{"id":"7b61da79-f6dc-4c24-9d58-47789a686b81","arxiv_id":"2411.17563","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For -div(A grad u)+aVu=0 with block-structured complex coefficients and reverse-Holder potentials, the Dirichlet, Regularity and Neumann problems are well-posed on p-intervals determined by critical numbers, with Hardy-space endpoint data for p less than or equal to 1.","lead":"This paper proves that boundary value problems for Schrodinger equations with rough, complex coefficients and singular potentials are uniquely solvable for boundary data in a whole range of L^p and adapted Hardy spaces, not just the previously known L^2 case. It is a substantial step in extending the modern operator-theoretic extrapolation theory to equations with singular lower-order terms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The extrapolation engine rests on Proposition 2.8, whose proof is only a sketch; if the improved Fefferman–Phong inequality fails in its stated scaling, density in \\dot V^{1,p} and the Calderón–Zygmund–Sobolev decomposition fail, and so do the identifications behind Theorems 1.2–1.4.","rationale":"The visible portion of the paper is coherent and shows a serious, careful adaptation of the Auscher–Egert machinery to singular potentials. The strategy is credible, and the paper gives credit where the argument is genuinely independent of [11]. However, the single most load-bearing point in the visible text is Proposition 2.8: it is imported with only a sketch, and it is used in the density of C_c^∞ in \\dot V^{1,p}, in the Calderón–Zygmund–Sobolev decomposition, and in the cancellation bound for the Riesz transform. The paper itself flags that the analogous density statement fails under weaker assumptions on V, so this is not a harmless technicality. The reader identified exactly this premise as the weakest assumption, and my read agrees: if Proposition 2.8 is correct with uniform constants, the identification of the adapted Hardy spaces in Theorem 6.1 has a solid foundation; if it is not, the extrapolation to L^p well-posedness collapses. The absence of Sections 9–11 means the final steps toward Theorems 1.2–1.4 remain unverified, which further supports keeping the conditional verdict rather than accepting unconditionally. I do not see evidence of circularity, data manipulation, or internal inconsistency beyond the already-noted reliance on Proposition 2.8 and the missing final sections.","tokens_in":89531,"tokens_out":29245,"duration_ms":260999,"concrete_test":"Independently write out the proof of Proposition 2.8 in the special case p = n, V ∈ RH^{n/2+ε}, following [9, Lemma 2.1] and [51, Proposition 2.3]. In particular, verify that the inequality (2.13) holds for the truncated functions f_N with constants uniform in R, and that the exponent of ℓ(Q) and the power of avg_Q V^{n/2} are exactly as stated. If the derived inequality is m_β(∫_Q V^{n/2}) ∫_Q |f|^n ≲ ℓ(Q)^n ∫_Q |∇_μ f|^n with a uniform β, the concern is settled; if the exponent of ℓ(Q) or the scaling of the m_β term differs, then the density argument in Proposition 2.9 and the Calderón–Zygmund decomposition in Lemma 6.2 lose their uniformity, and the identification in Theorem 6.1 needs repair before Theorems 1.2–1.4 can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.8 is the load-bearing premise of the paper. Its proof is one sentence ('We follow the arguments in the proof [51, Proposition 2.3] to extend [9, Lemma 2.1]'), but the statement is a parameter-dependent extension: it asserts a uniform β ∈ (0,1) such that m_β(ℓ(Q)^p avg_Q V^{p/2}) ∫_Q |f|^p ≲ ℓ(Q)^p ∫_Q |∇_μ f|^p for all cubes and all f ∈ V^{1,p}_{loc}, under only V^{p/2} ∈ RH^q. This inequality is used at three structurally essential places: Proposition 2.9 (density of C_c^∞ in \\dot V^{1,p} at p = n, via equation (2.13)), Lemma 6.2 (type-1 cube estimates (6.2) and (6.7) and property (viii) of the Calderón–Zygmund–Sobolev decomposition), and Theorem 1.1 (the Fefferman–Phong step in the cancellation bound (5.14)). The paper itself notes in §2.3 that mere V^{p/2} ∈ L^1_loc fails to make C_c^∞ dense in \\dot V^{1,p}, so the boundary between validity and failure of Proposition 2.8 is exactly where the extrapolation theory operates. Because the proof of Proposition 2.8 is not supplied, a reader cannot verify the m_β scaling in the critical case p = n, V ∈ RH^{n/2+ε}, nor the uniformity of the constants in the RH constant and in Q; if either fails, Proposition 2.9 and Lemma 6.2 no longer imply Theorem 6.1, and the L^p well-posedness in Theorems 1.2–1.4 lacks its identification step. The decisive proofs of Theorems 1.2–1.4 in Sections 9–11 are also not present in the reviewed excerpt, so no alternative route avoiding this dependency can be checked.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":89954,"tokens_out":29663,"duration_ms":238373,"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":[{"comment":"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.","section":"Section 2.3 (Proposition 2.8)"},{"comment":"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.","section":"Sections 7–11 (proofs of Theorems 1.2–1.4)"}],"minor_comments":[{"comment":"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.","section":"Section 1 and Theorem 1.1"},{"comment":"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).","section":"Section 1 and Theorem 1.4"},{"comment":"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.","section":"Section 5.2 (Step 4)"},{"comment":"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.","section":"Section 1 and Theorem 1.2"},{"comment":"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.","section":"Section 6.1 and Theorem 6.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope, and the citation pattern is appropriate: the novelty claims relative to [51] and [11] appear accurate for the visible portions, and the paper is careful about prior inaccuracies, for example the correction of [47, Lemma 4.4]. My main caveat for the editor is procedural: the copy I received stops inside Section 6, so the proofs of Theorems 1.2–1.4 (Sections 7–11) were not refereed; please send the full text in any further round of review. I would also ask the authors to expand the proof of Proposition 2.8 or to cite a statement that literally covers V^{p/2}∈RH^q with the stated m_β scaling; with those two items resolved, I would expect to be able to support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine advance, and the right move is to send it to a knowledgeable referee rather than desk reject. The visible core—Sections 2 through 6—is careful, detailed, and internally consistent. The new tools are real: molecules with arbitrary size cubes (Definition 3.11), the cancellation bound (5.12) for p>2 Riesz transforms, and the Hardy space identifications in Theorem 6.1. The extension of the V=0 theory of Auscher–Egert to non-symmetric block coefficients with a reverse-Hölder potential is exactly what the area needs, and the authors handle the loss of the conservation property (1+t^2H)^{-1}1=1 with an honest extra term.\n\nWhere I part company with a desk-reject instinct: the reliance on Morris–Turner for L^2 and Kato estimates is legitimate external benchmark. No fitted parameters, no circularity.\n\nThe real soft spot is Proposition 2.8, the improved Fefferman–Phong inequality imported from Auscher–Ben Ali. Its proof is one sentence, and the stress-test note is right that it is load-bearing: it drives density of C_c^∞ in \\dot V^{1,p} at p=n, the Calderón–Zygmund–Sobolev decomposition in Lemma 6.2, and the cancellation bound in Theorem 1.1. The paper itself admits density can fail for merely V^{p/2} in L^1_loc, so this is not cosmetic. However, this is a checkable technical bridge, not a conceptual flaw; standard practice in this literature is to import such inequalities, and the statement is plausible from [9, Lemma 2.1]. I would want the referee to verify the m_β scaling in the critical case p=n and uniformity in the RH constant before acceptance.\n\nSecond caveat: Sections 9–11, where Theorems 1.2–1.4 are proved, were not visible in the excerpt I reviewed. The outline there is clear and the auxiliary estimates in Section 8 look right, but the decisive extrapolation arguments need eyes.\n\nWho it is for: people working on boundary value problems, Riesz transforms, and Hardy spaces adapted to Schrödinger operators. It deserves a serious referee. My recommendation: send to review, with specific instruction to check Proposition 2.8 and Sections 9–11.","headline":"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.","tokens_in":90530,"tokens_out":2015,"would_cite":true,"duration_ms":21713,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J25","35J10","42B37","47D06","47A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["boundary value problems","singular Schrödinger equations","reverse Hölder potentials","Riesz transforms","Hardy spaces","Kato square root","extrapolation of L^p solvability","block structure coefficients"],"falsifier":"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.","tokens_in":89278,"feed_emoji":"📐","tokens_out":11375,"duration_ms":98056,"temperature":0.7,"pith_summary":"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.","feed_headline":"Singular Schrödinger boundary-value problems solved on L^p","feed_subtitle":"Dirichlet, Regularity and Neumann well-posedness for reverse-Hölder potentials extends from L^2 to an explicit range of p.","key_machinery":"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 $V$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the $L^2$ Regularity and Neumann solvability, the quadratic estimates, and the square-root domain identification $D(H^{1/2})=V^{1,2}$ that the paper extrapolates.","marker":"[51]"},{"why":"Provides the extrapolation machinery—critical numbers, operator-adapted Hardy spaces, and Calderón–Zygmund–Sobolev decomposition—that the paper adapts to singular potentials.","marker":"[11]"},{"why":"The Kato square root theorem for block-structure coefficients, which underpins $L^2$ solvability for non-symmetric $A$.","marker":"[12]"},{"why":"Gives the improved Fefferman–Phong inequality that powers density, decomposition, and cancellation bounds; the proof relies on this with only a sketch supplied.","marker":"[9]"},{"why":"Supplies reverse-Hölder properties, the critical radius function, and $L^p$ Riesz transform estimates for $-\\Delta+V$ used in the endpoint identifications.","marker":"[55]"},{"why":"Introduces the potential-adapted Hardy spaces $H^p_V$ with atomic decompositions used for boundary data at $p\\leq 1$.","marker":"[30]"},{"why":"Provides molecular and atomic boundedness criteria for $H^p_V$ and for the Riesz transform of $H_0$ that the identifications rely on.","marker":"[21]"}],"fun_headline_variants":["Block-structure Schrödinger BVPs: L^p extrapolation solved","Hardy-space method yields L^p well-posedness for singular equations","L^p extrapolation for Schrödinger BVPs with block coefficients","Singular Schrödinger BVPs: well-posedness from L^2 to Hardy spaces","Riesz bounds and Hardy spaces crack Schrödinger BVPs on L^p"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Block-structure Schrödinger BVPs: L^p extrapolation solved","Hardy-space method yields L^p well-posedness for singular equations","L^p extrapolation for Schrödinger BVPs with block coefficients","Singular Schrödinger BVPs: well-posedness from L^2 to Hardy spaces","Riesz bounds and Hardy spaces crack Schrödinger BVPs on L^p"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1517,"prompt_tokens":1194,"completion_tokens":323,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":810,"completion_tokens_details":{"reasoning_tokens":222}},"tokens_in":810,"tokens_out":323,"duration_ms":3814,"temperature":1.0,"reasoning_tokens":222,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:58:51.461072+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the $L^2$ Regularity and Neumann solvability, the quadratic estimates, and the square-root domain identification $D(H^{1/2})=V^{1,2}$ that the paper extrapolates."},{"cited_title":"Auscher and M","cited_arxiv_id":null,"evidence_quote":"Provides the extrapolation machinery—critical numbers, operator-adapted Hardy spaces, and Calderón–Zygmund–Sobolev decomposition—that the paper adapts to singular potentials."},{"cited_title":"Auscher, S","cited_arxiv_id":null,"evidence_quote":"The Kato square root theorem for block-structure coefficients, which underpins $L^2$ solvability for non-symmetric $A$."},{"cited_title":"Auscher and B","cited_arxiv_id":null,"evidence_quote":"Gives the improved Fefferman–Phong inequality that powers density, decomposition, and cancellation bounds; the proof relies on this with only a sketch supplied."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies reverse-Hölder properties, the critical radius function, and $L^p$ Riesz transform estimates for $-\\Delta+V$ used in the endpoint identifications."},{"cited_title":"Dziuba´ nski and J","cited_arxiv_id":null,"evidence_quote":"Introduces the potential-adapted Hardy spaces $H^p_V$ with atomic decompositions used for boundary data at $p\\leq 1$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides molecular and atomic boundedness criteria for $H^p_V$ and for the Riesz transform of $H_0$ that the identifications rely on."}],"review_version":1}