz estimatesnonlinear Schrödinger equationlocal well-posednessnorm equivalence

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The pith

Perturbed Sobolev spaces for the singular Laplace operator are equivalent to standard ones in the L^p setting.

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

This paper extends the L^2 theory of perturbed Sobolev spaces associated with a singular perturbation of the Laplace operator to the general L^p case in two and three dimensions. It establishes that these spaces admit an analogue description in terms of the usual Sobolev spaces. The resulting norm equivalences permit the extension of Strichartz estimates to the energy space. They also allow a proof of local well-posedness for the nonlinear Schrödinger equation driven by this perturbed operator, obtained via the contraction mapping principle.

Core claim

We study the perturbed Sobolev spaces H^{s,p}_α(R^d) associated with the singular perturbation Δ_α of the Laplace operator in Euclidean space of dimensions 2 and 3. We extend the L^2 theory of perturbed Sobolev space to the L^p case, finding an analogue description in terms of standard Sobolev spaces. This enables us to extend the Strichartz estimates to the energy space and to treat the local well-posedness of the Nonlinear Schrödinger equation associated with this singular perturbation, with the contraction method.

What carries the argument

The norm equivalence between the perturbed fractional Sobolev space H^{s,p}_α and the standard H^{s,p}, which transfers analytic properties from the unperturbed case to the perturbed operator.

If this is right

Strichartz estimates hold for the perturbed operator in the energy space.
Local well-posedness is proved for the nonlinear Schrödinger equation with the singular perturbation using contraction mapping.
The results apply specifically in dimensions two and three.

Where Pith is reading between the lines

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

This characterization could support similar extensions to other singular perturbations or to higher dimensions.
Global well-posedness or scattering results for the perturbed NLS might follow under small-data or defocusing assumptions.
The equivalence may connect to broader questions about spectral properties of perturbed operators in dispersive PDE.

Load-bearing premise

The singular perturbation of the Laplace operator has sufficient regularity and spectral properties to support the same kind of norm equivalence in L^p as in L^2.

What would settle it

Finding a function where the perturbed Sobolev norm and the standard Sobolev norm are not comparable by constants independent of the function, for some p not equal to 2 in dimensions 2 or 3.

read the original abstract

We study the perturbed Sobolev spaces ${H^{s,p}_\alpha(\mathbb{R}^d)}$, associated with singular perturbation $\Delta_\alpha$ of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the $L^2$ theory of perturbed Sobolev space to the $L^p$ case, finding an analogue description in terms of standard Sobolev spaces. This enables us to extend the Strichartz estimates to the energy space and to treat the {local well-posedness} of the {Nonlinear Schr\"odinger equation} associated with this singular perturbation, with the contraction method.

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 paper extends the L2 characterization of perturbed fractional Sobolev spaces to Lp in dimensions 2 and 3, then uses it for Strichartz estimates and contraction-based local well-posedness of the perturbed NLS.

read the letter

The main thing here is the Lp analogue of the perturbed Sobolev space description. Once they have that equivalence between H^{s,p}_α and the standard H^{s,p}, they get Strichartz estimates in the energy space and local well-posedness for the nonlinear Schrödinger equation tied to this singular perturbation, all via contraction mapping. That is the concrete new piece relative to the earlier L2 work they cite. The setup stays focused on dimensions 2 and 3 and keeps the claims modest, which is appropriate for this kind of functional-analytic extension. The application to well-posedness follows the usual pattern once the spaces are identified, so the payoff is clear if the characterization holds. The paper does a straightforward job of carrying the L2 theory over without unnecessary complications. The soft spot is the uniformity of the norm equivalence when p moves away from 2. In L2 the spectral theorem gives it immediately, but in Lp the functional calculus for the perturbed operator has to produce multipliers whose constants do not blow up. The singular perturbation adds a rank-one term to the resolvent whose symbol is not smooth at infinity, and Mihlin-type conditions can fail to hold uniformly for p far from 2. If the proofs only control the constants near p=2 or omit an explicit p-independent bound, the Strichartz constants and the contraction argument would be limited in scope. I would check the multiplier section carefully to see whether they address this or assume it carries over. This is aimed at people already working on singular perturbations of dispersive operators and on Lp theory for such problems. A reader who needs the explicit Lp description for further estimates in low dimensions will get direct value from it. The work is honest and technically grounded enough to deserve a serious referee, even if the Lp uniformity requires some extra verification in review. I would send it to peer review with a note to confirm the multiplier bounds.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the fractional Sobolev spaces H^{s,p}_α(R^d) associated to a singular perturbation Δ_α of the Laplacian in dimensions d=2,3. It extends the existing L² theory by establishing an equivalent description of these spaces in terms of the standard Sobolev spaces H^{s,p}, and applies the equivalence to obtain Strichartz estimates in the energy space and local well-posedness for the associated nonlinear Schrödinger equation via the contraction-mapping argument.

Significance. If the claimed norm equivalences hold with constants independent of the perturbation parameter α and uniform for 1<p<∞, the work would supply a functional-analytic tool that extends dispersive estimates and well-posedness results for singularly perturbed Schrödinger equations beyond the Hilbert-space setting. The explicit use of the equivalence to reach Strichartz and contraction-mapping conclusions is a direct and natural application.

major comments (1)

[proof of the main norm-equivalence theorem (presumably §3 or Theorem 2.3)] The central claim that ||(I−Δ_α)^{s/2}u||_p is equivalent to the standard H^{s,p} norm for 1<p<∞ rests on the functional calculus for Δ_α producing a Mihlin multiplier whose constants remain controlled independently of α. The resolvent of Δ_α differs from that of the free Laplacian by a rank-one term whose symbol fails to be smooth at infinity; without a separate verification that the resulting multiplier satisfies the Hörmander–Mihlin derivative conditions uniformly in p (and with constants independent of α), the equivalence constant may blow up for |p−2| large. This equivalence is load-bearing for all subsequent Strichartz and well-posedness statements.

minor comments (2)

[Introduction] Notation for the perturbed operator Δ_α and the associated spaces H^{s,p}_α should be introduced with an explicit reference to the precise definition of the singular perturbation (point interaction, etc.) already in the introduction.
[Strichartz section] The statement of the Strichartz estimates in the energy space should include the precise range of admissible (q,r) pairs and the dependence of the constants on α and p.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the key technical point underlying the norm equivalence. We address the concern about uniform control of the Mihlin constants below and are willing to strengthen the presentation.

read point-by-point responses

Referee: [proof of the main norm-equivalence theorem (presumably §3 or Theorem 2.3)] The central claim that ||(I−Δ_α)^{s/2}u||_p is equivalent to the standard H^{s,p} norm for 1<p<∞ rests on the functional calculus for Δ_α producing a Mihlin multiplier whose constants remain controlled independently of α. The resolvent of Δ_α differs from that of the free Laplacian by a rank-one term whose symbol fails to be smooth at infinity; without a separate verification that the resulting multiplier satisfies the Hörmander–Mihlin derivative conditions uniformly in p (and with constants independent of α), the equivalence constant may blow up for |p−2| large. This equivalence is load-bearing for all subsequent Strichartz and well-posedness statements.

Authors: We agree that uniform control of the multiplier constants is essential. In the proof of Theorem 2.3 we explicitly compute the symbol of (I−Δ_α)^{s/2} via the functional calculus and the rank-one resolvent correction. Because the perturbation is supported at a single point in dimensions 2 and 3, the correction term is a smooth function of ξ whose derivatives decay as O(|ξ|^{-2-k}) for |ξ| large, uniformly in the perturbation parameter α. Consequently the full symbol satisfies the Hörmander–Mihlin conditions with constants independent of α. The resulting operator-norm bound on L^p therefore depends on p only through the standard Mihlin theorem and does not blow up as |p−2| increases. We will add a short auxiliary lemma that isolates these derivative estimates to make the argument fully self-contained. revision: partial

Circularity Check

0 steps flagged

No circularity: extension of L2 theory to Lp relies on independent multiplier estimates and spectral properties

full rationale

The paper's central claim is an extension of prior L2 perturbed Sobolev space theory to the Lp setting via norm equivalence between H^{s,p}_α and standard H^{s,p}. The abstract and context describe this as an analogue description enabling Strichartz estimates and NLS well-posedness. No quoted equations or self-citations reduce the Lp equivalence to a fitted parameter, self-definition, or prior result by the same authors that is itself unverified. The derivation chain is presented as building on spectral theorem (for L2) plus separate multiplier analysis for Lp, which is not shown to collapse by construction. This is the common honest non-finding for extension papers whose core estimates remain externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of the singular perturbation Δ_α together with the assumption that its associated spaces admit a useful description via ordinary Sobolev norms; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The singular perturbation Δ_α of the Laplace operator admits a well-defined functional calculus and spectral properties that allow norm equivalences with standard Sobolev spaces in both L2 and Lp.
This background property is presupposed when the authors speak of extending the L2 theory and finding an analogue description.

pith-pipeline@v0.9.0 · 5628 in / 1436 out tokens · 103219 ms · 2026-05-22T18:50:43.746654+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Existence, structure, and robust- ness of ground states of a NLSE in 3D with a point defect

Riccardo Adami, Filippo Boni, Raffaele Carlone, and Lor enzo T entarelli. Existence, structure, and robust- ness of ground states of a NLSE in 3D with a point defect. Journal of Mathematical Physics, 63(7):071501, 07 2022

work page 2022

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Ground states for the planar NLSE with a point defect as minimizers of the constrained ene rgy .Calc

Riccardo Adami, Filippo Boni, Raffaele Carlone, and Lor enzo T entarelli. Ground states for the planar NLSE with a point defect as minimizers of the constrained ene rgy .Calc. Var . Partial Differential Equa- tions, 61(5):Paper No. 195, 32, 2022

work page 2022

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Point intera ctions as limits of short range interactions

Sergio Albeverio and Raphael Høegh-Krohn. Point intera ctions as limits of short range interactions. J. Operator Theory, 6(2):313–339, 1981

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On the cauchy problem for the reaction- diffusion system with point-interaction in R2

Daniele Barbera, Vladimir Georgiev, and Mario Rastrell i. On the cauchy problem for the reaction- diffusion system with point-interaction in R2. arXiv, 2504.08460, 2025

work page arXiv 2025

[6] [6]

F . A. Berezin and L. D. Faddeev. Remark on the Schrödingerequation with singular potential. Dokl. Akad. Nauk SSSR, 137:1011–1014, 1961

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Betancor, Alejandro J

Jorge J. Betancor, Alejandro J. Castro, and Pablo Raúl St inga. The fractional Bessel equation in Hölder spaces. J. Approx. Theory, 184:55–99, 2014

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M. S. Birman and M. Z. Solomjak. Spectral Theory of Self-Adjoint Operators in Hilbert Space , volume 5 of Mathematics and its Applications . D. Reidel Publishing Company, Dordrecht, Holland, Dordre cht, 1 edition, 1987. Originally published in Russian. Part of the Springer Book Archive. Copyright 1987

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We ll posedness of the nonlinear schrödinger equation with isolated singularities

Claudio Cacciapuoti, Domenico Finco, and Diego Noja. We ll posedness of the nonlinear schrödinger equation with isolated singularities. Journal of Differential Equations, 305:288–318, 2021

work page 2021

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Semilinear Schrödinger equations , volume 10 of Courant Lecture Notes in Mathemat- ics

Thierry Cazenave. Semilinear Schrödinger equations , volume 10 of Courant Lecture Notes in Mathemat- ics. New Y ork University, Courant Institute of Mathematical Sciences, New Y ork; American Mathematical Society, Providence, RI, 2003

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Cornean, Alessandro Michelangeli, and Kenji Y ajima

Horia D. Cornean, Alessandro Michelangeli, and Kenji Y ajima. T wo-dimensional Schrödinger operators with point interactions: threshold expansions, zero modes and Lp -boundedness of wave operators. Rev. Math. Phys., 31(4):1950012, 32, 2019

work page 2019

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Cornean, Alessandro Michelangeli, and Kenji Y ajima

Horia D. Cornean, Alessandro Michelangeli, and Kenji Y ajima. Erratum: Two-dimensional Schrödinger operators with point interactions: threshold expansions, zero modes and Lp -boundedness of wave op- erators. Rev. Math. Phys., 32(4):2092001, 5, 2020

work page 2020

[13] [13]

Lp - boundedness of wave operators for the three-dimensional mu lti-centre point interaction

Gianfausto Dell’Antonio, Alessandro Michelangeli, R affaele Scandone, and Kenji Yajima. Lp - boundedness of wave operators for the three-dimensional mu lti-centre point interaction. Ann. Henri Poincaré, 19(1):283–322, 2018. 38 V . GEORGIEV AND M.RASTRELLI

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Klaus-Jochen Engel and Rainer Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate T exts in Mathematics. Springer-Verlag, New Y ork, 2000. With contributions by S.Brendle, M. Campiti, T . Hahn, G. Metafune, G. Nickel, D. Pallara, C. Pe razzoli, A. Rhandi, S. Romanelli and R. Schnaubelt

work page 2000

[15] [15]

Blow-up and instability of standing waves for the nls with a point in- teraction in dimension two

Domenico Finco and Diego Noja. Blow-up and instability of standing waves for the nls with a point in- teraction in dimension two. Zeitschrift für angewandte Mathematik und Physik, 74(4), 2023

work page 2023

[16] [16]

Local well-pose dness and blow-up in the energy space for the 2d nls with point interaction

Luigi Forcella and Vladimir Georgiev. Local well-pose dness and blow-up in the energy space for the 2d nls with point interaction. arXiv, 2410.16039, 2024

work page arXiv 2024

[17] [17]

On stability and instability of standing waves for 2d-nonlinear schrödinger equations with point in teraction

Noriyoshi Fukaya, Vladimir Georgiev, and Masahiro Ike da. On stability and instability of standing waves for 2d-nonlinear schrödinger equations with point in teraction. Journal of Differential Equations , 321:258–295, 2022

work page 2022

[18] [18]

On fractional powers of singular perturbations of the Laplacian

Vladimir Georgiev, Alessandro Michelangeli, and Raff aele Scandone. On fractional powers of singular perturbations of the Laplacian. J. Funct. Anal., 275(6):1551–1602, 2018

work page 2018

[19] [19]

Standing waves and global well- posedness for the 2d hartree equation with a point interacti on

Vladimir Georgiev, Alessandro Michelangeli, and Raff aele Scandone. Standing waves and global well- posedness for the 2d hartree equation with a point interacti on. Communications in Partial Differential Equations, 49(3):242–278, 2024

work page 2024

[20] [20]

Sobolev spaces for singular perturbation of 2D Laplace operator

Vladimir Georgiev and Mario Rastrelli. Sobolev spaces for singular perturbation of 2D Laplace operator. Nonlinear Analysis, 251:113710, 2025

work page 2025

[21] [21]

Geometric theory of semilinear parabolic equations , volume 840 of Lecture Notes in Mathe- matics

Daniel Henry . Geometric theory of semilinear parabolic equations , volume 840 of Lecture Notes in Mathe- matics. Springer-Verlag, Berlin-New Y ork, 1981

work page 1981

[22] [22]

On nonlinear Schrödinger equations

T osio Kato. On nonlinear Schrödinger equations. II. H s -solutions and unconditional well-posedness. J. Anal. Math., 67:281–306, 1995

work page 1995

[23] [23]

Fractional powers of operators

Hikosaburo Komatsu. Fractional powers of operators. Paciﬁc J. Math., 19:285–346, 1966

work page 1966

[24] [24]

Fractional powers of operators

Hikosaburo Komatsu. Fractional powers of operators. II. Interpolation spaces. Paciﬁc J. Math., 21:89–111, 1967

work page 1967

[25] [25]

Singular Hartree equation in frac- tional perturbed Sobolev spaces

Alessandro Michelangeli, Alessandro Olgiati, and Raf faele Scandone. Singular Hartree equation in frac- tional perturbed Sobolev spaces. J. Nonlinear Math. Phys., 25(4):558–588, 2018

work page 2018

[26] [26]

E nergy methods for abstract nonlinear Schrödinger equations

Noboru Okazawa, T oshiyuki Suzuki, and T omomi Y okota. E nergy methods for abstract nonlinear Schrödinger equations. Evol. Equ. Control Theory, 1(2):337–354, 2012

work page 2012

[27] [27]

B. Simon. Essential self-adjointness of schrödinger o perators with singular potentials. Arch. Rational Mech. Anal., 52:44–48, 1973

work page 1973

[28] [28]

The initial value problem for some dispersive differential equations

Gigliola Stafﬁlani. The initial value problem for some dispersive differential equations. ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–The University of Chicago

work page 1995

[29] [29]

Michael E. T aylor. T ools for PDE, volume 81 of Mathematical Surveys and Monographs. American Mathe- matical Society, Providence, RI, 2000. Pseudodifferential operators, paradifferential operators, and layer potentials

work page 2000