{"id":"d38a151f-7e1f-4d76-b686-a02cd940a3c3","arxiv_id":"2411.15939","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New Strichartz estimates in similarity variables for potential-perturbed radial wave equations yield optimal regularity blowup stability for the quintic wave equation in all dimensions.","lead":"Wallauch proves new Strichartz estimates for radial wave equations with potentials in similarity coordinates, covering all dimensions d>=3 and a large range of Sobolev regularities. The estimates give a general route to optimal blowup stability for energy supercritical nonlinear wave equations, demonstrated on the quintic wave equation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.1's unproved spectral assertion is the load-bearing premise for Theorem 1.2; it needs a self-contained hypergeometric verification for all d≥3.","rationale":"The reader's weakest_assumption is exactly Lemma 6.1, and my reading agrees: the spectral assertion is the point where the headline application depends on a nontrivial, dimension-dependent fact whose proof is not shown. The rest of the paper, including the resolvent construction and interpolation framework behind Theorem 1.1, is extensive and internally coherent, but Theorem 1.2's conclusion is conditional on the one-dimensional unstable spectrum. I also note a secondary presentation issue: the abstract's 'almost all regularities' overreaches the ceil(s) ≤ d/2 restriction in odd dimensions, but that is a scope statement rather than a correctness risk. The concrete test proposed—an independent hypergeometric eigenvalue computation plus a numerical scan for small d—would settle whether Lemma 6.1 holds. Since the reader already assigned CONDITIONAL, my recommendation is UNCHANGED; the verdict should remain conditional pending the spectral verification.","tokens_in":80801,"tokens_out":20550,"duration_ms":181418,"concrete_test":"Independently derive the eigenvalue condition for Lemma 6.1: reduce the displayed ODE to hypergeometric form, compute the connection coefficient c_{2,4}(λ) (the criterion from Lemma 4.1), and prove that its only zero in Re λ ≥ 0 is λ = 1 with multiplicity one for every d ≥ 3. As a cross-check, implement a numerical shooting method for the ODE for d = 3,4,5,6,7 and scan for eigenvalues in Re λ ≥ 0. Any additional zero would invalidate Theorem 1.2 as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2 (optimal blowup stability for the quintic wave equation in all dimensions d≥3) rests on Lemma 6.1, which asserts that the linearized operator L has point spectrum in {Re z < 0} ∪ {1}, with 1 simple. The proof is a single sentence: the eigenvalue ODE 'can be transformed into a hypergeometric equation' and the conclusion follows by adapting Lemmas 4.10–4.11 of [16]. No hypergeometric eigenvalue calculation is displayed. The ODE in Lemma 6.1 contains d explicitly through the (d−1)/ρ term, so the hypergeometric parameters are dimension-dependent; [16] concerns odd space dimensions, while Theorem 1.2 claims all d≥3. If an additional eigenvalue with Re λ≥0 exists for some d (especially an even d), then the projection P used in §6 is not the full unstable spectral projection, the one-dimensional tail bound for P K_u in Lemma 6.3 fails, and the open-ball stability statement of Theorem 1.2 is false. This is structurally separate from Theorem 1.1, which is potential-independent and would survive, but the advertised blowup-stability application would not. The manuscript should provide the full hypergeometric spectral computation or cite a theorem that covers all d≥3 explicitly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops Strichartz estimates for radial wave equations with potentials in similarity variables on the unit ball, for dimensions d >= 3 and regularities 1 <= s < d/2 with ceiling(s) <= d/2. Theorem 1.1 asserts that, after removing a finite-rank component, the semigroup associated with the similarity-variable operator L satisfies homogeneous and inhomogeneous Strichartz estimates, including derivative versions and fractional regularity. The proof combines a resolvent construction from ODE asymptotics and Bessel functions, a Laplace representation of the semigroup, oscillatory-integral estimates, and interpolation. Theorem 1.2 applies these estimates to prove an optimal blowup stability result for the quintic nonlinear wave equation at scaling regularity, based on spectral information about the linearized operator.","tokens_in":1703,"tokens_out":2365,"duration_ms":229971,"significance":"If Theorem 1.1 is fully correct, it is a substantial advance: it extends the Strichartz framework in similarity variables to all dimensions d >= 3 and to a large range of non-integer regularities, thereby unifying earlier results at energy and at selected higher regularities. The resolvent/oscillatory-integral strategy is coherent, and Theorem 1.1 is largely independent of fine spectral information because the finite-rank projection absorbs the unstable directions. The application to quintic blowup stability is natural and demonstrates the utility of the framework. However, the advertised application currently rests on an unproved spectral assertion, and the manuscript also contains a sign inconsistency in the central resolvent equation and a questionable embedding in the nonlinear estimates; these issues must be addressed before the full claims can be accepted.","major_comments":[{"comment":"The proof of Theorem 1.2 depends on the assertion that the linearized operator has point spectrum contained in (Re z < 0) union {1}, with 1 simple. This is used in Lemma 6.3 to identify P as one-dimensional and to control the projected nonlinearity. The proof of Lemma 6.1 is a single sentence: the eigenvalue ODE 'can be transformed into a hypergeometric equation' and the conclusion follows by adapting Lemmas 4.10 and 4.11 of [16]. This is not a complete proof. The ODE contains d explicitly through the (d-1)/rho term, so the hypergeometric parameters depend on d, while [16] treats odd space dimensions and Theorem 1.2 claims all d >= 3. If an additional eigenvalue with Re lambda >= 0 exists for some d, the projection P in Section 6 is not the full unstable spectral projection, the estimate for P K_u in Lemma 6.3 fails, and the stability statement does not follow. Please provide the full hypergeometric spectral calculation for all d >= 3, including the simplicity of lambda = 1 and the absence of eigenvalues on the boundary of the essential spectrum, or cite an explicit theorem covering all dimensions.","section":"Section 6, Lemma 6.1"},{"comment":"The sign of the potential term is inconsistent across the central equations. Substituting f2 = rho f1' + (d - 2s + 2lambda)/2 f1 - g1 from the first component of (lambda - L)f = g into the second component gives a potential term -V(rho) f1, not +V(rho) f1. Eq. (3.1) and Lemma 6.1 indeed use the minus sign, but Eqs. (2.7), (2.8), and (4.3) display +V(rho) f1. Consequently, the resolvent R constructed in Lemma 4.6 and used in the oscillatory-integral estimate is the resolvent for L0 - (0, V f1), not for L = L0 + (0, V f1) as defined in (1.4). The argument is probably repairable by replacing V with -V consistently, but as written the derivation is not matched to the operator in Theorem 1.1. Please correct the sign throughout.","section":"Section 2 and Section 4"},{"comment":"The proof for even d relies on the embedding H^{(d-3)/2}(B^d_1) subset W^{d/2-1, 2d/(d+1)}(B^d_1). For d = 4 this asserts H^{1/2} subset W^{1, 8/5}, which is not a standard Sobolev embedding and is false for general functions; radial symmetry does not increase the differentiability order from s to s+1. The subsequent Holder/product estimate for u^5 in H^{(d-3)/2} is therefore not justified for even dimensions. Please provide a correct proof of the nonlinear estimate, for example by using the full H^{(d-1)/2} information available in the X-norm together with admissible Strichartz pairs, or by citing a proved radial embedding with the stated parameters.","section":"Section 6, Lemma 6.2"}],"minor_comments":[{"comment":"The free Strichartz estimates are stated with (I - P)f on the right-hand side, but no operator P has been defined for the free semigroup S0; the projector should presumably be the identity in that statement.","section":"Lemma 2.1"},{"comment":"The condition q in [2, d/n] is undefined when n = 0; it should read q in [2, infinity] for n = 0.","section":"Definition 5.1 and Lemma 6.2"},{"comment":"There are numerous typographical and formatting issues, including 'ad verbatim', a missing symbol in the proof of Lemma 4.15, undefined or inconsistent symbols in the definitions of kappa_j, and several repeated phrases such as 'we conclude this proof'. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"In the proof of Lemma 5.3, some operators are written as functions of lambda where the variable tau is intended, and the change of variables t = 1 - e^{-y} in the L^p L^infinity estimates should be stated consistently with the subsequent Young's inequality application.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The Strichartz part of the paper is long and technically ambitious, and the overall strategy appears credible; the main theorem is not circular and does not assume the Strichartz estimates it derives. The decisive issues are the missing spectral calculation for Lemma 6.1 covering all d >= 3, the sign inconsistency in the potential in the resolvent equation, and the unjustified embedding in Lemma 6.2. All three are fixable, but they are load-bearing for Theorem 1.2. If Lemma 6.1 cannot be proved for even d, the paper could be revised to state Theorem 1.2 only for the dimensions covered by the spectral result, while keeping Theorem 1.1 as the main contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a serious paper and the main theorem is a real advance. Wallauch proves Strichartz estimates, including derivative-involving ones, for radial wave operators with potentials in similarity variables across all d≥3 and a large range of regularities s in [1,d/2), with the caveat ceil(s)≤d/2. That this works for non-integer s with derivatives is new and genuinely unifies earlier dimension-by-dimension work.\n\nWhat is new: Theorem 1.1 is the first general-s, all-dimensions result of this kind. The proof architecture—resolvent via ODE asymptotics, Laplace representation, oscillatory integrals, interpolation—is coherent, and the paper does the heavy lifting in Sections 3–5. The free semigroup estimates match the known Strichartz range; the perturbed estimates are derived at two regularity levels and interpolated. I see no circularity: the Strichartz estimates are outputs, not inputs, and the cited prior framework is used for genuine structural pieces.\n\nSoft spots: two. First, the abstract says “almost all regularities above energy and below d/2”, but Theorem 1.1 actually requires ceil(s)≤d/2, which excludes s>(d-1)/2 in odd d. The restriction appears only in a remark, not the abstract. That overreach is minor but should be fixed. Second, and load-bearing: Theorem 1.2 depends on Lemma 6.1, which asserts the linearized quintic operator has point spectrum in {Re z<0}∪{1}, with 1 simple. The proof is one sentence: transform to hypergeometric and adapt Lemmas 4.10–4.11 of [16]. But [16] treats odd space dimensions, and the eigenvalue ODE contains d explicitly through the (d−1)/ρ term, so the hypergeometric parameters are dimension-dependent. If some even d admits another eigenvalue with Re λ≥0, the projection P in Section 6 is not the full unstable projection, the estimate for P K_u fails, and Theorem 1.2 collapses. This is not a defect in Theorem 1.1—it is structurally separate—but the advertised application is not proved as written. The fix is a self-contained hypergeometric spectral computation for all d≥3, or a citation to a theorem explicitly covering all dimensions.\n\nWho it is for: researchers working on blowup stability and dispersive estimates in similarity coordinates. Theorem 1.1 deserves to be read carefully and cited once verified; Theorem 1.2 should be treated as conditional.\n\nRecommendation: I would send this to peer review. A good referee can check the spectral lemma and the dimensional dependence; the main theorem is important enough and the manuscript is serious. The paper likely needs revision, not rejection.","headline":"Broad new Strichartz framework in similarity variables for potential-perturbed wave equations; the advertised quintic application hangs on an unproved spectral lemma that needs a real proof.","tokens_in":81573,"tokens_out":1863,"would_cite":true,"duration_ms":17214,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35L05","35B44","35L71","47D06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves Strichartz estimates for radial wave equations with potentials in similarity variables, for all d≥3 and almost all regularities 1≤s<d/2, and uses them to show that the quintic ODE blowup is stable at optimal regularity.","keywords":["Strichartz estimates","similarity variables","blowup stability","supercritical wave equations","radial potentials","quintic nonlinear wave equation","semigroup resolvent","Sobolev regularity"],"falsifier":"Numerically locate the zeros of the eigenvalue coefficient $c_{2,4}(\\lambda)$ for the ODE (2.8) with $V=\\tfrac{15}{4}$ in a concrete dimension, say $d=3$, $s=1$; finding any eigenvalue with $\\operatorname{Re}\\lambda\\ge0$ other than the simple $\\lambda=1$ would falsify Lemma 6.1 and hence Theorem 1.2, while confirming the spectrum would support the stability claim, and a direct numerical check of the asserted endpoint Strichartz pairs for the potential-free case would test the resolvent estimates behind Theorem 1.1.","tokens_in":80531,"feed_emoji":"💥","tokens_out":10806,"duration_ms":87969,"temperature":0.7,"pith_summary":"The paper establishes Strichartz estimates for radial wave equations with potentials in similarity variables, for every spatial dimension $d\\ge 3$ and almost every regularity $1\\le s<d/2$ with $\\lceil s\\rceil\\le d/2$. In similarity coordinates a blowup becomes an infinite-time evolution, and the linearized equation acquires a potential; these spacetime bounds are the missing dispersive tool for studying that evolution at supercritical regularities. The estimates hold after projecting off a finite-dimensional unstable subspace, and they include derivative and $L^pL^\\infty$ endpoint versions. As an application, the paper proves that the explicit quintic ODE blowup $u_T(t)=\\bigl(\\tfrac34\\bigr)^{1/4}(T-t)^{-1/2}$ is stable at the optimal Sobolev regularity: initial data in $H^{(d-1)/2}\\times H^{(d-3)/2}$ sufficiently close to the profile yield a solution that blows up like $u_T$, with the difference measured by $\\int_0^T\\|u(t)-u_T(t)\\|^2_{L^\\infty}dt\\le\\delta^2$. The same framework is designed to give optimal blowup stability for other supercritical nonlinear wave equations once their linearized spectra are known.","feed_headline":"Potential wave equations satisfy Strichartz estimates; blowup stable","feed_subtitle":"The new bounds hold for any smooth radial potential and give optimal stability for quintic wave blowup.","key_machinery":"The carrying mechanism is the similarity-variable operator $\\widehat L$ and the resolvent of its closure. The resolvent equation reduces to the singular second-order ODE (2.8); the paper constructs explicit fundamental solutions by a Liouville--Green transformation to a Bessel equation, with symbol-type expansions separately near $\\rho=0$ and $\\rho=1$, glued by a smooth cut-off $\\chi_\\lambda$. This gives enough control of $(\\lambda-\\widehat L)^{-1}$ to justify the Laplace representation $$S(\\tau)(I-P)f=\\frac1{2\\pi i}\\lim_{N\\to\\infty}\\int_{\\varepsilon+iN}^{\\varepsilon-iN}$e^{{\\lambda\\tau}}$(\\$\\lambda$-\\widehat L)^{-1}(I-P)f\\,d\\$\\lambda$,$$ and the desired Strichartz estimates follow from oscillatory-integral bounds on $e^{\\lambda\\tau}$ times resolvent differences. A two-level interpolation between $\\lfloor s\\rfloor$ and $\\lceil s\\rceil$ regularity then upgrades the bounds to all $s$, while the finite-rank projection $P$ removes the unstable part of the spectrum.","core_discovery":"On the cylinder $[0,\\infty)\\times B_1^d$ obtained from $\\tau=-\\log(T-t)+\\log T$, $\\rho=r/(T-t)$, the radial wave equation with potential $V$ becomes $\\partial_\\tau\\Psi=\\widehat L\\Psi+N(\\Psi)$, where $\\widehat L$ is the differential operator (1.4). Theorem 1.1 asserts that the closure of $\\widehat L$ on $H^s_{\\rm rad}\\times H^{s-1}_{\\rm rad}$ generates a semigroup $S$, and there is a finite-rank projection $P$ onto the unstable eigenspace such that $S(\\tau)(I-P)$ satisfies the homogeneous and inhomogeneous Strichartz estimates $$\\|[S(\\tau)(I-P)f]_1\\|_{L^p_\\tau\\dot $W^{{n,q}}$(B_1^d)}\\lesssim\\|(I-P)f\\|_{H^s\\times $H^{{s-1}}$}$$ for $0\\le n\\le s-1$ and scaling-admissible $p,q$, together with fractional-derivative versions and an $L^\\infty_\\tau H^s$ bound. With the quintic potential $V=\\tfrac{15}{4}$ and the spectral fact that the point spectrum of $\\widehat L$ lies in $\\{\\operatorname{Re}z<0\\}\\cup\\{1\\}$ with $1$ simple, Theorem 1.2 shows that the ODE blowup is stable at the critical regularity: for small $\\delta$ and data in $H^{(d-1)/2}_{\\rm rad}\\times H^{(d-3)/2}_{\\rm rad}(B_{1+\\delta}^d)$ within $\\delta/M$ of $u_1[0]$, there is a $T\\in[1-\\delta,1+\\delta]$ and a unique solution in the light cone satisfying (1.6) and (1.7). The proof identifies the finite-dimensional subspace $U$ with the unstable eigenspace introduced by $V$, so the Strichartz theorem itself is independent of the concrete spectrum.","pith_inferences":["If Theorem 1.1 is as general as it appears, the main remaining obstacle for optimal blowup stability of other explicit supercritical profiles is purely spectral: one must locate the eigenvalues of the corresponding $\\widehat L$ on $H^s\\times H^{s-1}$, not re-prove dispersive estimates.","The excluded regularities in odd dimensions, where $\\lceil s\\rceil>d/2$, are an artifact of the interpolation route; a direct resolvent estimate at regularity $(d+1)/2$, if found, would close the gap and likely extend stability to all $s<d/2$.","The endpoint $L^pL^\\infty$ bounds at $s=d/2-1/p$ may support stability statements in slightly weaker topologies, for instance by measuring the blowup profile in $L^p_tL^\\infty_x$ alone as in (1.6), and may give information about the rate at which solutions converge to the self-similar profile.","For potentials whose new eigenvalues sit on the boundary of the essential spectrum, the projection $P$ is a bounded part of a spectral projection; quantifying how $S(\\tau)P$ decays there could yield explicit rates in the stability estimate rather than a fixed small $\\delta$ ball."],"forward_implications":["Theorem 1.1 applies to any smooth radial potential, so the dispersive part of a blowup-stability proof no longer has to be re-derived for each nonlinearity.","For the quintic equation, the theorem yields an open ball around $u_1[0]$ in the critical space whose data all follow the ODE blowup profile, with a quantitative $L^2_t L^\\infty_x$ bound and control of $\\dot W^{n,d/n}$ norms.","Because the Strichartz range matches that of the free radial wave equation, including $L^pL^\\infty$ endpoints, the estimates are strong enough to work at the scaling-critical regularity $s=(d-1)/2$.","The inhomogeneous version of the estimates provides the contraction-mapping setup used in Section 6, so the stability theorem is a direct corollary once spectral data for the linearized operator are available.","The paper states that the same proof can be modified to other nonlinearities whenever the required spectral information can be obtained."],"supporting_citations":[{"why":"Introduces Strichartz estimates in similarity coordinates and the stable-blowup strategy at the critical wave equation, which this paper extends.","marker":"[10]"},{"why":"Provides the optimal-regularity blowup stability argument and the oscillatory-integral lemmas reused in Section 5.","marker":"[13]"},{"why":"Contains the hypergeometric spectral analysis that Lemma 6.1 adapts to prove the quintic point-spectrum statement.","marker":"[16]"},{"why":"Supplies the resolvent-construction template, including the eigenvalue criterion $c_{2,4}(\\lambda)=0$ and the inhomogeneous-estimate method used in Lemma 3.7.","marker":"[17]"},{"why":"Gives Theorem 2.1, the semigroup bounds for the free similarity operator from which the perturbed operator's essential spectrum and growth bounds are derived.","marker":"[34]"},{"why":"Extends the framework to small even dimensions and provides the coefficient-patching lemma and the blowup-time variation arguments reused in Section 6.","marker":"[36]"},{"why":"Supplies Theorem B.1, used to locate the essential spectrum and isolated eigenvalues of the perturbed operator in Lemma 2.2.","marker":"[19]"},{"why":"Provides the bounded-extension lemma for finite-rank operators used to justify the projections on a dense subset in Lemma 5.13.","marker":"[18]"}],"fun_headline_variants":["Strichartz estimates for radial wave equations with any potential","Optimal blowup stability from new Strichartz estimates","Strichartz estimates unlock optimal blowup stability for wave equations","Radial waves: blowup stability from Strichartz estimates","Supercritical wave blowup stabilized by new Strichartz estimates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 6.1, that the quintic linearized operator has exactly one nonnegative eigenvalue, the simple $\\lambda=1$, with the rest of the spectrum decaying; this spectral fact is quoted from an earlier hypergeometric calculation rather than derived in the paper, and if it failed the projection step in Theorem 1.2 would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Strichartz estimates for radial wave equations with any potential","Optimal blowup stability from new Strichartz estimates","Strichartz estimates unlock optimal blowup stability for wave equations","Radial waves: blowup stability from Strichartz estimates","Supercritical wave blowup stabilized by new Strichartz estimates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001245,"raw_usage":{"total_tokens":5159,"prompt_tokens":1047,"completion_tokens":4112,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":4028}},"tokens_in":663,"tokens_out":4112,"duration_ms":25564,"temperature":1.0,"reasoning_tokens":4028,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:43:11.721425+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically locate the zeros of the eigenvalue coefficient $c_{2,4}(\\lambda)$ for the ODE (2.8) with $V=\\tfrac{15}{4}$ in a concrete dimension, say $d=3$, $s=1$; finding any eigenvalue with $\\operatorname{Re}\\lambda\\ge0$ other than the simple $\\lambda=1$ would falsify Lemma 6.1 and hence Theorem 1.2, while confirming the spectrum would support the stability claim, and a direct numerical check of the asserted endpoint Strichartz pairs for the potential-free case would test the resolvent estimates behind Theorem 1.1.","supporting_citations":[{"cited_title":"Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation","cited_arxiv_id":null,"evidence_quote":"Introduces Strichartz estimates in similarity coordinates and the stable-blowup strategy at the critical wave equation, which this paper extends."},{"cited_title":"Blowup stability at optimal regula rity for the critical wave equation","cited_arxiv_id":null,"evidence_quote":"Provides the optimal-regularity blowup stability argument and the oscillatory-integral lemmas reused in Section 5."},{"cited_title":"Stable blowup for wa ve equations in odd space dimen- sions","cited_arxiv_id":null,"evidence_quote":"Contains the hypergeometric spectral analysis that Lemma 6.1 adapts to prove the quintic point-spectrum statement."},{"cited_title":"Optimal blowup stability for s upercritical wave maps","cited_arxiv_id":null,"evidence_quote":"Supplies the resolvent-construction template, including the eigenvalue criterion $c_{2,4}(\\lambda)=0$ and the inhomogeneous-estimate method used in Lemma 3.7."},{"cited_title":"Stable blowup for focusing semilinear wave equations in all dimensions","cited_arxiv_id":null,"evidence_quote":"Gives Theorem 2.1, the semigroup bounds for the free similarity operator from which the perturbed operator's essential spectrum and growth bounds are derived."},{"cited_title":"Strichartz estimates and blowup stability for ene rgy critical nonlinear wave equa- tions","cited_arxiv_id":null,"evidence_quote":"Extends the framework to small even dimensions and provides the coefficient-patching lemma and the blowup-time variation arguments reused in Section 6."},{"cited_title":"Stable blowup for the supercritical hyperbolic Y ang-Mills equations","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem B.1, used to locate the essential spectrum and isolated eigenvalues of the perturbed operator in Lemma 2.2."},{"cited_title":"Optimal blowup stability for t hree-dimensional wave maps, 2023","cited_arxiv_id":null,"evidence_quote":"Provides the bounded-extension lemma for finite-rank operators used to justify the projections on a dense subset in Lemma 5.13."}],"review_version":1}