Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Boundary value problems and Hardy spaces for singular Schr\"odinger equations with block structure

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read The paper proves that for singular Schrödinger equations with block-structured coefficients and reverse-Hölder potentials, the Dirichlet, Regularity, and Neumann boundary value problems are well-posed for boundary data in extrapolated…

desk verdict A serious and likely correct extension of the Auscher–Egert extrapolation machinery to singular Schrödinger equations with block-structured coefficients; the main caveat is a load-bearing imported Fefferman–Phong inequality whose proof is only sketched. read the letter →

arxiv 2411.17563 v2 pith:QNTDIHVT submitted 2024-11-26 math.AP

Arnaud Dumont , Andrew J. Morris This is my paper · ORCID
classification math.AP MSC 35J2535J1042B3747D0647A60
keywords boundaryvalueproblemssingularSchrödingerequationsreverseHölderpotentialsRiesztransformsHardyspacesKatosquarerootextrapolationofL^psolvabilityblockstructurecoefficients
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 aims to show that for singular Schrödinger equations $-\operatorname{div}(A\nabla u)+aVu=0$ in the upper half-space, with coefficients that are independent of the vertical direction, with $A$ of block-diagonal form and with $V$ in a reverse-Hölder class, the Dirichlet, Regularity, and Neumann boundary value problems are well-posed for boundary data in $L^p$ and in potential-adapted Hardy spaces. The known input is $L^2$-solvability of the Regularity and Neumann problems; the paper extrapolates this to explicit intervals of $p$ around $2$ determined by four critical numbers attached to the operator, and it proves the Riesz transform bounds that make the extrapolation work. It also identifies the abstract operator-adapted Hardy spaces with concrete spaces such as $L^p$, the potential-gradient Sobolev space $\dot V^{1,p}$, and the Hardy space $H^1_V$ adapted to $-\Delta+V$, allowing endpoint boundary data at $p=1$. If the results are correct, singular potentials and complex non-symmetric coefficients no longer stop at $L^2$: solutions exist, are unique, and their nontangential maximal function and square function are comparable to the boundary data norm in the stated ranges.

What carries the argument

The main engine is a chain of identifications: the operator-adapted Hardy spaces $H^p_H(\mathbb{R}^n)$, defined through the holomorphic functional calculus of $H$, are shown to coincide, with equivalent quasinorms, with potential-adapted spaces—$bH^p_{V,\mathrm{pre}}$, $\dot V^{1,p}\cap L^2$, or $\dot H^{1,p}_{V,\mathrm{pre}}$—on intervals determined by four critical numbers $p_\pm(H)$, $q_\pm(H)$ and by the reverse-Hölder exponent $q$. Those critical numbers record where the resolvent families $\{A_{\perp\perp}(1+t^2H)^{-1}A_{\perp\perp}^{-1}\}$ and $\{t\nabla_\mu(1+t^2H)^{-1}A_{\perp\perp}^{-1}\}$ are uniformly bounded on $H^p_V$. The proof is powered by $L^p$ Riesz transform bounds, a new cancellation bound for averages of $t\nabla_\mu(1+t^2H)^{-k}(1)$ on balls, the improved Fefferman–Phong inequality, and a Calderón–Zygmund–Sobolev decomposition adapted to $V$.

What would settle it

Find $n\geq 3$, $p\in[1,n]$, and $V\in\mathrm{RH}^q$ with $q\geq\max\{n/2,2\}$ for which the improved Fefferman–Phong inequality fails on a sequence of cubes, for instance by computing it for $V(x)=|x|^{-\alpha}$ at the critical range of $\alpha$; since the paper itself notes density of $C_c^\infty$ in $\dot V^{1,p}$ can fail when $V^{p/2}\in L^1_{\mathrm{loc}}$ but $V\notin\mathrm{RH}^{n/2}$, a failure inside $\mathrm{RH}^q$ would refute the density results and with them the main theorems.

Watch

Extended reading notes

Core claim

The central discovery is that the $L^2$ theory for singular Schrödinger operators with block-structure coefficients can be extrapolated to all $p$ in intervals governed by critical numbers $p_\pm(H)$ and $q_\pm(H)$, provided the operator-adapted Hardy spaces $H^p_H$ are identified with concrete potential-adapted spaces. Concretely, for $n\geq 3$ and $V\in\mathrm{RH}^q$ with $q\geq\max\{n/2,2\}$, the Dirichlet problem is well-posed for $p\in[1,p_+(H)_*)\cap(p_-(H),\infty)$, the Regularity problem for $p\in(p_-(H)_*,q_+(H))\cap(n/(n+1),2q]$, and the Neumann problem for $p\in(p_-(H),q_+(H))\cap[1,2q]$. In each case the appropriate norm of the solution is comparable to the norm of the boundary data, with endpoint data allowed in the Hardy space $H^1_V$ and in adapted Hardy–Sobolev spaces. The identification of abstract operator-adapted Hardy spaces with these concrete spaces, and the Riesz transform bounds that feed it, carry the proof.

Load-bearing premise

The argument leans on an improved Fefferman–Phong inequality—a weighted Poincaré-type bound relating variation over a cube to gradient and potential-weighted gradient—that is imported from an earlier paper with only a sketch of the proof; if that estimate fails for some admissible reverse-Hölder potential, the smooth-function density in the adapted Sobolev space and the decomposition step that the extrapolation depends on would collapse.

Editorial extensions

If this is right

  • The $L^2$ solvability of Regularity and Neumann problems, and the existing theory for $V\equiv 0$, are extended to $L^p$ solvability of all three boundary value problems in explicit extrapolation intervals around $p=2$.
  • For $p\leq 1$, boundary data can live in the potential-adapted Hardy space $H^1_V$ and in adapted Hardy–Sobolev spaces, giving endpoint solvability at $p=1$ rather than stopping at $L^p$ with $p>1$.
  • Solutions satisfy comparability of nontangential maximal function, conical square function, and boundary data norm in the stated ranges; the reverse bounds at $p=1$ require $V\in\mathrm{RH}^\infty$ in some cases.
  • The Riesz transform $\nabla_\mu H^{-1/2}$ is $L^p$-bounded for $p\in(p_-(H),q_+(H))\cap(1,2q]$, a result needed for the Hardy-space identifications and of independent interest.
  • The method gives a new self-contained proof even in the case $V\equiv 0$, no longer relying on two earlier technical results while retaining the same ranges.

Reading between the lines

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

  • Beyond the paper: the upper half-space is the standard prototype for Lipschitz graph domains, so the same extrapolation intervals should transfer to the region above a Lipschitz graph whenever the $L^2$ theory transfers by perturbation; the paper does not carry out that step.
  • Beyond the paper: the new cancellation bound controlling averages of $t\nabla_\mu(1+t^2H)^{-k}(1)$ may be the right tool for Riesz transform $L^p$ bounds for other operators whose semigroups lack the conservation property, such as magnetic or degenerate Schrödinger operators, since that was the main obstruction to extrapolating above $p=2$.
  • Beyond the paper: the special set of exponents $I(V)$ where reverse Riesz bounds hold at $p\leq 1$ is shown to include potentials of the form $|P|^a$; if the reverse bound were proved for all $V\in\mathrm{RH}^q$, the Hardy–Sobolev identifications and the $p\leq 1$ Regularity endpoint would hold in the full stated range without the extra condition.
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

2 major / 5 minor

Summary. This paper studies boundary value problems for the singular Schrödinger equation −div(A∇u)+aVu=0 in the upper half-space R^{1+n}_+, with t-independent complex coefficients (A,a) of block structure and non-negative potential V∈RH^q(R^n), n≥3, q≥max{n/2,2}. Building on the L^2 Kato-type estimates of Morris–Turner [51] and the extrapolation framework of Auscher–Egert [11], the authors prove L^p Riesz transform bounds for H=−b div(A‖‖∇)+baV (Theorem 1.1, p∈(p_-(H),q_+(H))∩(1,2q]), identify the adapted Hardy spaces H^p_H and H^{1,p}_H with Dziubański–Zienkiewicz spaces and adapted Sobolev spaces (Theorem 6.1), and state well-posedness results for the Dirichlet, Regularity and Neumann problems (Theorems 1.2–1.4) for L^p data and for H^1_V or \dot H^{1,p}_V data at p≤1, with comparability of nontangential maximal and square functions. The visible portion develops the Hardy-space theory in detail: maximal and square-function characterisations, atomic and molecular decompositions, interpolation and duality, the critical numbers p±(H), q±(H), the Riesz transform proof, and the identification theorem.

Significance. If correct, the main results constitute a substantial advance: L^p (and Hardy-space endpoint) solvability for singular Schrödinger equations with block-structured complex coefficients, previously known only at L^2 for the Regularity and Neumann problems. The paper delivers a genuinely new set of tools rather than a routine transcription of [11]: the square function characterisation of H^p_{V,pre} in the full range p>n/(n+1) (Theorem 3.14), the molecule class with unrestricted cube sizes (Definition 3.11 and Theorem 3.12), the conservation-property substitute (Lemma 5.2) and the new off-diagonal and cancellation bounds (Lemma 5.5 and (5.12)) used for the Riesz transform theorem, and the enriched Calderón–Zygmund–Sobolev decomposition (Lemma 6.2, properties (i)–(ix)). The proofs in Sections 2–6 are detailed and internally consistent, with explicit dependence of constants on n, q and the RH constant of V, and no fitted parameters; the L^2 anchor is the external parameter-free Kato estimate of [51]. The paper also carefully documents where the theory breaks (for example, the density failure for merely L^1_loc potentials in §2.3) and corrects an inaccuracy in [47, Lemma 4.4].

major comments (2)
  1. [Section 2.3 (Proposition 2.8)] Proposition 2.8 is the Fefferman–Phong engine of the extrapolation theory, but its proof is a single sentence that refers to the argument of [51, Proposition 2.3] to extend [9, Lemma 2.1] (Section 2.3). The statement is parameter-dependent: it asserts a uniform β∈(0,1) and constants uniform in the cube Q and in the RH constant for every p∈[1,∞) under V^{p/2}∈RH^q, and it is used at three structurally essential places: (2.13) in the density of C_c^∞ in \dot V^{1,n} (Proposition 2.9), the type-1/type-2 inequalities (6.2) and (6.7) together with property (viii) of the Calderón–Zygmund–Sobolev decomposition (Lemma 6.2), and the cancellation bound (5.14) inside the proof of Theorem 1.1. Since Section 2.3 itself shows that the density conclusion fails for merely V^{p/2}∈L^1_loc, the boundary between validity and failure of the inequality is exactly where the extrapolation theory operates, and the m_β scaling in the critical case p=n with V∈RH^{n/2+ε}, together with the uniformity of the constants, cannot be checked by the reader. The proof should be supplied in full, or the statement should be replaced by a citation of a published result with exactly this parameter dependence; I am raising this as a completeness issue, not claiming that the inequality is false.
  2. [Sections 7–11 (proofs of Theorems 1.2–1.4)] The main well-posedness theorems (Theorems 1.2–1.4) are proved in Sections 9–11, according to the roadmap at the end of Section 1 (Theorem 9.1, Theorems 9.2–9.3, Theorem 10.1, Theorem 10.2, and Section 11 with Remark 11.2). The text made available for review ends in the middle of the proof of Proposition 6.12 in Section 6.5, so Sections 7–11, including the solvability, uniqueness and Neumann arguments, could not be examined. I verified the machinery in Sections 2–6 (up to the cut-off point) and found it internally consistent, but the decisive pieces for the paper's central claims are exactly the missing parts, and no alternative route avoiding them can be checked from the visible text. The editor should ensure that the complete version, with Sections 7–11, is provided in any further round of review, and my recommendation must be read as conditional on that material.
minor comments (5)
  1. [Section 1 and Theorem 1.1] In Theorem 1.1 the letter q denotes both the reverse-Hölder exponent and the right critical number q_+(H), as in p∈(p_-(H),q_+(H))∩(1,2q]; the paper acknowledges this clash in Section 1, but readers and any later citation of the theorem would benefit from renaming one of the two, for example q_H for the critical number.
  2. [Section 1 and Theorem 1.4] The introduction and abstract present H^1_V boundary data as a general feature at p=1, but the Neumann problem at p=1 is solved in Theorem 1.4 only under the additional assumption 1∈I(V), and the only class for which I(V) is proved non-empty is V∈RH^q∩S_α with q>n (Theorem 4.10 and Section 4.2.3). The theorem statements themselves are explicit about this conditionality, but a sentence in Section 1 should flag which endpoint conclusions are unconditional and which require I(V).
  3. [Section 5.2 (Step 4)] There are several small typographical lapses in the typesetting, such as 'Holder's inequality' instead of 'Hölder's inequality' in the paragraph following (5.14); a careful proofread would remove these.
  4. [Section 1 and Theorem 1.2] The Dirichlet problem (D)^H_p is stated with boundary data in A_{\perp\perp}^{-1}H^1_V(R^n) for p=1, whereas Theorem 1.2 states the result for data in H^1_V(R^n); since b=A_{\perp\perp}^{-1} is bounded with bounded inverse the two spaces are isomorphic, but the equivalence constants are not recorded.
  5. [Section 6.1 and Theorem 6.1] Theorem 6.1 assumes q>max{n/2,2}, while the main theorems assume q≥max{n/2,2}; the reduction via self-improvement of the reverse-Hölder class (Lemma 2.1(i)) is standard, but it would be helpful to state it explicitly at the point where Theorem 6.1 is applied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: imported L2/Kato estimates and operator bounds are external, and the Lp/BVP conclusions are not encoded in the definitions.

full rationale

The paper's derivation chain imports its L2 foundations (Kato square root estimates, bounded H-infinity calculus, L2 Regularity/Neumann solvability) from Morris-Turner [51] and Auscher-Ben Ali [9]. These are prior parameter-free theorems with stated assumptions that do not include the target Lp extrapolation; [51] is co-authored by one of the present authors, but the hard rule limits self-citation circularity to cases where the load-bearing argument reduces to a self-citation that is itself unverified, which is not the case here. The critical numbers p_+(H), p_-(H), q_+(H), q_-(H) are defined by the boundedness of resolvent families on the adapted Hardy spaces, not by the boundary value problems; the main theorems then prove BVP solvability on intervals expressed in terms of these operator-determined exponents, which is a genuine implication rather than a definitional equivalence. The set I(V) is explicitly flagged as a conditional hypothesis governing reverse Riesz bounds, and the paper states that it is not required for the unconditional existence part of the Regularity problem. Proposition 2.8 (Fefferman-Phong) is indeed proved only by a one-sentence reduction to [51, Prop 2.3] and [9, Lemma 2.1], and the paper itself notes that density of C_c^infinity in \dot V^{1,p} can fail without it; however this is a verifiability/assumption-strength concern, not circularity, because no equation used in the proof is shown to be identical to the statement being derived. The full proofs of Sections 9-11 are not in the reviewed excerpt, but absence of material is not evidence of circularity. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors is invoked to forbid alternatives, and no known result is merely relabelled. Consequence: the central claims have independent content beyond their inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the ellipticity constants lambda and Lambda in (1.4)-(1.5) are inputs, and the critical numbers p+-(H) and q+-(H) are defined by boundedness properties, not tuned to make the conclusion hold. The paper relies on standard harmonic analysis and PDE machinery, plus cited deep results from [9], [11], and [51]. No new physical entities are postulated.

assumptions (5)
  • domain assumption V is in RH^q(R^n) with q at least max(n/2,2) (abstract and equation (1.6))
    Defines the class of singular potentials; used for the critical radius function (2.3), Lemma 2.1, the Fefferman-Phong inequality, and the Kato estimates imported from [51].
  • standard math Improved Fefferman-Phong inequality (Proposition 2.8), sourced to [9, Lemma 2.1]
    Engine for density in the homogeneous adapted Sobolev space, the Calderon-Zygmund-Sobolev decomposition, and cancellation estimates; proof is only sketched by reference to [51] and [9].
  • standard math Kato square root estimate D(H^{1/2}) = V^{1,2} with equivalence (1.9), from [51, Corollary 3.21]
    Gives the L^2 Riesz transform bound and identification of the square root domain; central to the first-order approach and to Theorem 1.1.
  • domain assumption Block structure and t-independence of (A,a,V) (Section 1, equations (1.1)-(1.3))
    Reduces the boundary value problem to a boundary operator H = -b div(A|| grad) + baV and allows the Poisson semigroup construction; without t-independence the whole method collapses.
  • standard math Tent space theory and complex interpolation of tent spaces (Section 2.1.5, Lemma 3.20)
    Provides atomic decomposition and interpolation used to extrapolate Hardy-space boundedness and to transfer estimates between exponents.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Boundary value problems and Hardy spaces for singular Schr\"odinger equations with block structure." pith.science (2026). https://pith.science/paper/QNTDIHVT

@misc{pith2026241117563,
  author       = {Pith},
  title        = {Pith review of: Boundary value problems and Hardy spaces for singular Schr\"odinger equations with block structure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QNTDIHVT}},
  note         = {Machine review of arXiv:2411.17563}
}
abstract

We obtain Riesz transform bounds and characterise operator-adapted Hardy spaces to solve boundary value problems for singular Schr\"odinger equations $-\mathrm{div}(A\nabla u)+aVu=0$ in the upper half-space $\mathbb{R}^{1+n}_{+}$ with boundary dimension $n\geq 3$. The coefficients $(A,a,V)$ are assumed to be independent of the transversal direction to the boundary, and consist of a complex-elliptic pair $(A,a)$ that is bounded and measurable with a certain block structure, and a non-negative singular potential $V$ in the reverse H\"older class $\mathrm{RH}^{q}(\mathbb{R}^{n})$ for $q\geq \max\{\frac{n}{2},2\}$. This block structure is significant because it allows for coefficients that are not symmetric but for which $\mathrm{L}^{2}(\mathbb{R}^{n})$-solvability persists due to recently obtained Kato square root type estimates. We find extrapolation intervals for exponents $p$ around $2$ on which the Dirichlet problem is well-posed for boundary data in $\mathrm{L}^{p}(\mathbb{R}^{n})$, and the associated Regularity problem is well-posed for boundary data in Sobolev spaces $\dot{\mathcal{V}}^{1,p}(\mathbb{R}^{n})$ that are adapted to the potential $V$, when $p>1$. The well-posedness of these Dirichlet problems and related estimates then allow us to solve the corresponding Neumann problem with boundary data in $\mathrm{L}^{p}$. The results permit boundary data in the Dziuba\`{n}ski--Zienkiewicz Hardy space $\mathrm{H}^{1}_{V}(\mathbb{R}^{n})$ and adapted Hardy--Sobolev spaces $\dot{\mathrm{H}}^{1,p}_{V}(\mathbb{R}^{n})$ when $p\leq 1$. We also obtain comparability of square functions and nontangential maximal functions for the solutions with their boundary data.

Figures

Figures reproduced from arXiv: 2411.17563 by the authors.

Figure 1
Figure 1. Relations between various adapted Hardy spaces All spaces appearing on the diagram are intersections of some operator-adapted Hardy space with one of its dense subsets, for both the L2 norm and the corresponding Hardy space norm. Moreover, each arrow is a bijection which is bounded from above and below for the correponding p-(quasi)norms. We shall follow the construction outlined in [11, Section 8.6], starting with … view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. $\mathrm{L}^p$ bounds for parabolic Riesz transforms with rough coefficients: The case $1<p \leq 2$

    math.CA 2026-07 accept novelty 8.0 of 10

    The parabolic Riesz transform for non-autonomous divergence-form operators with bounded measurable coefficients is bounded on L^p for the maximal open range 1<p≤2, with sharpness in dimension n≥2.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · cited by 1 Pith paper

  1. [51]

    A. J. Morris and A. J. Turner. Solvability for non-smoot h Schr¨ odinger equations with singular potentials and square integrable data. J. Funct. Anal. , 288(1):Paper No. 110680, 95, 2025

    work page 2025

  2. [11]

    Auscher and M

    P. Auscher and M. Egert. Boundary Value Problems and Hardy Spaces for Elliptic Syste ms with Block Structure. Birkh¨ auser Cham, 2023

    work page 2023

  3. [1]

    A. Amenta. Tent spaces over metric measure spaces under d oubling and related assumptions. In Opera- tor Theory in Harmonic and Non-commutative Analysis , pages 1–29, Cham, 2014. Springer International Publishing

    work page 2014

  4. [2]

    A. Amenta. Interpolation and embeddings of weighted tent spaces. J. Fourier Anal. Appl. , 24:108–140, 2018

    work page 2018

  5. [3]

    Amenta and P

    A. Amenta and P. Auscher. Elliptic Boundary Value Problems with Fractional Regulari ty Data. CRM Mono- graph Series. American Mathematical Society, 2018

    work page 2018

  6. [4]

    Arendt and A

    W. Arendt and A. V. Bukhvalov. Integral representations of resolvents and semigroups. Forum Math. , 6(1):111–136, 1994

    work page 1994

  7. [5]

    P. Auscher. On L p estimates for square roots of second order elliptic operators on Rn. Publ. Mat., 48(1):159– 186, 2004

    work page 2004

  8. [6]

    P. Auscher. On Necessary and Sufficient Conditions for Lp-Estimates of Riesz Transforms Associated to Elliptic Operators on Rn and Related Estimates . Memoirs of the American Mathematical Society. American Mathematical Society, 2007

    work page 2007

Show all 62 references
  1. [7]

    P. Auscher. On the Calder´ on–Zygmund lemma for Sobolev functions, 2008

    2008

  2. [8]

    Auscher, A

    P. Auscher, A. Axelsson, and A. McIntosh. On a quadratic e stimate related to the Kato conjecture and boundary value problems. Contemp. Math. , 505:105–129, 2010

    2010

  3. [9]

    Auscher and B

    P. Auscher and B. Ben Ali. Maximal inequalities and Riesz transform estimates on Lp spaces for Schr¨ odinger operators with nonnegative potentials. Ann. Inst. Fourier (Grenoble) , 57(6):1975–2013, 2007

    1975

  4. [10]

    Auscher and M

    P. Auscher and M. Egert. On uniqueness results for Diric hlet problems of elliptic systems without de Giorgi–Nash–Moser regularity. Anal. PDE, 13(6):1605 – 1632, 2020

    2020

  5. [12]

    Auscher, S

    P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, and P. Tchamitchian. The solution of the Kato square root problem for second order elliptic operators on Rn. Ann. of Math. (2) , 156(2):633–654, 2002

    2002

  6. [13]

    Auscher, A

    P. Auscher, A. McIntosh, and A. Morris. Calder´ on reproducing formulas and applications to Hardy spaces. Rev. Mat. Iberoam, 31(3):865–900, 2015

    2015

  7. [14]

    Auscher, A

    P. Auscher, A. McIntosh, and M. Mourgoglou. On L2 solvability of BVPs for elliptic systems. J. Fourier Anal. Appl., 19(3):478–494, 2013

    2013

  8. [15]

    Auscher, A

    P. Auscher, A. McIntosh, and A. Nahmod. The square root p roblem of Kato in one dimension, and first order elliptic systems. Indiana Univ. Math. J. , 46(3):659–695, 1997

    1997

  9. [16]

    Auscher and M

    P. Auscher and M. Mourgoglou. Representation and uniqueness for boundary value elliptic problems via first order systems. Rev. Mat. Iberoam., 35(1):241–315, 2019

    2019

  10. [17]

    Auscher and S

    P. Auscher and S. Stahlhut. Functional calculus for firs t order systems of Dirac type and boundary value problems. M´ em. Soc. Math. Fr. (N.S.), 2016

    2016

  11. [18]

    Bongioanni, E

    B. Bongioanni, E. Harboure, and O. Salinas. Weighted in equalities for negative powers of Schr¨ odinger oper- ators. J. Math. Anal. Appl , 348(1):12–27, 2008

    2008

  12. [19]

    Bortz, S

    S. Bortz, S. Hofmann, J. L. Luna Garc´ ıa, S. Mayboroda, a nd B. Poggi. Critical perturbations for second order elliptic operators—Part I: Square function bounds fo r layer potentials. Anal. PDE, 15(5):1215–1286, 2022

    2022

  13. [20]

    Bortz, S

    S. Bortz, S. Hofmann, J. L. Luna Garc´ ıa, S. Mayboroda, a nd B. Poggi. Critical perturbations for second order elliptic operators—Part II: Non-tangential maximal function estimates. Arch. Ration. Mech. Anal. , 248(3), June 2024

    2024

  14. [21]

    T. Bui, J. Li, and F. Ly. T1 criteria for generalised Cald er´ on–Zygmund type operators on Hardy and BMO spaces associated to Schr¨ odinger operators and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 18(1):203–239, 2018

    2018

  15. [22]

    L. A. Caffarelli, E. B. Fabes, and C. E. Kenig. Completely singular elliptic-harmonic measures. Indiana Univ. Math. J. , 30(6):917–924, 1981

    1981

  16. [23]

    R. R. Coifman, Y. Meyer, and E. M. Stein. Some new functio n spaces and their applications to harmonic analysis. J. Funct. Anal. , 62(2):304–335, 1985

    1985

  17. [24]

    E. B. Davies. Heat Kernels and Spectral Theory . Cambridge University Press, 1990

    1990

  18. [25]

    Dekel, G

    S. Dekel, G. Kerkyacharian, G. Kyriazis, and P. Petrush ev. Hardy spaces associated with non-negative self-adjoint operators. Studia Math. , 239:17–54, 2017

    2017

  19. [26]

    Deny and J.-L

    J. Deny and J.-L. Lions. Les espaces du type de Beppo Levi . Ann. Inst. Fourier (Grenoble), 5:305–370, 1954

    1954

  20. [27]

    Dindoˇ s, S

    M. Dindoˇ s, S. Hofmann, and J. Pipher. Regularity and Neumann problems for operators with real coefficients satisfying Carleson conditions. J. Funct. Anal. , 285(6):Paper No. 110024, 32, 2023

    2023

  21. [28]

    X. T. Duong and J. Li. Hardy spaces associated to operato rs satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus. J. Funct. Anal. , 264(6):1409–1437, 2013

    2013

  22. [29]

    X. T. Duong and D. W. Robinson. Semigroup kernels, Poiss on bounds, and holomorphic functional calculus. J. Funct. Anal. , 142(1):89–128, 1996

    1996

  23. [30]

    Dziuba´ nski and J

    J. Dziuba´ nski and J. Zienkiewicz. H p spaces for Schr¨ odinger operators. In Fourier analysis and related topics (Bpolhkedlewo, 2000) , volume 56 of Banach Center Publ. , pages 45–53. Polish Acad. Sci. Inst. Math., Warsaw, 2002. BOUNDARY V ALUE PROBLEMS FOR SINGULAR SCHR ¨OD...

    2000

  24. [31]

    Dziuba´ nski, G

    J. Dziuba´ nski, G. Garrig´ os, T. Mart´ ınez, J. L. Torrea, and J. Zienkiewicz. BMO spaces related to Schr¨ odinger operators with potentials satisfying a reverse H¨ older inequality. Math. Z. , 249:329–356, 2005

    2005

  25. [32]

    Dziuba´ nski and J

    J. Dziuba´ nski and J. Zienkiewicz. H p spaces associated with Schr¨ odinger operators with potent ials from reverse H¨ older classes.Colloq. Math. , 98:5–38, 2003

    2003

  26. [33]

    M. Egert. On Kato’s conjecture and mixed boundary conditions . PhD thesis, Sierke Verlag, G¨ ottingen, 2015

    2015

  27. [34]

    Geng and Z

    J. Geng and Z. Xu. On the schr¨ odinger equations with b∞ potentials in the region above a lipschitz graph, 2024

    2024

  28. [35]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order . Classics in Mathe- matics. Springer Berlin Heidelberg, 2001

    2001

  29. [36]

    Goldberg

    D. Goldberg. A local version of real Hardy spaces. Duke Math. J. , 46(1):27 – 42, 1979

    1979

  30. [37]

    Gong and L

    R. Gong and L. Yan. Weighted Lp estimates for the area integral associated to self-adjoint operators. Manuscripta Math. , 144:25–49, 2014

    2014

  31. [38]

    M. Haase. The Functional Calculus for Sectorial Operators . Birkh¨ auser Basel, 2006

    2006

  32. [39]

    M. Haase. Lectures on Functional Calculus . 21st International Internet Seminar, 2018

    2018

  33. [40]

    Haj/suppress lasz and A

    P. Haj/suppress lasz and A. Ka/suppress lamajska. Polynomial asymptotics and approximation of Sobolev functions. Studia Math., 113(1):55–64, 1995

    1995

  34. [41]

    C. Heil. A Basis Theory Primer: Expanded Edition . Applied and Numerical Harmonic Analysis. Birkh¨ auser Boston, 2011

    2011

  35. [42]

    Hille and R

    E. Hille and R. S. Phillips. Functional Analysis and Semi-groups , volume 31 of American Mathematical Society: Colloquium publications . American Mathematical Society, 1996

    1996

  36. [43]

    Hofmann, C

    S. Hofmann, C. Kenig, S. Mayboroda, and J. Pipher. Squar e function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elli ptic operators. J. Amer. Math. Soc. , 28(2):483– 529, 2015

    2015

  37. [44]

    Hofmann, G

    S. Hofmann, G. Lu, D. Mitrea, M. Mitrea, and L. Yan. Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Amer. Math. Soc. , 214(1007):1–84, Nov. 2011

    2011

  38. [45]

    Kerkyacharian and P

    G. Kerkyacharian and P. Petrushev. Heat kernel based de composition of spaces of distributions in the framework of Dirichlet spaces. Trans. Amer. Math. Soc. , 367(1):121–189, 2015

    2015

  39. [46]

    K. Kurata. An estimate on the heat kernel of magnetic Schr¨ odinger operators and uniformly elliptic operators with non-negative potentials. J. Lond. Math. Soc. (2) , 62, 1999

    1999

  40. [47]

    T. Ma, P. R. Stinga, J. L. Torrea, and C. Zhang. Regularit y estimates in H¨ older spaces for Schr¨ odinger operators via a T 1 theorem. Ann. Mat. Pura Appl. (4) , 2014

    2014

  41. [48]

    Mayboroda and B

    S. Mayboroda and B. Poggi. Exponential decay estimates for fundamental solutions of Schr¨ odinger-type operators. Trans. Amer. Math. Soc. , 372(6):4313–4357, Sept. 2019

    2019

  42. [49]

    McIntosh

    A. McIntosh. Operators which have an H∞ calculus. Proc. Centre Math. Anal. Austral. Nat. Univ. , 14:210– 231, 1986

    1986

  43. [50]

    Meyer-Nieberg

    P. Meyer-Nieberg. Banach Lattices. Universitext (Berlin. Print). Springer Berlin Heidelber g, 1991

    1991

  44. [52]

    E.-M. Ouhabaz. Analysis of Heat Equations on Domains . Princeton University Press, 2009

    2009

  45. [53]

    W. Rudin. Functional Analysis. International series in Pure and Applied Mathematics. McG raw-Hill, 1991

    1991

  46. [54]

    Z. Shen. On the Neumann problem for Schr¨ odinger operators in Lipschitz domains. Indiana Univ. Math. J. , 43(1):143–176, 1994

    1994

  47. [55]

    Z. Shen. Lp estimates for Schr¨ odinger operators with certain potenti als. Ann. Inst. Fourier (Grenoble) , 45(2):513–546, 1995

    1995

  48. [56]

    Z. Shen. On the number of negative eigenvalues for a Schr ¨ odinger operator with magnetic field. Comm. Math. Phys. , 182(3):637–660, 1996

    1996

  49. [57]

    Song and L

    L. Song and L. Yan. A maximal function characterization for Hardy spaces associated to nonnegative self- adjoint operators satisfying Gaussian estimates. Adv. Math., 287:463–484, 2016

    2016

  50. [58]

    E. M. Stein. Singular Integrals and Differentiability Properties of Fun ctions (PMS-30). Princeton University Press, 1970

    1970

  51. [59]

    E. M. Stein. Harmonic Analysis (PMS-43): Real-Variable Methods, Ortho gonality, and Oscillatory Integrals. With the assistance of Timothy S.Murphy. Princeton Univers ity Press, 1993

    1993

  52. [60]

    X. Tao. The regularity problems with data in Hardy—Sobo lev spaces for singular Schr¨ odinger equation in Lipschitz domains. Potential Anal., 36:405–428, 2012

    2012

  53. [61]

    Tao and H

    X. Tao and H. Wang. On the Neumann problem for the Schr¨ od inger equations with singular potentials in Lipschitz domains. Canad. J. Math. , 56(3):655–672, 2004

    2004

  54. [62]

    D. Yang, D. Yang, and Y. Zhou. Localized Morrey–Campana to spaces on metric measure spaces and appli- cations to Schr¨ odinger operators.Nagoya Math. J. , 198:77–119, 2010. A. Dumont, School of Mathematics, University of Birmingham, E dgbaston, B15 2TT, UK Email address: axd46...

    2010

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.