{"id":"a816a424-729f-45b9-a5f1-269fcae7527c","arxiv_id":"2608.12734","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Radial symmetry and existence of nonconstant periodic singular (Delaunay-type) solutions are established for the critical fractional Hartree equation with a Riesz convolution.","lead":"This paper proves two structural results for the critical fractional Hartree equation: every positive solution with a non-removable singularity at the origin is radially symmetric, and periodic Delaunay-type singular solutions exist for all sufficiently large periods. The interest is that this equation combines two nonlocal effects, the fractional Laplacian and a Riesz convolution, so the usual ODE reduction fails.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"L∞-regularity step for minimizers (Prop 4.5) rests on an unverified flattening of the curved cylinder that was only proved for the local Yamabe RHS; the Hartree term's fate under that coordinate change is not demonstrated.","rationale":"The reader's weakest_assumption identifies exactly the step I find most load-bearing: the reduction in Proposition 4.5 from the curved cylinder to the flat Euclidean problem. Theorem 1.2's conclusion that the periodic profile is a smooth, nonconstant solution depends on the L∞ bound for the minimizer; without it, the bootstrap to C^∞ in Lemma 5.1 has no starting point. The paper imports [18, Prop 3.4] without displaying the coordinate computation in the doubly nonlocal Hartree setting. This is a genuine gap in presentation and possibly in substance, because the Neumann condition now contains B(t), which is nonlocal in t and depends on the full period of v; the cited Yamabe result does not automatically cover such a term. I agree with the reader that this is the weakest assumption, and I do not see a way to certify it from the text alone. Other issues are less severe: the formula for c_L in (5.6) is algebraically wrong, but since the Rayleigh quotient is homogeneous and (5.7) is the value for any positive constant, the energy comparison in Proposition 5.4 is unaffected. The implicit restriction to nonnegative functions is a notational gap, not a mathematical one, because absolute values preserve the functional and the Euler–Lagrange equation. The rest of the variational argument, including the kernel estimates, the compact embedding, and the De Giorgi iteration after the flattening, is coherent and largely checkable. The conditional-accept verdict is therefore appropriate: the central claim is plausible and well supported except for this one unverified bridge, which should be made explicit and checked before the result can be considered fully established.","tokens_in":35856,"tokens_out":29531,"duration_ms":290263,"concrete_test":"Derive explicitly the local boundary-flattening change of variables for the metric g* in (3.8) near a point (t0,0), compute the transformed extension equation and Neumann condition for the Hartree term B(t)v^{p*-1}, and verify that all error terms are zeroth order (controlled by a bounded function times V or v) as asserted before (4.14). In particular, check whether any ∂_t v or ∂_t V term appears with a coefficient depending on the curvature of S^{n-1}; if such a term appears, test whether the De Giorgi energy estimate (4.17) still closes by re-running Claims 1-4 with that term present. A simpler sufficient check: reproduce the L∞ bound of Prop 4.5 working directly in normal coordinates on the cylinder without flattening, and confirm that the weighted trace inequality [30, Cor 5.3] applies with constants independent of the Hartree convolution.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.2) requires the minimizer from Lemma 5.1 to be smooth, and the only bridge from H^s_L to C^∞ is Proposition 4.5. That proposition hinges on the paragraph before (4.14), which asserts that after flattening the boundary of the cylinder (R×S^{n-1}×(0,ρ*),g*), the problem (4.10) becomes the Euclidean weighted problem (4.14), with curvature terms 'absorbed into lower-order perturbations' and a citation to [18, Prop 3.4]. The cited proposition was established for the fractional Yamabe equation, where the Neumann nonlinearity is the local function v^{p*}. Here the Neumann data is B(t)v^{p*-1} with B(t)=∫_0^L K^L_{-α/2}(t-τ)v(τ)^{p*}dτ, a nonlocal functional of the whole period. The authors do not give the flattened metric, the explicit lower-order terms, or the transformed boundary condition. If the coordinate change produces first-order tangential terms (for instance a(t)∂_t v) or introduces a ρ*-dependence into B, then the energy identity (4.15) and the estimates (4.18)-(4.19) are not justified, and the De Giorgi iteration collapses. No independent check of this reduction is supplied for the doubly nonlocal setting. Since every subsequent bootstrap (Hölder continuity, C^∞) uses the L∞ bound, this is the least secured step in the proof of Theorem 1.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the critical fractional Hartree equation (−Δ)^s u = (R_α ∗ u^{p*}) u^{p*−1} in R^n \\ {0}, a doubly nonlocal conformally invariant equation. The first main result (Theorem 1.1) asserts that every positive solution with a non-removable isolated singularity is radially symmetric; the proof combines the Caffarelli–Silvestre extension with the method of moving spheres. The second main result (Theorem 1.2) asserts the existence, for n≥2, s∈(0,1), α∈(α*,n), of nonconstant periodic-in-log solutions u(x)=|x|^{-(n−2s)/2} v(ln|x|) for all sufficiently large periods. The proof follows the variational framework of DelaTorre–del Pino–González–Wei for the fractional Yamabe problem: after the Emden–Fowler transformation the equation becomes a nonlocal periodic equation, which is attacked by minimizing a Rayleigh-type quotient in the periodic fractional Sobolev space H^s_L. The main technical step is an L∞ bound for minimizers via a De Giorgi truncation argument in the Caffarelli–Silvestre extension, followed by a bootstrap to smoothness and an energy comparison that shows the minimizer is nonconstant for large periods.","tokens_in":36156,"tokens_out":24843,"duration_ms":228178,"significance":"The results are significant if correct: Theorem 1.2 would provide the first Delaunay-type (periodic-in-log, singular) solutions for a critical equation with two independent nonlocalities, and Theorem 1.1 extends the radial symmetry theory of isolated singularities from the fractional Yamabe equation to the Hartree setting. The variational construction with a genuinely nonlocal Hartree energy, the analysis of the periodic kernel K^L_{−α/2}, and the De Giorgi-type regularity for a nonlocal nonlinearity go beyond a routine adaptation of [18]. The paper is generally well organized and the statements are precise. The main weakness is that the L∞-regularity step rests on an unverified local reduction of the curved extension problem to a Euclidean one; this is the load-bearing point for the smoothness of the minimizer and hence for Theorem 1.2.","major_comments":[{"comment":"The proof of the L∞ bound for minimizers, which is the essential bridge from H^s_L to C^∞ in Lemma 5.1, is not complete as written. The text asserts that after flattening the boundary the cylindrical problem (4.10) is 'locally equivalent' to the Euclidean weighted problem (4.14), with curvature terms 'absorbed into lower-order perturbations,' citing [18, Proposition 3.4]. That proposition was established for the fractional Yamabe equation, whose boundary nonlinearity is the local function v^{p*}. Here the nonlinearity is B(t)v^{p*−1} with B(t)=∫_0^L K^L_{−α/2}(t−τ)v(τ)^{p*}dτ, a nonlocal functional of the whole period. The paper does not specify the coordinate change, the transformed metric, the resulting boundary condition, or the precise form of the lower-order terms. If the flattening introduces t-dependent factors into B(t) or first-order tangential terms into the energy identity, then the identity (4.15) and the estimates (4.18)–(4.19) are not justified and the De Giorgi iteration collapses. Since every subsequent bootstrap (Hölder continuity, C^∞ regularity) uses the L∞ bound, this gap is load-bearing for Theorem 1.2. The authors should either prove the reduction in the Hartree setting with all terms tracked, or perform the De Giorgi iteration directly on the cylinder using the appropriate weighted trace inequality.","section":"Section 4.2, Proposition 4.5 (paragraph before Eq. (4.14))"},{"comment":"The application of Proposition 4.5 to the minimizer v_L is not fully justified: the paper does not verify the hypothesis (4.12), namely that ∫_0^L |v_L|^{2/(1−2s)} dt is finite. This is not a consequence of the compact embedding stated in Proposition 4.2, which covers only q < 2/(1−2s) when s ≤ 1/2. The missing fact is the continuous (non-compact) Sobolev embedding H^s(S^1) ⊂ L^{2/(1−2s)}(S^1) for s < 1/2, which does hold; the authors should state this explicitly and apply it to v_L before invoking Proposition 4.5. As written, the proof jumps from membership in H^s_L to the L∞ bound without checking the required integrability hypothesis.","section":"Section 5.1, Lemma 5.1 Claim 5"}],"minor_comments":[{"comment":"The displayed formula for c_L appears to be (c_{n,s} ∫_{−∞}^{∞} K_{−α/2}(τ) dτ)^{1/(2(p*−1))}, i.e., the product of c_{n,s} and the integral. Solving the constant equation c_{n,s} c = c^{2p*−1} ∫ K gives c = (c_{n,s}/∫K)^{1/(2(p*−1))}, so the printed formula should show a quotient. The proof text and the subsequent energy formula (5.7) are consistent with the quotient version, so this is a typographical/algebraic error that should be corrected.","section":"Section 5.2, Lemma 5.2, Eq. (5.6)"},{"comment":"The sentence 'Such a choice is possible under the assumption (4.12) on α' mis-references (4.12), which is the hypothesis ∫|v|^{2/(1−2s)} = ζ < ∞ and contains no condition on α. The non-emptiness of the interval (4.26) depends on the condition α > α* from (1.10) (or, for s ≥ 1/2, on Remark 4.7); the reference should be corrected.","section":"Section 4.2, Prop. 4.5 Claim 4"},{"comment":"In the proof of the gradient estimate (2.6), the case (x,t) ∈ Ω_2 with |y−x| ≥ |x|/2 is explicitly omitted 'for brevity' with a reference to the proof of (2.8). Since the W^{1,2} regularity of the extension is used in the moving spheres argument, this missing case should be supplied or the reference made precise; as written, the estimate is not fully verifiable.","section":"Section 2.2, Lemma 2.1 Claim 3"},{"comment":"The in-text citation 'Ma, Shang and Zhang [31]' does not match the reference list entry [31], which is by Guo, Hu, Peng and Shuai (Choquard equation). The authors should correct the citation and ensure that the intended Ma–Shang–Zhang paper is properly referenced.","section":"Section 1, references"},{"comment":"The statement of the compact embedding gives the range q ∈ (1, 2/(1−2s)) for s ≤ 1/2, but when s = 1/2 the denominator vanishes; the case s = 1/2 should be treated separately (e.g., q ∈ (1,∞)).","section":"Section 4.1, Proposition 4.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses a timely and relevant problem and the main results are likely correct, but the proof of the central regularity proposition needs substantial completion before the paper can be accepted. The reliance on [18, Proposition 3.4] in the doubly nonlocal setting is currently not justified; this is the main obstacle. I also note a citation mismatch for reference [31] and recommend the authors verify all references, including their own preprints [1] and [24]. The paper is within the journal's scope and the contributions are significant, so with a careful revision it could become a valuable addition to the literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends the Caffarelli–Gidas–Spruck/Delaunay picture to the doubly nonlocal critical Hartree equation, and if the proofs hold up, it is a genuine step forward. The two main theorems are new for 0 \\alpha_*$. A direct algebraic check over a grid of $(n,s)$ settles the sharpness of the range in Theorem 1.2: a nonempty interval below $\\alpha_*$ means the range is not sharp, an empty interval above $\\alpha_*$ means the $L^\\infty$ proof has a gap. A complementary numerical check is to evaluate $F_L$ on the constant profile versus the localized bump for representative parameters (e.g. $n=2$, $s=0.4$, $\\alpha=1$): the theorem requires $F_L(c_L) \\sim L^{1-1/p^*}$ to diverge while $F_L(\\tilde w_L)$ stays $O(1)$, and the crossing of the two curves pins down $T_0$.","tokens_in":35617,"feed_emoji":"🔁","tokens_out":29420,"duration_ms":243707,"temperature":0.7,"pith_summary":"The paper studies singular solutions of the critical fractional Hartree equation, a conformally invariant equation that is doubly nonlocal: the fractional Laplacian acts alongside a Riesz convolution potential. It establishes two results. Theorem 1.1 proves that every positive solution with a non-removable isolated singularity at the origin is radially symmetric, using the Caffarelli–Silvestre extension (a standard device replacing the nonlocal operator by a degenerate local problem in one extra dimension) and the method of moving spheres. Theorem 1.2 proves that for $\\alpha \\in (\\alpha_*, n)$, a range that for $s \\geq 1/2$ is just $0 < \\alpha < n$, the equation admits nonconstant Delaunay-type solutions $u(x) = |x|^{-(n-2s)/2} v(\\ln|x|)$ with $v$ periodic, for every period $T$ above a threshold $T_0$. The key point is that after the Emden–Fowler change of variables $r = e^t$ the Riesz convolution survives as a genuine nonlocal integral, so the periodic problem is not an ODE and no phase-plane analysis is available; the proof instead minimizes a Rayleigh-type quotient (the ratio of energy to Hartree interaction) in a periodic fractional Sobolev space and shows that the minimizer beats the constant solution for large periods.","feed_headline":"Log-periodic singular profiles exist for critical Hartree equation","feed_subtitle":"Even with the Riesz convolution term surviving the change of variables, periodic singular solutions still exist.","key_machinery":"The paper runs on two pieces of machinery. For Theorem 1.1, the load-bearing device is the Caffarelli–Silvestre extension, which turns the nonlocal equation into the degenerate elliptic problem $-\\mathrm{div}(t^{1-2s}\\nabla U) = 0$ in the upper half-space with the Neumann condition $\\partial U/\\partial \\nu_s = (R_\\alpha \\ast u^{p^*}) u^{p^*-1}$; the method of moving spheres applied to $U$ gives invariance under Kelvin inversions about arbitrary centers, hence radial symmetry. For Theorem 1.2, the central object is the Emden–Fowler transformation $r = e^t$, which rewrites the equation as $L_s v(t) = \\bigl(\\int_{-\\infty}^{+\\infty} K_{-\\alpha/2}(t-\\tau) v(\\tau)^{p^*} d\\tau\\bigr) v(t)^{p^*-1}$, where $L_s$ is the fractional operator with kernel $K_s$ and $K_{-\\alpha/2}$ is the kernel left by the Riesz potential on the sphere; this surviving convolution is exactly why the problem cannot be reduced to an ODE. On $T$-periodic functions the problem becomes the minimization of the Rayleigh-type quotient $F_L$ over the periodic fractional Sobolev space $H^s_L$, and the new technical tool is a De Giorgi truncation through the extension problem on the cylinder $\\mathbb{R} \\times S^{n-1} \\times (0, \\rho_*)$, flattened to the Euclidean weighted problem $-\\mathrm{div}(y^{1-2s}\\nabla V) = 0$, which yields the $L^\\infty$ bound on minimizers for $\\alpha > \\alpha_*$. A final energy comparison — the constant solution growing like $L^{1-1/p^*}$ versus a localized bump of energy $O(1)$ — makes the minimizer nonconstant for all large periods.","core_discovery":"The paper's central discovery is that the critical fractional Hartree equation $(-\\Delta)^s u = (R_\\alpha \\ast u^{p^*}) u^{p^*-1}$ in $\\mathbb{R}^n \\setminus \\{0\\}$, with Riesz kernel $R_\\alpha(x) = |x|^{\\alpha-n}$ and Hardy–Littlewood–Sobolev critical exponent $p^* = (n+\\alpha)/(n-2s)$, shares the two central qualitative features of the classical Yamabe equation. Theorem 1.1 states that every positive solution with a non-removable singularity is radially symmetric about the origin, under the natural regularity assumption $u \\in C^{1,1}_{\\mathrm{loc}}(\\mathbb{R}^n \\setminus \\{0\\}) \\cap L_s(\\mathbb{R}^n) \\cap L^1_{\\mathrm{loc}}(\\mathbb{R}^n)$. Theorem 1.2 states that for $n \\geq 2$, $s \\in (0,1)$ and $\\alpha \\in (\\alpha_*, n)$, with $\\alpha_* = \\max\\{0, (n(1-6s)+8s^2)/(2n-1-2s)\\}$, there is a threshold $T_0 < \\infty$ such that for every period $T \\geq T_0$ the equation admits a positive, nonconstant solution $u(x) = |x|^{-(n-2s)/2} v(\\ln|x|)$ with $v(t+T) = v(t) > 0$: a Delaunay-type profile. The mechanism is that the Hartree convolution survives the Emden–Fowler change of variables as a genuinely nonlocal integral term, so the reduced problem is an integro-differential equation rather than an ODE; the authors prove existence by minimizing a Rayleigh-type quotient over a periodic fractional Sobolev space, establish $L^\\infty$ regularity via a De Giorgi truncation, and force nonconstancy for large periods by an energy comparison.","pith_inferences":["The threshold $\\alpha_*$ likely marks a genuine integrability boundary rather than an artifact of the proof: below it the periodized kernel $K^L_{-\\alpha/2}$ no longer satisfies the convolution estimates that close the De Giorgi iteration, and one would expect the variational functional to lose coercivity; a testable conjecture is that no positive singular solutions exist for $0 < s < 1/2$ with $\\","The same recipe — Emden–Fowler reduction, a surviving convolution kernel, a Rayleigh-type quotient, De Giorgi truncation — should port to other doubly nonlocal critical problems (for instance Choquard-type equations with exponential nonlinearities in the plane), where comparable regularity theories are absent.","The energy comparison identifies $T_0$ implicitly as the period at which $F_L(c_L) \\sim L^{1-1/p^*}$ overtakes the $O(1)$ energy of the localized profile; should uniqueness of minimizers ever be proved, the equation $c(L) = F_L(c_L)$ would characterize the critical period explicitly."],"forward_implications":["Every positive solution with a non-removable isolated singularity is radially symmetric (Theorem 1.1), so any future classification of singular profiles can restrict to radial solutions.","For $\\alpha \\in (\\alpha_*, n)$ — which for $s \\geq 1/2$ is simply $0 < \\alpha < n$ — and every period $T \\geq T_0$, the equation admits a smooth positive Delaunay-type solution satisfying the self-similar invariance $u(e^T x) = e^{-(n-2s)T/2}u(x)$.","The minimizers are proven smooth and positive, and the De Giorgi regularity argument is the first for Hartree-type equations, widening the admissible range of $\\alpha$ beyond Moser iteration.","The paper leaves open whether nonconstant profiles exist for small periods $T < T_0$ and whether the lower bound $\\alpha > \\alpha_*$ for small $s$ can be removed entirely."],"supporting_citations":[{"why":"Backbone of Theorem 1.2: supplies the Emden–Fowler form of the fractional Laplacian with kernels K_s and K_m, the periodic kernel periodization, the flattening reduction (their Prop 3.4), the Hölder and Schauder regularity results (their Props 3.8–3.9), and the energy-comparison strategy.","marker":"[18]"},{"why":"The Caffarelli–Silvestre extension turning the fractional Laplacian into a degenerate local Neumann problem; foundational for both the radial symmetry proof and the regularity theory.","marker":"[5]"},{"why":"Model for the moving-spheres argument on the extension problem for a fractional Yamabe-type equation with an isolated singularity, including the Kelvin-transform invariance at the critical exponent.","marker":"[4]"},{"why":"Supplies the maximum principle (their Prop 3.1, cited here as Lemma A) that forces the extension to stay positive at a non-removable singularity, initiating the moving spheres.","marker":"[36]"},{"why":"Source of the kernel estimates for K_m (their Lemma 2.1) that Lemma 3.1 uses for positivity, exponential decay, and the |t|^{alpha-1} singularity behavior.","marker":"[37]"},{"why":"Provides the weighted Sobolev trace inequality and strong maximum principle for the extension problem, used in the De Giorgi truncation and in the positivity of minimizers.","marker":"[2]"},{"why":"Fabes–Kenig–Serapioni degenerate-elliptic theory for A_2 weights, invoked as the foundation of the weighted De Giorgi truncation in Proposition 4.5.","marker":"[23]"},{"why":"Supplies the trace Sobolev embedding (their Corollary 5.3) that turns the gradient bound from the truncation into the psi(k) iteration in Claim 4 of Proposition 4.5.","marker":"[30]"},{"why":"Gives the geometric extension setup on the cylinder (metric g and normalized defining function rho*) used to write the periodic extension problems (3.6)–(3.8).","marker":"[17]"}],"fun_headline_variants":["Log-periodic singular solutions for critical Hartree","Radial symmetry and Delaunay profiles in fractional Hartree","Riesz convolution survives, yet periodic solutions exist","Critical Hartree: periodic Delaunay singular states","Fractional Hartree: symmetric singularities, periodic orbits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction stands on the claim that, near the boundary, the curved cylinder formed by the time axis and the sphere can be replaced by flat Euclidean space, with curvature effects too small to change the outcome even while the nonlocal Hartree boundary term is acting.","fun_headline_variants_meta":{"raw":{"variants":["Log-periodic singular solutions for critical Hartree","Radial symmetry and Delaunay profiles in fractional Hartree","Riesz convolution survives, yet periodic solutions exist","Critical Hartree: periodic Delaunay singular states","Fractional Hartree: symmetric singularities, periodic orbits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000831,"raw_usage":{"total_tokens":3697,"prompt_tokens":1083,"completion_tokens":2614,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":699,"completion_tokens_details":{"reasoning_tokens":2537}},"tokens_in":699,"tokens_out":2614,"duration_ms":21346,"temperature":1.0,"reasoning_tokens":2537,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:25:33.203430+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed pair with $0 < s < 1/2$, set $\\alpha$ exactly at the threshold $\\alpha_*$ of (1.10) and check whether the admissible interval for $q'$ in (4.26) is nonempty: the derivation says the two bounds coincide precisely at $\\alpha_*$, so the interval should be empty there and nonempty for every $\\alpha > \\alpha_*$. A direct algebraic check over a grid of $(n,s)$ settles the sharpness of the range in Theorem 1.2: a nonempty interval below $\\alpha_*$ means the range is not sharp, an empty interval above $\\alpha_*$ means the $L^\\infty$ proof has a gap. A complementary numerical check is to evaluate $F_L$ on the constant profile versus the localized bump for representative parameters (e.g. $n=2$, $s=0.4$, $\\alpha=1$): the theorem requires $F_L(c_L) \\sim L^{1-1/p^*}$ to diverge while $F_L(\\tilde w_L)$ stays $O(1)$, and the crossing of the two curves pins down $T_0$.","supporting_citations":[{"cited_title":"DelaTorre, M","cited_arxiv_id":null,"evidence_quote":"Backbone of Theorem 1.2: supplies the Emden–Fowler form of the fractional Laplacian with kernels K_s and K_m, the periodic kernel periodization, the flattening reduction (their Prop 3.4), the Hölder and Schauder regularity results (their Props 3.8–3.9), and the energy-comparison strategy."},{"cited_title":"Caffarelli and L","cited_arxiv_id":null,"evidence_quote":"The Caffarelli–Silvestre extension turning the fractional Laplacian into a degenerate local Neumann problem; foundational for both the radial symmetry proof and the regularity theory."},{"cited_title":"Caffarelli, T","cited_arxiv_id":null,"evidence_quote":"Model for the moving-spheres argument on the extension problem for a fractional Yamabe-type equation with an isolated singularity, including the Kelvin-transform invariance at the critical exponent."},{"cited_title":"Jin, Y.Y","cited_arxiv_id":null,"evidence_quote":"Supplies the maximum principle (their Prop 3.1, cited here as Lemma A) that forces the extension to stay positive at a non-removable singularity, initiating the moving spheres."},{"cited_title":"Jin and J","cited_arxiv_id":null,"evidence_quote":"Source of the kernel estimates for K_m (their Lemma 2.1) that Lemma 3.1 uses for positivity, exponential decay, and the |t|^{alpha-1} singularity behavior."},{"cited_title":"Cabr´ e and Y","cited_arxiv_id":null,"evidence_quote":"Provides the weighted Sobolev trace inequality and strong maximum principle for the extension problem, used in the De Giorgi truncation and in the positivity of minimizers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Fabes–Kenig–Serapioni degenerate-elliptic theory for A_2 weights, invoked as the foundation of the weighted De Giorgi truncation in Proposition 4.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the trace Sobolev embedding (their Corollary 5.3) that turns the gradient bound from the truncation into the psi(k) iteration in Claim 4 of Proposition 4.5."},{"cited_title":"DelaTorre and M","cited_arxiv_id":null,"evidence_quote":"Gives the geometric extension setup on the cylinder (metric g and normalized defining function rho*) used to write the periodic extension problems (3.6)–(3.8)."}],"review_version":1}