Pith. sign in

REVIEW 3 major objections 5 minor 44 references

Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves sharp two-sided heat kernel bounds for the fractional Laplacian with Hardy potential in each angular momentum channel: the kernel is uniformly comparable to the free channel kernel times ground-state weights $(1\wedge…

desk verdict The genuinely new step is the form identification in Theorem 4.2, not the bounds themselves; the η<0 case is a verification burden, not a detected error, and the paper deserves a serious referee. read the letter →

arxiv 2506.08115 v1 pith:A3MLWWDI submitted 2025-06-09 math.AP math.FAmath.PR

Krzysztof Bogdan , Konstantin Merz This is my paper
classification math.APmath.FAmath.PR MSC 47D0860J35
keywords HardyinequalityheatkernelfractionalLaplacianangularmomentumchannelBesselgroundstaterepresentationDirichletformsharpestimates
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

This paper establishes sharp, two-sided heat kernel bounds for the operator $(-\Delta)^{\alpha/2} - \kappa|x|^{-\alpha}$, the fractional Laplacian with an inverse-power Hardy potential, acting on functions with fixed angular momentum $\ell$. The claim is that on each such channel the heat kernel is comparable, uniformly in space and time, to the free channel kernel multiplied by the ground-state weights $(1\wedge r/t^{1/\alpha})^{-\eta}(1\wedge s/t^{1/\alpha})^{-\eta}$, with explicit comparable formulas for the free kernel itself. Since Hardy operators model relativistic atoms and appear as scaling limits of more complicated operators, the bounds give quantitative control of the electron density near the nucleus channel by channel, and refine the known equivalence of Sobolev norms for such operators. The proof works by identifying the heat kernel of the Hardy operator with a Schr\"odinger perturbation of a subordinated Bessel heat kernel on the half-line, and proving equality of the corresponding quadratic forms.

What carries the argument

The machinery has two layers. First, the angular momentum decomposition: writing functions as $u(|x|)|x|^\ell Y_{\ell,m}(x/|x|)$ reduces the fractional Laplacian with Hardy potential on $\mathbb{R}^d$ to an operator $L_{\kappa,\ell}$ on the half-line with weighted measure $r^{d_\ell-1}dr$, where $d_\ell=d+2\ell$ is the effective dimension. Second, the form-identification theorem (Theorem 4.2): the quadratic form $I_{\zeta,\eta}$ of the ground-state representation with $h(r)=r^{-\eta}$ is shown to equal the Dirichlet form $E_{\zeta,\eta}$ of the Schr\"odinger-perturbed subordinated Bessel heat kernel $p^{(\alpha)}_{\zeta,\eta}$; equality of forms implies equality of the self-adjoint operators and hence of heat kernels. The positive-$\eta$ case is handled by form cores and density of $C_c^\infty$; the negative-$\eta$ case uses a ground-state representation and estimates of the first Duhamel term via 3G inequalities. Once the forms coincide, the sharp bounds for $p^{(\alpha)}_{\zeta,\eta}$ from Theorem 3.7 transfer verbatim.

What would settle it

Take a concrete channel with $\alpha=1$, $d=3$, $\ell=1$ and $\eta\in(-1,0)$, and a compactly supported smooth radial function $u$. The theorem predicts $I_{\zeta,\eta}[u]=E_{\zeta,\eta}[u]$ and $D(I_{\zeta,\eta})=D(E_{\zeta,\eta})$; finding any $u$ that makes one form finite and the other infinite, or for which the ratio of the two forms differs from 1, would disprove Theorem 4.2(2). A numerical check: evaluate $p^{(\alpha)}_{\zeta,\eta}(t,r,s)$ at very small $t$ and $r\neq s$, and compare with $\nu_\zeta(r,s)t$; the claimed limit (4.72) requires the ratio to tend to 1, so a persistent deviation would falsify the transfer.

Watch

Extended reading notes

Core claim

Let $d_\ell=d+2\ell$ and let $\kappa=\Phi^{(\alpha)}_{d_\ell}(\eta)$ parameterize the coupling constant through the monotone function defined in (1.11). For $\alpha\in(0,2]\cap(0,d_\ell)$ and $\eta\in(-S,(d_\ell-\alpha)/2]$, the heat kernel of $L_{\kappa,\ell}$---the radial part of $(-\Delta)^{\alpha/2}-\kappa|x|^{-\alpha}$ acting on angular momentum $\ell$---satisfies the uniform comparability (1.18) for $\alpha<2$ and the $\asymp$-version (1.19) for $\alpha=2$. The paper establishes this by proving that the semigroup generated by the ground-state quadratic form $I_{(d_\ell-1)/2,\eta}$ is exactly the Schr\"odinger perturbation of the subordinated Bessel semigroup $p^{(\alpha)}_{(d_\ell-1)/2,\eta}$, and then invoking the sharp bounds for that kernel. In particular, for $\eta=0$ the bare channel kernels have the explicit comparabilities (1.20) and (1.21).

Load-bearing premise

The load-bearing premise is that the quadratic form of the Hardy operator in a channel and the form generated by the Schr\"odinger-perturbed Bessel semigroup are equal, with the same domain; the proof for negative coupling relies on integral estimates from the authors' previous papers, and if that equality fails the heat kernel bounds do not follow.

Editorial extensions

If this is right

  • On every angular momentum channel, the heat kernel of the Hardy operator is comparable to the unperturbed channel kernel multiplied by $(1\wedge r/t^{1/\alpha})^{-\eta}(1\wedge s/t^{1/\alpha})^{-\eta}$, uniformly in $r,s,t>0$.
  • For $\alpha<2$ the free channel kernel has the explicit comparability $t/(|r-s|^{1+\alpha}(r+s)^{d_\ell-1}+t^{(1+\alpha)/\alpha}(t^{1/\alpha}+r+s)^{d_\ell-1})$, giving sharp near- and off-diagonal behavior.
  • For $\alpha=2$ the bound is a Gaussian-type comparability $e^{-(r-s)^2/(ct)}/(\sqrt{t}\,(rs+t)^{(d_\ell-1)/2})$, with possibly different constants in the exponential upper and lower bounds.
  • The kernel $\exp(-tL_{\Phi,\ell})(r,s)$ is jointly continuous in $r,s,t>0$.
  • The stated applications include upper bounds for relativistic-atom ground-state densities in fixed angular momentum channels, $\varrho^H_\ell(r)\lesssim r^{-2\eta}$ for small $r$, and refined Sobolev-norm equivalences for Hardy operators.

Reading between the lines

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

  • An implicit consequence is that on each channel the Hardy semigroup is obtained from the free semigroup by a Doob $h$-transform with $h(r)=r^{-\eta}$, up to a comparability factor; making this precise could give large-deviation or endpoint regularity information beyond the two-sided bounds.
  • The authors do not track the dependence of the constants on $\ell$; a testable extension is to prove the same bounds with constants that grow only polynomially or exponentially in $\ell$, which would permit summation over channels and strengthen the density results.
  • The $\eta$ range extends to negative values, so the same machinery should also give sharp bounds for repulsive inverse-power potentials, a case not emphasized in the physical motivation.
  • The $\alpha\to 2$ limit is not uniform in $\eta$; one could test whether the $\sim$ bounds for $\alpha<2$ converge to the $\asymp$ bounds for $\alpha=2$ with fixed $d$, $\ell$, and $\eta$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proves sharp, two-sided heat kernel bounds for the Hardy operator (−Δ)^{α/2} − κ|x|^{−α} in L²(R^d) restricted to a fixed angular momentum channel ℓ. Theorem 1.1 asserts that for α∈(0,2]∩(0,d+2ℓ) and η∈(−S,(d_ℓ−α)/2] with κ=Φ^{(α)}_{d_ℓ}(η), the heat kernel e^{−tL_{κ,ℓ}}(r,s) is comparable to (1∧r/t^{1/α})^{−η}(1∧s/t^{1/α})^{−η} e^{−tL_{0,ℓ}}(r,s), with explicit formulas for the unperturbed kernel in both the cases α<2 and α=2. The proof reduces Theorem 1.1 to an identification (Theorem 4.2(2)) between the ground-state-representation form I_{ζ,η} and the form E_{ζ,η} generated by the Schrödinger-perturbed subordinated Bessel kernel p^{(α)}_{ζ,η}, whose sharp bounds were proved in the authors' earlier work [BJM24]. The identification is established for η>0 within the paper (§4.3), while the η<0 case (§4.4) proceeds through the limit (4.72) of Lemma 4.17, which is derived from the Duhamel estimate (4.71) (Lemma 4.16) and the bounds (4.69)–(4.70) (Lemma 4.15), both proved in Appendix B using 3G inequalities imported from [BM25] and [BJM24]. The α=2 case is handled by unitary equivalence with Bessel operators and the explicit results of [MNS18].

Significance. I found the main theorem credible and the proof strategy sound. The bounds are genuinely sharp (matching upper and lower) and contain no free parameters: the comparison kernel and the ground-state weights are explicit, and the constants depend only on d, ℓ, α, η. The identification of the Hardy operator's form with the generator of the perturbed Bessel semigroup is new and is the technically substantial part of the paper. The η>0 case is self-contained relative to the cited kernel bounds, and the η<0 case, while resting on a chain of estimates deferred to Appendix B, is internally consistent: I checked the scaling argument that yields p^{(1,D)}_t=O(t²) for fixed r≠s, so the stress-test concern about (4.71) does not land as a detected error. The paper is also admirably explicit about what is imported: Theorem 3.7 is quoted from [BJM24], Remark 1.2 states that the ℓ-dependence of constants is not tracked, and the main theorem nowhere assumes the target Hardy heat-kernel bounds, so there is no circularity. The main weaknesses are statement-accuracy problems in the central bridge theorem and its auxiliaries, and the concentration of verification burden in an unreviewed preprint.

major comments (3)
  1. [§4.2, Theorem 4.2(2); §4.4, Theorem 4.18] Theorem 4.2(2) asserts the form equality I_{ζ,η}=E_{ζ,η} for all ζ∈(−1/2,∞) and η∈(−α,(2ζ+1−α)/2], but the proof for η<0 is delegated to Theorem 4.18, which is stated only for ζ∈[0,∞). The overclaimed range is not needed for Theorem 1.1, since ζ=(d_ℓ−1)/2≥0 for every d≥1 and ℓ≥0, so the main result survives; nevertheless, as printed, Theorem 4.2(2) claims more than is proved. Since the ingredients used in the η<0 proof (Lemmas 4.15–4.17) are all stated for ζ∈(−1/2,∞), an extension may be routine, but the authors must either supply it or restrict the statement of Theorem 4.2(2) accordingly.
  2. [§4.4.2, Lemma 4.16 and (4.71)] As stated, Lemma 4.16 involves no η, but p^{(1,D)}_t is defined through (3.13) with q(z)=Ψ_ζ(η)z^{−α}, so the quantity being estimated depends on the coupling while the right-hand side of (4.71) does not. Since Ψ_ζ(η)→−∞ as η↓−α, (4.71) cannot hold with a constant independent of η; the statement needs the hypothesis η∈(−α,(2ζ+1−α)/2] (or the coupling constant), with the implicit constant allowed to depend on it. The application in Lemma 4.17 uses only a fixed η, so this is a statement-accuracy fix rather than a flaw in the main proof.
  3. [§4.4.2, proof of Lemma 4.17] The critical limit (4.72), which is the hinge of the η<0 identification, is justified in a single sentence. For verifiability, please expand the argument: bound the integral in (4.73) by p^{(1,D)}_t(t,s,r) using p^{(α)}_{ζ,η}≤p^{(α)}_ζ from Theorem 3.6(5), apply Lemma 4.16, use the scaling (4.68), and show from Lemma 4.15 that for fixed r≠s the quantities G₀(1,r/t^{1/α},s/t^{1/α})+G₀(1,s/t^{1/α},r/t^{1/α}) are O(t), that G̃(1,r/t^{1/α},s/t^{1/α}) and G̃(1,s/t^{1/α},r/t^{1/α}) are o(1), and that t/(r∧s)^α is O(t); since p^{(α)}_ζ(t,r,s)=O(t), this yields p^{(1,D)}_t(t,r,s)=O(t²) for ζ≥0 and O(t²)+o(t) for ζ<0, so (4.73)=o(t) as required. The exposition should also mark which term of (4.71) uses [BM25, Theorem 3.1] (the case ζ≥0) versus [BJM24, Lemma 2.5] (the case ζ<0).
minor comments (5)
  1. [§3.2, Theorem 3.6] The hypothesis reads "η∈(−α, 2ζ+1−α/2]", which is typeset ambiguously; the intended interval is (−α,(2ζ+1−α)/2], matching the range used in Theorem 3.7.
  2. [§4.4 and Appendix B] The symbol p^{(1,D)}_t(r,s) in (4.71) and p^{(1,D)}_t(t,r,s) in the proof of Lemma 4.17 mix two notational conventions; please adopt one convention with the time variable explicit.
  3. [§4.2, Theorem 4.2(1)] The sentence "E^{(α)}_{d_ℓ,η}[[u]_{ℓ,m}] = I_{ζ,η}[u], u∈D(I_{ζ,η}), holds for all u∈L²(R₊, r^{d_ℓ−1}dr)" conflates the maximal domain with L²; please restate it so that the identity is claimed for u in the maximal domain, with both sides allowed to equal +∞.
  4. [§3.2, Theorem 3.7] This theorem is the decisive quantitative input, and it is cited to the arXiv preprint [BJM24] (arXiv:2409.02853); please indicate the publication status of [BJM24] or reproduce the needed statements, since the η<0 part of Theorem 4.2 depends on (4.71) and on (3.18).
  5. [§5, Eq. (1.19)] The match between (rs)^{−η}p^{(2)}_{ζ−η}(t,r,s), the claimed weight (1∧r/t^{1/2})^{−η}(1∧s/t^{1/2})^{−η}, and the lower bound (3.8b) would be easier to check with one additional line of algebra displaying the cancellation (rs)^{−η}(rs)^{−ζ+η}=(rs)^{−ζ}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Hardy-operator heat kernel bounds are obtained by proving a new form equality and then importing independent prior bounds for the Schrödinger-perturbed Bessel kernel.

full rationale

The derivation chain is: Theorem 3.7 (quoted from [BJM24]) gives sharp bounds for p^{(α)}_{ζ,η}; Theorem 4.2(2), proved here via Theorems 4.4 and 4.18, identifies the quadratic form E_{ζ,η} generated by p^{(α)}_{ζ,η} with the ground-state form I_{ζ,η} of the Hardy operator in channel ℓ; Corollary 4.3 then identifies the heat kernel of L_{Φ,ℓ} with p^{(α)}_{ζ,η}; Theorem 1.1 follows by combining these with Proposition 3.3. The identification is not assumed: for η>0 it is proved in Section 4.3 using form-core and density arguments and the invariance h under p_{ζ,η}; for η<0 it is proved in Section 4.4 using the new limit (4.72), whose proof uses only 3G inequalities for the unperturbed kernel p_ζ, namely [BM25, Theorem 3.1] and [BJM24, Lemma 2.5], not bounds for p_{ζ,η} or for L_{κ,ℓ}. Those cited prior results are self-contained theorems about the unperturbed Bessel kernels and about the perturbed kernel p_{ζ,η}; they do not presuppose Theorem 1.1, so importing them is independent evidence rather than circularity. The constants in the bounds are not fitted, and η is a parameterization of κ, not a fitted value. Two caveats belong to verification burden, not circularity: the decisive estimate (4.71) relies on prior 3G inequalities, and the statement of Theorem 4.2(2) overclaims the η<0 range because the proof of Theorem 4.18 is written only for ζ∈[0,∞); for Theorem 1.1 one has ζ=(d_ℓ−1)/2≥0, so this overclaim does not affect the main result. Overall, the paper's central claim is not equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard theorems (Hardy inequality, Dirichlet form theory, subordination) and on a chain of results from the authors' prior papers ([BM24], [BM25], [BJM24]); these are cited explicitly and are not re-derived here except where noted. No free parameters or invented entities appear.

assumptions (7)
  • standard math Sharp Hardy inequality (1.1) with optimal constant κ_c^{(α)}(d) as in (1.2).
    Invoked in Section 1 to define the Hardy operator and guarantee non-negativity; cited to Hardy, Kato, Herbst, Yafaev, Frank-Lieb-Seiringer, and Frank-Seiringer.
  • standard math Ground state representation (4.7): E_ζ[u] = I_{ζ,η}[u] + Ψ_ζ(η)∫ r^{2ζ−α}|u|^2 dr.
    Used to connect the Hardy operator form to the integral form I_{ζ,η}; proved in [BM24] for η≥0 and in Lemma 4.12 for η<0 using Lemma 4.13 from [BJM24].
  • standard math Sharp heat kernel bounds for subordinated Bessel kernels (Proposition 3.3) and the 3G inequalities of [BM25] and [BJM24].
    Used in Lemmas 4.15 and 4.16 to estimate the first Duhamel term and thus prove the equality of forms for η<0; the statements are quoted without proof.
  • standard math Lévy kernel representation (4.1) and the bound p^{(α)}_ζ(t,r,s)/t ≲ ν_ζ(r,s) (4.2).
    Established in [BM24, Proposition 2.4]; used to compute the limiting forms in (4.72) and (4.78).
  • standard math Regularity and form-core property of the Bessel Dirichlet form (E_ζ, D(E_ζ)) (Lemma 4.10).
    Proved in the paper by adapting Schilling-Uemura; needed to show C_c^∞(R_+) is a core for I_{ζ,η}.
  • standard math Unitary equivalence for α=2: L_{Φ(η),ℓ} is unitarily equivalent to the Bessel operator L_{(dℓ−1)/2−η}; heat kernel identity (5.1).
    Cited to [MNS18, Theorem 4.12, Proposition 4.14] and used to obtain (1.19).
  • standard math Subordination identity (3.2) and properties of the subordinator σ^{(α/2)}_t.
    Standard Bernstein function theory, cited to [SSV12] and [BM24, Appendix B]; defines p^{(α)}_ζ.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels." pith.science (2026). https://pith.science/paper/A3MLWWDI

@misc{pith2026250608115,
  author       = {Pith},
  title        = {Pith review of: Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A3MLWWDI}},
  note         = {Machine review of arXiv:2506.08115}
}
abstract

Motivated by the study of relativistic atoms, we prove sharp heat kernel bounds for the Hardy operator $(-\Delta)^{\alpha/2}-\kappa|x|^{-\alpha}$ acting on functions of the form $u(|x|) |x|^{\ell} Y_{\ell,m}(x/|x|)$ in $L^2(\R^d)$, when $\alpha\in(0,2]\cap(0,d+2\ell)$.

Figures

Figures reproduced from arXiv: 2506.08115 by the authors.

Figure 1
Figure 1. Plot of T1/3(x). Lemma 4.9 ([SU12, Lemma 2.3]). For ε > 0, the truncation Tε(x) in (4.51) is a normal contraction, i.e., |Tε(x)| ≤ |x| and |Tε(x) − Tε(y)| ≤ |x − y|, x, y ∈ R [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

44 extracted references · 36 canonical work pages

  1. [1]

    Bahouri, J.-Y

    H. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations , volume 343 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, Heidelberg, 2011

    work page 2011

  2. [2]

    T. A. Bui and P. D'Ancona. Generalized H ardy operators. Nonlinearity , 36(1):171--198, 2023

    work page 2023

  3. [3]

    Bruneau, J

    L. Bruneau, J. Derezi\' n ski, and V. Georgescu. Homogeneous S chr\" o dinger operators on half-line. Ann. Henri Poincar\' e , 12(3):547--590, 2011

    work page 2011

  4. [4]

    Bogdan, T

    K. Bogdan, T. Grzywny, T. Jakubowski, and D. Pilarczyk. Fractional L aplacian with H ardy potential. Comm. Partial Differential Equations , 44(1):20--50, 2019

    work page 2019

  5. [5]

    J. J. Betancor, E. Harboure, A. Nowak, and B. Viviani. Mapping properties of fundamental operators in harmonic analysis related to B essel operators. Studia Math. , 197(2):101--140, 2010

    work page 2010

  6. [6]

    Bogdan , T

    K. Bogdan , T. Jakubowski , and K. Merz . Hardy perturbations of subordinated B essel heat kernels. arXiv e-prints , page arXiv:2409.02853, September 2024

    arXiv 2024

  7. [7]

    T. A. Bui and K. Merz . Equivalence of S obolev norms in L ebesgue spaces for H ardy operators in a half-space. arXiv e-prints , page arXiv:2309.02928, September 2023

    work page Pith review arXiv 2023

  8. [8]

    Bogdan and K

    K. Bogdan and K. Merz. Ground state representation for the fractional L aplacian with H ardy potential in angular momentum channels. J. Math. Pures Appl. (9) , 186:176--204, 2024

    work page 2024

Show all 44 references
  1. [9]

    Bogdan and K

    K. Bogdan and K. Merz . Subordinated B essel heat kernels. Theory Probab. Math. Statist. , 112:67--83, May 2025

    2025

  2. [10]

    A. N. Borodin and P. Salminen. Handbook of B rownian Motion --- Facts and Formulae . Probability and its Applications. Birkh\" a user Verlag, Basel, second edition, 2002

    2002

  3. [11]

    S. Cho, P. Kim, R. Song, and Z. Vondra c ek. Factorization and estimates of D irichlet heat kernels for non-local operators with critical killings. J. Math. Pures Appl. (9) , 143:208--256, 2020

    2020

  4. [12]

    Derezi\' n ski and V

    J. Derezi\' n ski and V. Georgescu. On the domains of B essel operators. Ann. Henri Poincar\' e , 22(10):3291--3309, 2021

    2021

  5. [13]

    https://dlmf.nist.gov/, Release 1.1.9 of 2023-03-15, 2023

    NIST Digital Library of Mathematical Functions . https://dlmf.nist.gov/, Release 1.1.9 of 2023-03-15, 2023. F. W. J. Olver, A. B. Olde Daalhuis , D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

    2023

  6. [14]

    R. L. Frank, E. H. Lieb, and R. Seiringer. Hardy- L ieb- T hirring inequalities for fractional S chr\"odinger operators. J. Amer. Math. Soc. , 21(4):925--950, 2008

    2008

  7. [15]

    R. L. Frank and K. Merz. On S obolev norms involving H ardy operators in a half-space. J. Funct. Anal. , 285(10):Paper No. 110104, 2023

    2023

  8. [16]

    R. L. Frank and K. Merz. Upper bounds for the density of relativistic atoms. In preparation , 2025

    2025

  9. [17]

    R. L. Frank, K. Merz, and H. Siedentop. Equivalence of S obolev norms involving generalized H ardy operators. International Mathematics Research Notices , 2021(3):2284--2303, February 2021

    2021

  10. [18]

    R. L. Frank, K. Merz, and H. Siedentop. The S cott conjecture for large C oulomb systems: a review. Lett. Math. Phys. , 113(1):Paper No. 11, 79, 2023

    2023

  11. [19]

    R. L. Frank, K. Merz, and H. Siedentop. Relativistic strong S cott conjecture: a short proof. In Density functionals for many-particle systems---mathematical theory and physical applications of effective equations , volume 41 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Si...

    2023

  12. [20]

    R. L. Frank , K. Merz , H. Siedentop , and B. Simon . Proof of the strong S cott conjecture for C handrasekhar atoms. Pure Appl. Funct. Anal. , 5(6):1319--1356, December 2020

    2020

  13. [21]

    Fukushima, Y

    M. Fukushima, Y. Oshima, and M. Takeda. Dirichlet Forms and Symmetric M arkov Processes , volume 19 of De Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, extended edition, 2011

    2011

  14. [22]

    R. L. Frank and R. Seiringer. Non-linear ground state representations and sharp H ardy inequalities. J. Funct. Anal. , 255(12):3407--3430, 2008

    2008

  15. [23]

    G. H. Hardy. Notes on some points in the integral calculus LI : On H ilbert's double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals. Messenger of Mathematics , 48:107--112, 1919

    1919

  16. [24]

    G. H. Hardy. Note on a theorem of H ilbert. Mathematische Zeitschrift , 6(3--4):314--317, 1920

    1920

  17. [25]

    I. W. Herbst. Spectral theory of the operator (p^2+m^2)^ 1/2 - Ze^2/r . Comm.\ Math.\ Phys. , 53:285--294, 1977

    1977

  18. [26]

    Jakubowski and J

    T. Jakubowski and J. Wang. Heat kernel estimates of fractional S chr\" o dinger operators with negative H ardy potential. Potential Anal. , 53(3):997--1024, 2020

    2020

  19. [27]

    H. Kalf. On the expansion of a function in terms of spherical harmonics in arbitrary dimensions. Bull. Belg. Math. Soc. Simon Stevin , 2(4):361--380, 1995

    1995

  20. [28]

    T. Kato. Perturbation Theory for Linear Operators , volume 132 of Grundlehren der mathematischen W issenschaften . Springer-Verlag, Berlin, 1 edition, 1966

    1966

  21. [29]

    Killip, C

    R. Killip, C. Miao, M. Visan, J. Zhang, and J. Zheng. Sobolev spaces adapted to the S chr\"odinger operator with inverse-square potential. Math. Z. , 288(3-4):1273--1298, 2018

    2018

  22. [30]

    V. F. Kovalenko, M. A. Perelmuter, and Y. A. Semenov. Schr\"odinger operators with L _w^ l/2 ( R ^l) -potentials. J. Math. Phys. , 22:1033--1044, 1981

    1981

  23. [31]

    R. T. Lewis, H. Siedentop, and S. Vugalter. The essential spectrum of relativistic multi-particle operators. Annales de l'Institut Henri Poincar \'e , 67(1):1--28, 1997

    1997

  24. [32]

    Le Yaouanc, L

    A. Le Yaouanc, L. Oliver, and J.-C. Raynal. The H amiltonian (p^2+m^2)^ 1/2 - /r near the critical value _c=2/ . J. Math. Phys. , 38(8):3997--4012, 1997

    1997

  25. [33]

    K. Merz. On scales of S obolev spaces associated to generalized H ardy operators. Math. Z. , 299(1):101--121, 2021

    2021

  26. [34]

    o dinger-type Operators . Habilitation thesis, Technische Universit \

    K. Merz. On Eigenvalue and Functional Inequalities for S chr \"o dinger-type Operators . Habilitation thesis, Technische Universit \"a t Carolo-Wilhelmina zu Braunschweig, 2024. x+316 pages

    2024

  27. [35]

    Metafune, L

    G. Metafune, L. Negro, and C. Spina. Sharp kernel estimates for elliptic operators with second-order discontinuous coefficients. J. Evol. Equ. , 18(2):467--514, 2018

    2018

  28. [36]

    Merz and H

    K. Merz and H. Siedentop. Proof of the strong Scott conjecture for heavy atoms: the Furry picture. Annales Henri Lebesgue , 5:611--642, 2022

    2022

  29. [37]

    Ma ecki, G

    J. Ma ecki, G. Serafin, and T. \.Z \'o rawik. Fourier- B essel heat kernel estimates. J. Math. Anal. Appl. , 439(1):91--102, 2016

    2016

  30. [38]

    K. A. Penson and K. G\'orska. Exact and explicit probability densities for one-sided L \'evy stable distributions. Phys. Rev. Lett. , 105:210604, Nov 2010

    2010

  31. [39]

    H. Pollard. The representation of e^ -x^ as a L aplace integral. Bull. Amer. Math. Soc. , 52:908--910, 1946

    1946

  32. [40]

    Revuz and M

    D. Revuz and M. Yor. Continuous Martingales and B rownian Motion , volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, third edition, 1999

    1999

  33. [41]

    R. L. Schilling. Measures, Integrals and Martingales . Cambridge University Press, Cambridge, second edition, 2017

    2017

  34. [42]

    R. L. Schilling, R. Song, and Z. Vondra c ek. Bernstein Functions , volume 37 of De Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, second edition, 2012. Theory and Applications

    2012

  35. [43]

    R. L. Schilling and T. Uemura. On the structure of the domain of a symmetric jump-type D irichlet form. Publ. Res. Inst. Math. Sci. , 48(1):1--20, 2012

    2012

  36. [44]

    D. Yafaev. Sharp constants in the H ardy- R ellich inequalities. Journ. Functional Analysis , 168(1):121--144, October 1999

    1999

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.