{"id":"6d84e8e4-d163-4f87-b601-0a0c606ad188","arxiv_id":"2507.12577","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Small-data solutions of the semiclassical Hartree equation with long-range interaction satisfy the uniform-in-hbar optimal density decay ||rho(t)||_{L∞} ≲ <t>^{-3}.","lead":"This paper proves that small-data solutions of the semiclassical Hartree equation with smooth long-range interactions have density decay at the optimal rate, uniformly in the semiclassical parameter. The result matters as a step toward showing that quantum modified scattering states converge to classical Vlasov scattering states in the semiclassical limit.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 4 of Proposition 7.1 needs an FL1 estimate for the phase-corrected propagator, but Lemma 3.3 covers only the free propagator. The cited combination of Lemmas 3.3 and 5.8 does not supply the required bound, so the bootstrap estimate for ∇ρ in FL1 is unsupported.","rationale":"The reader's identified weakness is the same one I find most load-bearing. The manuscript's own text shows that Step 4 of Proposition 7.1 combines Lemma 3.3, which is expressly a free-propagator FL1 estimate, with Lemma 5.8, which gives phase-conjugated weighted Schatten norms. Because the phase Ψ is genuinely large in the long-range regime, these two bounds are not interchangeable, and no lemma in Sections 3 or 5 supplies an FL1 estimate for Uℏ(t)e^{-iΨ}. The FL1 derivative bound is not decorative: it is one of the norms in Y^{a,b}_T and enters directly into the bootstrap argument in Section 7.3, so the central uniform decay statement is not proved by the cited results. I do not regard this as evidence of falsity; the paper contains substantial original ingredients, and a phase-corrected FL1 estimate may well be provable. But the present text omits it, and the omission is load-bearing rather than cosmetic. The lack of a proof for Lemma 2.7, used later in Section 6.2, reinforces the assessment that the paper needs a substantive revision rather than minor correction. I therefore see no reason to alter the reader's rejection verdict, though the appropriate framing is 'not established as written' rather than 'central claim false.'","tokens_in":72303,"tokens_out":9284,"duration_ms":97222,"concrete_test":"Prove or disprove the phase-corrected FL1 estimate needed in Step 4: for B = e^{iΨ(t,-iℏ∇)}[x/ℏ,Wγ0W*]e^{-iΨ(t,-iℏ∇)}, show that ||ρℏ(Uℏ(t)e^{-iΨ}B e^{iΨ}Uℏ(t)*)||_{FL1} ≤ C ℏ^{-3/2}|t|^{-3} ||⟨x⟩^σ B ⟨x⟩^σ||_{S2_ℏ} under the hypotheses of Proposition 3.5, by rerunning the duality and MDFM argument of Lemma 3.3 with phase t|ξ|^2/2 + Ψ(t,ℏξ). If this estimate cannot be obtained from the stated assumptions, replace Lemma 3.3 in Step 4 by this new lemma; without such a lemma the Y^{a,b}_T bootstrap for the FL1 derivative term does not close.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Step 4 of Proposition 7.1 claims that ||∇ρℏ(γℏ(t))||_{FL1} is bounded by ℏ^{-3/2}⟨t⟩^{-4} times the S2 norms of ⟨ℏ∇⟩^σ[∇,Wγ0W*]⟨ℏ∇⟩^σ and ⟨x⟩^σ e^{iΨ}[x/ℏ,Wγ0W*]e^{-iΨ}⟨x⟩^σ, citing Lemmas 3.3 and 5.8. This does not close. Lemma 3.3 bounds ρℏ(Uℏ(t)γ0'Uℏ(t)*) in FL1 with weights on γ0' itself, namely ⟨ℏ∇⟩^σγ0'⟨ℏ∇⟩^σ or ⟨x⟩^σγ0'⟨x⟩^σ. Lemma 5.8, however, controls the phase-conjugated expression ⟨x⟩^σ e^{iΨ} C e^{-iΨ} ⟨x⟩^σ, not ⟨x⟩^σ C ⟨x⟩^σ. Since Ψ is large (of order ℏ^{-1} log t), the Fourier multiplier e^{iΨ} does not commute harmlessly with ⟨x⟩^σ, and no commutator estimate supplied in the paper converts one norm into the other. What is needed is an FL1 analogue of Proposition 3.5 for the propagator Uℏ(t)e^{-iΨ}, or equivalently a phase-corrected version of Lemma 3.3; the text proves neither. This FL1 bound is exactly the component of the Y^{a,b}_T norm used in the bootstrap closure, so Theorem 1.4 is not established as written. A secondary gap is that Lemma 2.7 is stated without proof and is used in the double commutator estimates of Section 6.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the semi-classical Hartree equation (NLH) with a smooth long-range interaction w satisfying assumption (A), and claims an optimal uniform-in-hbar bound on the density: sup_{hbar in (0,1]} sup_{t>=0} ( ||rho_hbar(gamma_hbar(t))||_{L^1} + <t>^3 ||rho_hbar(gamma_hbar(t))||_{L^infty} ) <= C, provided the initial data are small in an hbar-independent way in the norm X^sigma_hbar. The proof is organized around a phase-corrected wave operator (Proposition 4.1), an L^1-L^infty estimate for the phase-corrected propagator (Proposition 3.5 and Corollary 3.8), a new FL1 bound for the free propagator (Lemma 3.3), and long commutator estimates (Sections 5 and 6). The bootstrap is carried out in a space Y^{a,b}_T whose norm includes L^1, L^infty, FL1, and L^2 decay information on derivatives of the density. The author also claims a new proof of modified scattering for the fixed-hbar long-range Hartree equation, with the caveat that the convergence rate in the scattering statement may depend on hbar.","tokens_in":72709,"tokens_out":14366,"duration_ms":152200,"significance":"If the main result were established, it would be a substantial contribution: it would give the first uniform-in-hbar optimal t^{-3} density decay for long-range Hartree interactions, filling the gap between the short-range uniform results and the fixed-hbar modified scattering result of Nguyen and You. The paper contains several interesting and potentially reusable ingredients: a self-contained proof of an L^1-L^infty dispersive estimate for a phase-corrected propagator, a uniform boundedness statement for phase-corrected wave operators, and a large set of Schatten-class commutator estimates. The bootstrap itself is a standard self-consistency argument and is not circular in the sense of fitting constants. However, the proof of the key a priori estimate contains a load-bearing gap: the FL1 component of the Y^{a,b}_T norm is obtained by applying the free-propagator FL1 estimate Lemma 3.3 to an expression involving the phase-corrected propagator, for which no FL1 estimate is proved. As a result, the main theorem is not established as written.","major_comments":[{"comment":"The displayed estimate for ||nabla rho_hbar(gamma_hbar(t))||_{FL1} applies Lemma 3.3 to the phase-corrected expression, but Lemma 3.3 is a free-propagator estimate: it bounds rho_hbar(U_hbar(t) B U_hbar(t)*) by weighted norms of B itself, specifically ||<hbar nabla>^sigma B <hbar nabla>^sigma||_{S^2_hbar} or ||<x>^sigma B <x>^sigma||_{S^2_hbar}. In the second term of Step 4, the only available control is Lemma 5.8 (5.4), which bounds ||<x>^sigma e^{iPsi} [x/hbar, W V gamma0 W V*] e^{-iPsi} <x>^sigma||_{S^2_hbar}. To insert this into the free-propagator estimate one would have to apply Lemma 3.3 to B = e^{-iPsi} A e^{iPsi} and hence to the propagator U_hbar(t) e^{-iPsi}, i.e. one needs an FL1 estimate for the phase-corrected modified propagator. No such estimate is stated or proved: Proposition 3.5 gives only L^1 -> L^infty, and Corollary 3.8 gives L^r_x bounds but not FL1 bounds. Since Psi is of size hbar^{-1} log t, the phase factors do not commute with <x>^sigma, and no commutator estimate is supplied that converts the Lemma 5.8 norm into the norm required by Lemma 3.3. This FL1 bound is exactly the component of the Y^{a,b}_T norm that closes the bootstrap, so Theorem 1.4 is not established as written. The author needs to prove an FL1 analogue of Corollary 3.8 for U_hbar(t) e^{-iPsi}.","section":"Prop. 7.1, Step 4 (Section 7.1)"},{"comment":"Lemma 2.7 is stated without proof. The paragraph 'Proof of Lemmas 2.5, 2.6, and 2.7' explicitly proves only Lemma 2.5 and says that 'the same proof works for Lemma 2.6'; it does not discuss Lemma 2.7, which is a distinct mixed double commutator identity involving one derivative and one x/hbar commutator. Lemma 2.7 is used in Section 6.2.2 to estimate the mixed double commutator in the proof of Lemma 6.2. Without a proof of this identity, the double commutator estimates that feed into Proposition 7.1 are unsupported. Since the argument is algebraic, this item may be repairable, but it is not a presentation issue.","section":"Lemma 2.7 (Section 2.3.3)"}],"minor_comments":[{"comment":"The claim that (3.9) is equivalent to (2.19) is correct only because e^{+-iPsi} is a Fourier multiplier and therefore commutes with <hbar nabla>; this should be stated explicitly, since the same equivalence is false for the <x>-weighted estimate (3.10).","section":"Corollary 3.8 and Remark 3.9"},{"comment":"In the last line of Step 1 the notation switches from ||gamma0||_{X^sigma_hbar} to ||gamma0||_{X_hbar}; please standardize the notation for the initial-data norm.","section":"Section 7.1, Step 1"},{"comment":"In the proof of Remark 1.5, the convergence statement (7.1) is proved in B(L^2) after conjugation by <x>^{-1}, and the passage to convergence in S^1_hbar is compressed. Since the uniform S^1 bound is stated but not fully justified at that point, please expand this step for clarity.","section":"Section 7.3"}],"recommendation":"major_revision","confidential_remarks":"The novelty of the paper is genuine and the author has assembled a substantial set of tools. The FL1 gap in Step 4 of Proposition 7.1 is, however, essential: it is the only place where the FL1 component of the bootstrap norm is bounded, and no phase-corrected FL1 estimate is present in the manuscript. I do not see a quick local correction; the author needs to add a new estimate or restructure the proof. If that estimate can be supplied, the paper could be a valuable contribution. The missing proof of Lemma 2.7 is a second, more easily repairable gap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a serious look, but there is a load-bearing hole in the proof. The problem is genuinely new: uniform-in-hbar optimal density decay for long-range Hartree, where previous uniform results were short-range and previous long-range results were fixed-hbar. The paper also ships genuinely new machinery—the phase-corrected wave-operator boundedness in Proposition 4.1, the L1–L∞ estimate for the modified propagator in Proposition 3.5, and the commutator estimates in Sections 5–6. The framing around the semiclassical scattering-state diagram is clear and the relation to Nguyen–You, Hadama–Hong, and Smith is spelled out honestly. The citation pattern looks careful, and the self-reliance on [28] is not an abuse given the tools are imported rather than fitted.\n\nThe soft spot is exactly where the reader says it is. Step 4 of Proposition 7.1 claims an FL1 bound for ∇ρ_hbar by invoking Lemmas 3.3 and 5.8. Lemma 3.3 is a free-propagator FL1 estimate: it requires control of the weighted operator itself, e.g. ||<x>^σ A <x>^σ||_{S2}, where A is the operator being propagated. But the density derivative is ρ_hbar(U_hbar [x/hbar, W γ0 W*] U_hbar*), so A = [x/hbar, W γ0 W*]. Lemma 5.8 controls the phase-conjugated norm ||<x>^σ e^{iΨ} A e^{-iΨ} <x>^σ||, not ||<x>^σ A <x>^σ||. Since Ψ is of order hbar^{-1} log t, the phase conjugation does not commute harmlessly with the x-weights. What is needed is an FL1 estimate for U_hbar e^{-iΨ}, or equivalently a phase-corrected analogue of Lemma 3.3; the paper provides neither. This FL1 bound feeds directly into the Y^{a,b}_T bootstrap norm, so Theorem 1.4 is not established. The missing proof of Lemma 2.7 is a lesser concern—it looks like a routine albeit tedious commutator identity.\n\nThe gap is specific and localized. The paper is not a random collection of estimates; the architecture is coherent and the missing piece is identifiable. If the author can supply a phase-corrected FL1 estimate, the rest of the argument may well go through. This deserves referee time, not a desk reject. My recommendation: send it to peer review, with the expectation of major revision focused on Step 4.","headline":"Uniform long-range Hartree decay is a meaningful target and the paper brings new tools, but Step 4 of Proposition 7.1 uses an FL1 estimate for the free propagator where a phase-corrected version is required, so the main theorem is not closed as written.","tokens_in":73282,"tokens_out":3649,"would_cite":false,"duration_ms":39381,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","35B40","35Q40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For long-range Hartree interactions, the density of small-data solutions decays at the optimal $t^{-3}$ rate uniformly in $\\hbar$.","keywords":["Hartree equation","semi-classical limit","dispersive estimates","long-range interaction","modified scattering","density operators","Schatten norms","phase correction"],"falsifier":"Compute exactly $\\|\\rho_\\hbar(U^\\hbar(t)e^{-i\\Psi}Ae^{i\\Psi}U^\\hbar(t)^*)\\|_{FL^1}$ for a Gaussian rank-one $A$ and a phase $\\Psi(t,\\xi) = \\hbar^{-1}\\int_0^t V(\\tau,\\tau\\xi)\\,d\\tau$ with $V = w*\\rho$ built from the solution's own density; if this quantity decays slower than $\\hbar^{-3/2}\\langle t\\rangle^{-4+a}$, the Step 4 estimate of Proposition 7.1 fails and Theorem 1.4 cannot hold as stated.","tokens_in":72051,"feed_emoji":"⚛️","tokens_out":7444,"duration_ms":75572,"temperature":0.7,"pith_summary":"This paper establishes a uniform-in-$\\hbar$ dispersive estimate for the Hartree equation with a smooth long-range interaction in three dimensions. It proves that if the initial density operators satisfy an $\\hbar$-independent smallness condition in the semi-classical norm $X^\\sigma_\\hbar$, then the solution's density decays as $\\|\\rho_\\hbar(\\gamma_\\hbar(t))\\|_{L^\\infty_x} \\lesssim \\langle t\\rangle^{-3}$ for every $\\hbar \\in (0,1]$, with constants independent of $\\hbar$. For fixed $\\hbar$ this was essentially known, but the prior smallness condition shrank to zero as $\\hbar \\to 0$ unless the data was trivial; the uniformity here is what permits a nontrivial semi-classical limit. The argument also supplies a new proof of modified scattering for the long-range nonlinear Schr\\\"odinger equation with Hartree nonlinearity, via phase-corrected wave operators and a phase-corrected dispersive estimate.","feed_headline":"Long-range Hartree density decays at optimal rate, uniformly in ℏ","feed_subtitle":"Small data yield the optimal t^-3 density decay, with all smallness and bounds independent of Planck's constant.","key_machinery":"The engine of the proof is the phase-corrected wave operator $e^{i\\Psi(t,-i\\hbar\\nabla)} W^\\hbar_V(t)\\langle x\\rangle^{-s}$, whose uniform-in-$\\hbar$ boundedness (Proposition 4.1) replaces the ordinary weighted wave-operator boundedness that fails for long-range potentials. The phase $\\Psi(t,\\xi) = \\hbar^{-1}\\int_0^t V(\\tau,\\tau\\xi)\\,d\\tau$ is chosen so that commutators of $x$ with $e^{i\\Psi}W^\\hbar_V$ nearly cancel the terms produced by $[x, W^\\hbar_V]$. A second ingredient is the $L^1$--$L^\\infty$ dispersive estimate $\\|U^\\hbar(t)e^{-i\\Psi(t,-i\\hbar\\nabla)}\\|_{L^1 \\to L^\\infty} \\lesssim |t\\hbar|^{-3/2}$ (Proposition 3.5), proved by stationary phase with a uniqueness-of-critical-point argument; it feeds Corollary 3.8, the phase-corrected density estimates. A large family of single and double commutator estimates in Schatten norms (Sections 5--6) then controls derivatives of the density, which the bootstrap needs because the identity $\\partial_{x_j}\\rho_\\hbar(U^\\hbar(t)A(t)U^\\hbar(t)^*) = t^{-1}\\rho_\\hbar(U^\\hbar(t)[x_j/(i\\hbar), A(t)]U^\\hbar(t)^*)$ forces commutators to be tracked. The identity (1.14), writing the nonlinear evolution as $U^\\hbar_{w*\\rho}(t)\\gamma_0^\\hbar U^\\hbar_{w*\\rho}(t)^*$, converts the Duhamel $1/\\hbar$ loss into a closed equation for the density.","core_discovery":"The central claim is Theorem 1.4: for $3/2 < \\sigma < 2$ and long-range interaction $w \\in C^3$ satisfying $|\\partial^\\alpha w(x)| \\lesssim \\langle x\\rangle^{-1-|\\alpha|}$, there exists $\\varepsilon_0 > 0$ such that any family of self-adjoint initial data $\\gamma_0^\\hbar$ with $\\sup_{\\hbar \\in (0,1]} \\|\\gamma_0^\\hbar\\|_{X^\\sigma_\\hbar} \\le \\varepsilon_0$ gives a unique global solution $\\gamma_\\hbar(t)$ to the semi-classical Hartree equation whose density satisfies $\\sup_{t \\ge 0} (\\|\\rho_\\hbar(\\gamma_\\hbar(t))\\|_{L^1_x} + \\langle t\\rangle^3 \\|\\rho_\\hbar(\\gamma_\\hbar(t))\\|_{L^\\infty_x}) \\lesssim 1$ uniformly in $\\hbar$. This is the optimal decay rate, matching the free solution, and it holds for long-range interactions such as the regularized Coulomb potential $w(x) = \\pm\\langle x\\rangle^{-1}$ (though not the singular Coulomb $|x|^{-1}$). The same solution exhibits modified scattering: after a phase correction $e^{i\\Psi(t,-i\\hbar\\nabla)}$ with $\\Psi(t,\\xi) = \\hbar^{-1}\\int_0^t V(\\tau,\\tau\\xi)\\,d\\tau$, the propagated density operator converges to a scattering state $\\gamma_+^\\hbar$ in operator norm, although the rate of that convergence is not uniform in $\\hbar$.","pith_inferences":["If the phase-corrected dispersive estimate extends to the Fourier--Lebesgue norm that Step 4 of Proposition 7.1 needs, the same bootstrap scheme would likely give uniform bounds for slightly more singular long-range potentials such as the exact Coulomb interaction cut off only at the origin.","The uniform density decay proven here is the natural long-range analogue of the short-range semi-classical scattering diagram: it should let the Wigner transforms of the modified scattering states converge to the corresponding Vlasov scattering states as $\\hbar \\to 0$, but the paper only establishes the quantum-side ingredient.","The method suggests a route to uniform-in-$\\hbar$ modified scattering: replace the scalar phase $e^{i\\Psi}$ with a richer class of phase corrections so that the $1/\\hbar$ factor in the convergence-rate estimate (7.1) is cancelled."],"forward_implications":["The smallness condition and all constants in the density bound are independent of $\\hbar$, so nontrivial families of initial data can pass to the semi-classical limit without shrinking to zero.","At $\\hbar = 1$ with rank-one data, the theorem gives a new proof of modified scattering for the long-range Hartree-type nonlinear Schr\\\"odinger equation.","The density decays at the free rate $\\langle t\\rangle^{-3}$, while its first derivatives decay at $\\langle t\\rangle^{-4+\\varepsilon}$ and second derivatives at $\\langle t\\rangle^{-7/2+b}$, so derivatives decay faster than the density itself.","The construction of modified scattering states covers long-range interactions satisfying (A), including the regularized Coulomb potential, but the convergence rate of the scattering state is not uniform in $\\hbar$, leaving the fully uniform modified scattering statement open."],"supporting_citations":[{"why":"Supplies the short-range predecessor: the solution identity (1.14), wave-operator boundedness, and the bootstrap scheme that this paper adapts to long-range potentials.","marker":"[28]"},{"why":"Gives the fixed-$\\hbar$ modified scattering result for long-range Hartree equations that Theorem 1.4 makes uniform in $\\hbar$.","marker":"[49]"},{"why":"Establishes the optimal $\\langle t\\rangle^{-3}$ density decay for $\\hbar = 1$ with short-range interaction, the benchmark the present theorem matches for long-range interactions.","marker":"[52]"},{"why":"Gives uniform-in-$\\hbar$ density decay near stationary solutions for regular short-range interactions, the existing uniformity result that this paper extends to the long-range vacuum case.","marker":"[57]"},{"why":"Supplies the O'Neil convolution inequality used to turn decay of the density into decay of the potential $V = w * \\rho$ and its derivatives.","marker":"[50]"},{"why":"Supplies the Dunford--Pettis identification of $L^1 \\to L^\\infty$ operator norms with kernel bounds, used to pass from the phase-corrected dispersive estimate to density estimates.","marker":"[16]"},{"why":"Provides the Kato--Seiler--Simon inequality used throughout the commutator and Schatten-norm estimates.","marker":"[56]"}],"fun_headline_variants":["Hartree density decay uniform in ℏ","Optimal Hartree decay uniform in ℏ","Uniform in ℏ: optimal Hartree decay","Semi-classical Hartree: ℏ-uniform decay"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof depends on the free-propagator Fourier--Lebesgue estimate continuing to hold when the free propagator is replaced by the phase-corrected propagator; that extension is asserted in Step 4 of Proposition 7.1 but not proved.","fun_headline_variants_meta":{"raw":{"variants":["Hartree density decay uniform in ℏ","Optimal Hartree decay uniform in ℏ","Uniform in ℏ: optimal Hartree decay","Semi-classical Hartree: ℏ-uniform decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001514,"raw_usage":{"total_tokens":6145,"prompt_tokens":1100,"completion_tokens":5045,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":716,"completion_tokens_details":{"reasoning_tokens":4985}},"tokens_in":716,"tokens_out":5045,"duration_ms":38292,"temperature":1.0,"reasoning_tokens":4985,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:47:51.202310+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute exactly $\\|\\rho_\\hbar(U^\\hbar(t)e^{-i\\Psi}Ae^{i\\Psi}U^\\hbar(t)^*)\\|_{FL^1}$ for a Gaussian rank-one $A$ and a phase $\\Psi(t,\\xi) = \\hbar^{-1}\\int_0^t V(\\tau,\\tau\\xi)\\,d\\tau$ with $V = w*\\rho$ built from the solution's own density; if this quantity decays slower than $\\hbar^{-3/2}\\langle t\\rangle^{-4+a}$, the Step 4 estimate of Proposition 7.1 fails and Theorem 1.4 cannot hold as stated.","supporting_citations":[{"cited_title":"Hadama and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the short-range predecessor: the solution identity (1.14), wave-operator boundedness, and the bootstrap scheme that this paper adapts to long-range potentials."},{"cited_title":"Modified scattering for long-range Hartree equations of infinite rank near vacuum","cited_arxiv_id":"2408.15860","evidence_quote":"Gives the fixed-$\\hbar$ modified scattering result for long-range Hartree equations that Theorem 1.4 makes uniform in $\\hbar$."},{"cited_title":"Pusateri and I","cited_arxiv_id":null,"evidence_quote":"Establishes the optimal $\\langle t\\rangle^{-3}$ density decay for $\\hbar = 1$ with short-range interaction, the benchmark the present theorem matches for long-range interactions."},{"cited_title":"O’Neil, Convolution operators and L(p, q) spaces, Duke Math","cited_arxiv_id":null,"evidence_quote":"Supplies the O'Neil convolution inequality used to turn decay of the density into decay of the potential $V = w * \\rho$ and its derivatives."},{"cited_title":"Dunford and B","cited_arxiv_id":null,"evidence_quote":"Supplies the Dunford--Pettis identification of $L^1 \\to L^\\infty$ operator norms with kernel bounds, used to pass from the phase-corrected dispersive estimate to density estimates."},{"cited_title":"Simon, Trace ideals and their applications, Second edition , Math","cited_arxiv_id":null,"evidence_quote":"Provides the Kato--Seiler--Simon inequality used throughout the commutator and Schatten-norm estimates."}],"review_version":1}