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REVIEW 2 major objections 4 minor 46 references

The spectral picture of self-similar collapse in the Constantin-Lax-Majda equation

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that the linearized self-similar collapse of the CLM equation has a clean spectral gap of 1/2 on a carefully chosen function space, with only the two symmetry modes as spectrum above that gap.

desk verdict The a=0 spectral theorems are real and mostly self-contained; the stability interpretation is conditional on an admitted modeling choice, and the paper is honest about that. read the letter →

arxiv 2607.19762 v1 pith:T6FQW6NW submitted 2026-07-22 physics.flu-dyn math-phmath.MP

Jie Xu This is my paper
classification physics.flu-dynmath-phmath.MP MSC 35Q3535B4435P0547A10
keywords Constantin-Lax-Majdaequationself-similarblow-upessentialspectrumspectralgapHilberttransformoperatorrealizationlinearstabilityfractionaldissipation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's target is the linearized operator that governs small perturbations of the exact self-similar collapse profile Ω(y) = −y/(y² + 1/4) of the Constantin–Lax–Majda (CLM) equation, the a = 0 member of the generalized family. On the origin-H² space X of odd functions that are square-integrable together with their second derivative, the author proves that the essential spectrum meets the half-plane Re λ ≥ −1/2 in exactly the vertical line Re λ = −1/2, and that the only eigenvalues are 0 and 1, the scaling and time-shift modes. Consequently the collapse is linearly stable in the spectral sense with a gap of 1/2, and the linearized semigroup decays exactly as e^{−τ/2} on a conjugated weighted space reached by a bounded transfer map. A realization dichotomy shows this clean picture is space-dependent: on the maximal L² realization the whole strip is essential spectrum, which is what naive discretizations display. Conditional statements for a > 0 give a two-line essential-spectrum inclusion and a formal scaling-relevance exponent s*(a) = 1/c_l(a).

What carries the argument

The central object is the origin-H² space X = {odd φ : φ, φ'' ∈ L², φ(y) = a₁y + o(y)}; choosing it is what converts the dense L² spectrum into a single essential line. The carrying identity is the Hardy reduction due to the exact profile: on the upper-half-plane Hardy space the nonlocal term collapses to the scalar first-order operator L0⁺ = −1 − y∂_y + i/(y + i/2), with explicit solutions u_λ(y) = y^{1−λ}/(y + i/2)². The far-field essential line is placed by a log-widening approximate-eigenfunction (Weyl) sequence in the Mellin variable, and the strip is emptied by an explicit resolvent kernel controlled by the Hardy–Mellin operator, whose exact norm is 1/(Re z + 1/2). Conjugation by (y +

What would settle it

Compute, at a = 0, the second-derivative integral of the explicit candidate eigenfunction u_λ(y) = y^{1−λ}/(y + i/2)² for a value of λ in (−1/2, 0) other than 0. The paper's completeness lemma predicts ∫₀^δ |u_λ''(y)|² dy = ∞ for every such λ, so if the integral is finite for any λ in that range, the origin-H² completeness claim collapses.

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Extended reading notes

Core claim

At a = 0, with the exact profile Ω, the linearized CLM operator L0, realized on the space X, has essential spectrum equal to the single vertical line {Re λ = −1/2} inside the closed half-plane Re λ ≥ −1/2, and full point spectrum exactly {0, 1}; the open strip contains no discrete spectrum. The proof is constructive: a log-widening Weyl sequence places the essential line, an explicit Hardy–Mellin resolvent with norm governed by O(α^{−3/2}) empties the strip, and a Hardy diagonalization reduces the operator to a scalar first-order ODE whose only L², odd, smooth-at-origin solutions are the two symmetry modes. The paper further shows that the semigroup is explicit after conjugation v = (y + i/2

Load-bearing premise

The load-bearing premise is the choice of origin-H² space X as the physical realization — the paper states in Section 3.1 that this is a modeling choice, not derived from a dynamical well-posedness principle, and the spectral gap disappears on the maximal L² realization; separately, the a = 0 exact decay is proven on the weighted space Y_{3/2} but its transfer to the X norm, or to the nonlinear flow, like the a > 0 discrete exclusion, is left open in Section 8.

Editorial extensions

If this is right

  • The CLM collapsing profile is linearly stable at a = 0 in the spectral sense: no eigenvalues lie in the open strip (−1/2, 0), so a gap of 1/2 separates the two symmetry modes from the essential spectrum.
  • The full point spectrum over C is exactly {0, 1}; in particular the essential line carries no embedded eigenvalues, so nothing further contaminates the spectral picture.
  • The linearized semigroup has the exact closed-form decay e^{−τ/2} on the conjugated weighted space Y_{3/2}, reached from X by a bounded transfer; this is the linear input for a persistence proof, which the paper leaves open.
  • The realization dichotomy means that numerical methods without an origin condition are faithfully computing the maximal L² spectrum, a whole filled strip, rather than the physical X spectrum; methods enforcing the origin regularity should see the single line.
  • The recomputed branch c_l(a) reproduces the known critical advection to 0.04%, and the exponent s*(a) = 1/c_l(a) marks the formal threshold below which fractional dissipation is asymptotically subdominant in self-similar variables.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the realization dichotomy suggests a practical numerical test — adding the origin second-derivative condition to a log-Mellin discretization should collapse the filled strip to the line Re λ = −1/2; observing this collapse would independently corroborate the X picture.
  • Editorial inference: since the semigroup decay is exact and profile-independent in the conjugated variable, one can try to prove the missing quadratic estimate for N(φ) = φHφ and a lower bound for the dissipation form in Y_{3/2}; if those hold, the paper's linear machinery would likely close a nonlinear persistence theorem for 0 < s < 1 at a = 0.
  • Editorial inference: the two-line inclusion for a > 0 leaves an exactness question; a numerical scan of the rightmost essential edge along the branch, compared with the maximum of the two predicted lines, would indicate whether the gap is exactly 1 − c_l/2 minus the protrusion term.
  • Editorial inference: the observation that the origin line depends on the second-derivative weight suggests that stronger H³ realizations should move the origin line further left uniformly in a; verifying this would make the a > 0 essential picture realization-robust.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper analyzes the linearization of the generalized Constantin-Lax-Majda (gCLM) family at the exact a=0 self-similar collapse profile Ω(y) = -y/(y^2+1/4). The main theorems, all at a=0, are: Theorem 1 (on an origin-H2 space X, the essential spectrum of L0 in the half-plane {Re λ ≥ -1/2} is exactly the vertical line {Re λ = -1/2}); Theorem 2 (the discrete spectrum in the open strip is exactly {0,1}, the scaling and time-shift symmetry modes); and Theorem 3 (the full point spectrum over C is exactly {0,1}, so the essential line carries no embedded eigenvalues). Proposition 2 shows that on the maximal L2 realization the whole strip is essential spectrum, and the paper identifies the numerical smear with this maximal realization. Appendix A gives an explicit conjugated semigroup with exact decay e^{-τ/2} on a weighted space Y_{3/2}, reached from X by a bounded transfer map with no reverse estimate. For a>0, Proposition 1 gives a conditional two-line essential-spectrum inclusion under admissibility hypotheses, and the paper recomputes the branch c_l(a) as a numerical cross-check and records the formal scaling-relevance exponent s*(a)=1/c_l(a). The paper is explicit that the a>0 results and the semigroup transfer are not completed nonlinear-stability statements.

Significance. If the a=0 theorems are correct, this is a significant contribution: it gives a rigorous spectral picture of the CLM collapse profile, including the exact location of the essential spectrum, completeness of the two symmetry modes, and a closed-form linear semigroup. The paper's strengths are its explicit and constructive arguments — Mellin diagonalization, Weyl sequences, an explicit resolvent majorant, a Hardy-space completeness lemma, and machine-verified algebraic identities — and its unusually honest treatment of the distinction between spectral gaps and dynamical decay. The realization dichotomy (Proposition 2) is a valuable conceptual contribution because it explains a numerical artifact as the faithful spectrum of a weaker realization. The a>0 and s*(a) material is appropriately labeled conditional or formal; it does not carry the main claim.

major comments (2)
  1. [§3.1, Eq. (3.2); Proposition 2 (Eq. (3.5))] The spectral gap and the resulting statement 'linearly stable ... with a gap of 1/2 on X' are properties of the chosen origin-H2 realization X. Proposition 2 shows that on the maximal L2 realization the essential spectrum contains the full strip {-1/2 ≤ Re λ ≤ 3/2}, and §3.1 explicitly calls X 'a modeling choice, not derived here from a dynamical well-posedness principle.' Since no theorem shows that the nonlinear flow, or even the linearized evolution from natural data, keeps perturbations in X, the stability claim is conditional on an unproven modeling choice. This is not a technical error, but it is load-bearing for the physical interpretation. I recommend that the abstract and conclusion state prominently that the stability theorem is a spectral statement about the X-realization, and that the dynamical selection of X is an open problem.
  2. [Appendix A, Eqs. (A.21)-(A.24)] The exact semigroup decay e^{-τ/2} is proved only on the conjugated weighted space Y_{3/2}, reached from X by the bounded transfer map J_X; there is no reverse estimate controlling the X-norm of e^{L0 τ} φ. Because L0 is non-normal, the spectral gap on X does not by itself imply X-norm decay, as the paper acknowledges. Nevertheless, the abstract's 'exact decay rate e^{-τ/2}' could be misread as a decay statement in the physical space X. I recommend adding an explicit caveat, both in the abstract and in the semigroup theorem, that this decay holds in Y_{3/2} and that X-norm decay and nonlinear persistence remain open.
minor comments (4)
  1. [§4 (Numbering convention)] The convention that numbered results run independently within each section, so that 'Lemma 4.5' can live in 'Section 4.4,' is needlessly confusing and invites mis-citation. Please renumber the results or provide a mapping between result numbers and subsection numbers.
  2. [§3.1] The definition of X includes the condition φ(y)=a1 y+o(y), which is automatic for odd functions in H2(R) and therefore redundant. Since the 'modeling choice' discussion depends on the second-derivative condition, the redundancy should be stated explicitly to prevent a reader from thinking an extra origin condition is being imposed.
  3. [Introduction] There is a typo in 'classicalconecalculus[23,24]' — missing spaces. Please also check other instances of concatenated words in the manuscript.
  4. [§3.2 / §7] The numerical spectral classification relies on a heuristic Nyquist filter (<N/16) and other diagnostic thresholds. The paper is honest that these are exploratory, but a sentence in Section 7 summarizing which conclusions are robust to the filter choice would strengthen the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the a=0 results are proved from the exact profile and resolvent estimates, with the realization-dependence explicitly acknowledged.

full rationale

The derivation at a=0 is self-contained. It starts from the exact profile Ω(y)=−y/(y²+1/4), uses the exact Hardy identity HΩ−iΩ=i/(y+i/2) to reduce L0 to a first-order scalar operator, and proves Theorems 1–3 by explicit Weyl sequences, a Hardy–Mellin resolvent bound, and a domain/quantization argument. No fitted parameter enters the a=0 spectral statements. Proposition A.1 derives the semigroup by direct characteristics and the decay rate on Y_{3/2} is an exact operator norm, not an assumed rate. The a>0 quantities are explicitly labeled conditional or formal: the branch c_l(a) is an independent numerical recomputation cross-checked against published results, and s*(a)=1/c_l(a) is called a 'formal scaling diagnostic' rather than a prediction. The main caveat—that the spectral gap is realization-dependent—is openly stated: §3.1 says X is 'a modeling choice, not derived here from a dynamical well-posedness principle,' and Proposition 2 shows the maximal L2 realization has the full strip. This conditions the physical interpretation but does not make any theorem reduce to its own input by construction. There are no load-bearing self-citations and no fitted value is renamed as a prediction. The honest finding is therefore no significant circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The a=0 central spectrum is parameter-free: the exact profile and analytic resolvent require no fitted constants. The main non-derived inputs are the function-space choice X and the standard Hardy/Mellin facts it relies on; the a>0 statements additionally consume Adm(a) and the numerically fitted branch cl(a). No new physical entities, forces, or particles are introduced.

free parameters (1)
  • cl(a) focusing-exponent branch on (0,ac) = cl(0)=1 exact; cl(0.1)=0.8730, cl(0.3)=0.6178, cl(0.5)=0.3333, cl(0.65)=0.0775; zero crossing a*≈0.6888
    Numerically continued by Newton iteration from the exact cl(0)=1 anchor; two-grid difference ≲5×10^-4 and only the endpoint ac is compared quantitatively to [4]. Used to define s*(a)=1/cl(a) and the conditional Proposition 1 coefficients; not a certified continuum computation.
assumptions (6)
  • domain assumption The origin-H2 space X is the physically relevant realization of the linearized collapse dynamics
    Section 3.1: 'This is a modeling choice, not derived here from a dynamical well-posedness principle.' Proposition 2 shows the spectrum and the gap change under this choice, so the stability conclusion is realization-dependent.
  • domain assumption Admissibility hypotheses Adm(a) = (H1)-(H4) hold for a>0 profiles
    Definition 4.2 assumes C4 odd profiles, origin expansions, far-field tail bounds, and transport-weight comparability. Proposition 1 and the a>0 inclusion are conditional on these, and whether the Huang-Qin-Wang-Wei profiles satisfy them is explicitly left open (Section 4.1).
  • standard math Paley-Wiener/Hardy boundary uniqueness (Facts T1, T3)
    Used in Lemma 1 to link the half-line constants and force a Hardy component to vanish identically if it vanishes on a positive-measure set.
  • standard math H∞ multipliers preserve H2 and the Paley-Wiener-Schwartz tube statement (Facts T2, T5)
    Used to keep resolvent kernels and eigenfunctions in H2+ and to pass from tempered boundary values to Hardy-class membership.
  • standard math Mellin transform unitarily diagonalizes y∂_y
    Section 4.1-4.2: M[y∂_y f](s) = -s M[f](s); used to compute the far-field spectrum and the exact Hardy-Mellin operator norm.
  • standard math Distributional facts T4: W^{1,1}_loc = AC_loc and absolute continuity consequences
    Used in Step 5.4 to reduce the scalar ODE to a constant-coefficient form on each half-line.

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Cite this review

Pith. "Pith review of The spectral picture of self-similar collapse in the Constantin-Lax-Majda equation." pith.science (2026). https://pith.science/paper/T6FQW6NW

@misc{pith2026260719762,
  author       = {Pith},
  title        = {Pith review of: The spectral picture of self-similar collapse in the Constantin-Lax-Majda equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6FQW6NW}},
  note         = {Machine review of arXiv:2607.19762}
}
abstract

We give a spectral description of the self-similar collapse profile of the Constantin-Lax-Majda (CLM) equation, the $a=0$ anchor of the generalized family $w_t + a\,u\,w_x = u_x\,w$, $u_x = Hw$. Linearizing about the exact profile $\Omega(y) = -y/(y^2+1/4)$ and realizing $L_0$ as a closed operator on the origin-$H^2$ space, we prove three things at $a=0$. Its essential spectrum meets the closed half-plane $\{\mathrm{Re}\,\lambda \ge -1/2\}$ in the single vertical line $\{\mathrm{Re}\,\lambda = -1/2\}$: the line is placed by a log-widening Weyl sequence, and an explicit Hardy-Mellin resolvent bound constructively empties the rest of the half-plane apart from $0$ and $1$. Its full point spectrum over $\mathbb{C}$, on the odd realization, is exactly $\{0,1\}$, the scaling and time-shift symmetry modes, with no embedded eigenvalues; removing these by the standard modulation leaves a spectral gap of $1/2$ on $X$. The linear semigroup and its exact decay rate $e^{-\tau/2}$ are computed in closed form, but on a weighted space of the conjugated variable reached from $X$ by a bounded transfer map; we keep the two separate, since $L_0$ is non-normal and a spectral gap does not by itself give a decay rate in the $X$ norm. A realization dichotomy identifies the in-strip smear of generic discretizations as the faithful spectrum of the maximal $L^2$ realization, which origin-$H^2$ removes. For $a>0$ we prove a conditional two-line inclusion for each admissible smooth focusing profile, recompute the branch $c_l(a)$ of Lushnikov, Silantyev, and Siegel as a cross-check, and record the formal scaling-relevance exponent $s^*(a) = 1/c_l(a)$, below which fractional dissipation is asymptotically subdominant in self-similar variables for fixed sufficiently regular data. The contribution is the realization-dependent spectral picture of the collapse profile itself.

Figures

Figures reproduced from arXiv: 2607.19762 by the authors.

Figure 1
Figure 1. The recomputed branch cl(a) from the compactified-grid Newton continuation at N = 2048, plotted at the advections where the Newton solve succeeded, decreasing monotonically from the exact CLM anchor cl(0) = 1 to its zero crossing at a ∗ ≈ 0.6888, which reproduces the published critical advection ac = 0.6890665 of Lushnikov, Silantyev, and Siegel (dashed line) to 0.04%; the reference branch is theirs, this is an inde… view at source ↗
Figure 2
Figure 2. The two essential-spectrum lines of Proposition 1 (for an admissible profile). Panel [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The scaling-relevance exponent s ∗ (a) = 1/cl(a) read off the recomputed branch of Section 2 (log scale), rising from s ∗ (0) = 1 (Schochet’s supercritical full-Laplacian blow-up) through s ∗ (1/2) = 3 (J. Chen’s subcritical s = 2 blow-up) to about 54 near the branch endpoint (beyond the tabulated points), and crossing the ordinary-Laplacian value s = 2 at a ≈ 0.39 where cl = 1/2. This is a formal scaling diagnostic… view at source ↗

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Works this paper leans on

46 extracted references · 15 linked inside Pith

  1. [1]

    Pure Appl

    Constantin P, Lax P D and Majda A 1985Comm. Pure Appl. Math.38715–724

  2. [2]

    De Gregorio S 1990J. Stat. Phys.591251–1263

  3. [3]

    Okamoto H, Sakajo T and Wunsch M 2008Nonlinearity212447–2461

  4. [4]

    Nonlinear Sci.31art

    Lushnikov P M, Silantyev D A and Siegel M 2021J. Nonlinear Sci.31art. 82 (Preprint 2010.01201)

    arXiv 2010

  5. [5]

    Pure Appl

    Chen J, Hou T Y and Huang D 2021Comm. Pure Appl. Math.741282–1350 (Preprint 1905.06387) 39

    arXiv 1905

  6. [6]

    Huang D, Qin X, Wang X and Wei D 2024Arch. Ration. Mech. Anal.248art. 22 (Preprint 2305.05895)

    arXiv

  7. [7]

    Huang D, Tong J and Wang X 2026 Self-similar finite-time blowups with singular profiles of the generalized Constantin–Lax–Majda model: theoretical and numerical investigations (Preprint2603.25104)

    arXiv 2026

  8. [8]

    Chen J and Hou T Y 2022 Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis (Preprint2210.07191)

    arXiv 2022

Show all 46 references
  1. [9]

    Simul.2325–130 (Preprint2305.05660)

    Chen J and Hou T Y 2025Multiscale Model. Simul.2325–130 (Preprint2305.05660)

    arXiv

  2. [10]

    Wang Y, Lai C Y, Gómez-Serrano J and Buckmaster T 2023Phys. Rev. Lett.130244002 (Preprint2201.06780)

    arXiv

  3. [11]

    Wang Y, Bennani M, Martens J, Racanière S, Blackwell S, Matthews A, Nikolov S, Cao- Labora G, Park D S, Arjovsky M, Worrall D, Qin C, Alet F, Kozlovskii B, Tomašev N, Davies A, Kohli P, Buckmaster T, Georgiev B, Gómez-Serrano J, Jiang R and Lai C Y 2025 Discovery of unstable s...

    2025

  4. [12]

    Wang Y, Léger T, Lai C Y and Buckmaster T 2025 Resolving sharp gradients of unstable singularities to machine precision via neural networks (Preprint2511.22819)

    2025

  5. [13]

    Hou T Y, Wang Y and Yang C 2025 Nonuniqueness of Leray–Hopf solutions to the unforced incompressible 3D Navier–Stokes equation (Preprint2509.25116)

    2025

  6. [14]

    Ambrose D M, Lushnikov P M, Siegel M and Silantyev D A 2024Nonlinearity37025004 (Preprint2207.07548)

    arXiv

  7. [15]

    PDE14891–908 (Preprint1906

    Elgindi T M, Ghoul T E and Masmoudi N 2021Anal. PDE14891–908 (Preprint1906. 05811)

  8. [16]

    Elgindi T M and Jeong I J 2020Arch. Ration. Mech. Anal.2351763–1817 (Preprint 1701.04050)

    arXiv

  9. [17]

    Pure Appl

    Schochet S 1986Comm. Pure Appl. Math.39531–537

  10. [18]

    Sakajo T 2003Nonlinearity161319–1328

  11. [19]

    Sakajo T 2003J. Math. Sci. Univ. Tokyo10187–207

  12. [20]

    Chen J 2020Nonlinearity332502–2532 (Preprint1908.09385)

    arXiv

  13. [21]

    Silantyev D A, Lushnikov P M, Siegel M and Ambrose D M 2025Stud. Appl. Math.155 e70115 (Preprint2411.01891)

  14. [22]

    Chapman S J, Kavousanakis M, Charalampidis E G, Kevrekidis I G and Kevrekidis P G 2024Nonlinearity37095034 (Preprint2310.13770)

    arXiv

  15. [23]

    Moscow Math

    Kondrat’ev V A 1967Trans. Moscow Math. Soc.16227–313 english transl.; Russian orig- inal Tr. Mosk. Mat. Obs. 16 (1967) 209–292

    1967

  16. [24]

    Global Anal

    Coriasco S, Schrohe E and Seiler J 2007Ann. Global Anal. Geom.31223–285 (Preprint math/0401395)

    arXiv

  17. [25]

    Math.vol 327) (Amer

    Shvidkoy R and Latushkin Y 2003 The essential spectrum of the linearized 2D Euler operator is a vertical bandAdvances in Differential Equations and Mathematical Physics (Contemp. Math.vol 327) (Amer. Math. Soc.) pp 299–304 (Preprintmath-ph/0306027) 40

    2003

  18. [26]

    Huang D, Tong J and Wei D 2023Comm. Math. Phys.4022791–2829 (Preprint2209. 08232)

  19. [27]

    Huang D, Qin X and Wang X 2025SIAM J. Math. Anal.574068–4096 (Preprint2401. 14615)

  20. [28]

    Levitin M and Shargorodsky E 2004IMA J. Numer. Anal.24393–416 (Preprintmath/ 0212087)

  21. [29]

    Davies E B and Plum M 2004IMA J. Numer. Anal.24417–438 (Preprintmath/0302145)

  22. [30]

    Colbrook M J, Horning A and Townsend A 2021SIAM Rev.63489–524 (Preprint2006. 01766)

  23. [31]

    Jia H, Stewart S and Sverak V 2019Arch. Ration. Mech. Anal.2311269–1304 (Preprint 1710.02737)

    arXiv

  24. [32]

    Lei Z, Liu J and Ren X 2020Comm. Math. Phys.375765–783 (Preprint1811.09754)

    arXiv

  25. [33]

    Guo J and Jiu Q 2025 Stability and instability on the De Gregorio modification of the Constantin–Lax–Majda model (Preprint2506.02800)

    2025 arXiv

  26. [34]

    Scuola Norm

    Lockhart R B and McOwen R C 1985Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)12409–447

  27. [35]

    Melrose R B 1993The Atiyah–Patodi–Singer Index Theorem(Research Notes in Mathe- maticsvol 4) (A K Peters)

  28. [36]

    Lesch M 1997Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods (Teubner-Texte zur Mathematikvol 136) (Teubner)

  29. [37]

    Rabinovich V S, Roch S and Silbermann B 2004Limit Operators and Their Applications in Operator Theory(Operator Theory: Advances and Applicationsvol 150) (Birkhäuser)

  30. [38]

    Duduchava R 1979Integral Equations with Fixed Singularities(Teubner-Texte zur Mathe- matikvol 24) (Teubner)

  31. [39]

    Koosis P 1998Introduction toH p Spaces2nd ed (Cambridge Tracts in Mathematicsvol

  32. [40]

    Duren P L 1970Theory ofHp Spaces(Pure and Applied Mathematicsvol 38) (Academic Press)

  33. [41]

    Garnett J B 2007Bounded Analytic Functions(Graduate Texts in Mathematicsvol 236) (Springer)

  34. [42]

    Brezis H 2011Functional Analysis, Sobolev Spaces and Partial Differential EquationsUni- versitext (Springer)

  35. [43]

    Rudin W 1987Real and Complex Analysis3rd ed (McGraw-Hill)

  36. [44]

    Stein E M and Weiss G 1971Introduction to Fourier Analysis on Euclidean Spaces(Prince- ton Mathematical Seriesvol 32) (Princeton University Press)

  37. [45]

    Hörmander L 1990The Analysis of Linear Partial Differential Operators I2nd ed (Grundlehren der mathematischen Wissenschaftenvol 256) (Springer) 41

  38. [115]

    (Cambridge University Press)

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