{"id":"11456b92-0f81-4fdd-88cd-8b293ff6e64b","arxiv_id":"1908.07079","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The higher-dimensional Benjamin-Ono equation is locally well-posed in weighted Sobolev spaces up to an optimal decay threshold, with sharp unique continuation properties.","lead":"This paper proves local well-posedness in weighted Sobolev spaces for a higher-dimensional Benjamin-Ono equation and derives sharp unique continuation properties that determine the optimal decay rate. A new commutator estimate for Riesz transforms is introduced as the key tool.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claim 1 in §5.1 rests on unproved product estimate (5.11): the weighted derivative term ||⟨x⟩^3 u ∂x1 u||_{L2} is not controlled by the bound (5.12) proved there.","rationale":"The reader's weakest-assumption identification points to the same spot: Section 5.1, equations (5.11)-(5.12). My independent check confirms that the displayed proof of (5.11) proves a bound for ||⟨x⟩^3 u^2||_{L2} and then silently substitutes it for the needed ||⟨x⟩^3 u∂x1u||_{L2}. This is not merely a typographical slip: the needed estimate has one more power of x on a derivative of u, and the two data bounds ||⟨x⟩^2u||_{L2} and ||u||_{H^{2+ε}} do not imply it by standard interpolation, as the bump-family example shows. Because Claim 1 is the mechanism that converts the second-time hypothesis u(t2) ∈ Z_{3,3}(R^2) into u0(0)=0, the failure of the product estimate would invalidate Theorem 1.2 and therefore the sharp decay conclusions drawn in Remarks (iii) and (iv). The concern is load-bearing, but it is a gap in the written proof rather than evidence that the theorem is false; the author may be able to supply a correct estimate using the equation or an additional hypothesis. This leaves the reader's CONDITIONAL verdict unchanged. The recommendation is to require the missing weighted derivative estimate or a re-derivation of Claim 1 before accepting the unique-continuation and sharp-decay conclusions.","tokens_in":43181,"tokens_out":8589,"duration_ms":84648,"concrete_test":"Recompute the passage after (5.11) with u∂x1u in place of u^2: start from (5.10) with f = u∂x1u and attempt to close the L2(⟨ξ⟩^{-4}) bound using only Lemma 2.1, (2.21), ||u||_{H^{2+ε}}, and ||⟨x⟩^2 u||_{L2}. If the chain requires ||⟨x⟩^3 u ∂x1 u||_{L2} at the outset, the proof is circular. Separately, test the asserted implication by the explicit family u_N(x) = N^{-3} φ(x_1 - N, x_2), with φ ∈ C_c^∞ used as a solution-data model: compute the ratio ||⟨x⟩^3 u_N ∂x1 u_N||_{L2} / (||⟨x⟩^2 u_N||_{L2} + ||u_N||_{H^{2+ε}}); if this ratio is unbounded as N→∞, then (5.11) does not follow from the stated hypotheses and Claim 1 requires a genuinely new argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2 for d=2 depends on Claim 1, whose integral term requires (5.11): u∂x1u ∈ L∞([0,T]; Z_{1,3}(R^2)), in particular ||⟨x⟩^3 u ∂x1 u||_{L2} < ∞. The text asserts that H^1 follows because H^2(R^2) is a Banach algebra, and then proves (5.12), the bound ||⟨x⟩^3 u^2||_{L2}. But (5.12) controls the wrong expression: ∂x1(u^2) = 2u∂x1u, and the L2 norm of ⟨x⟩^3 ∂x1(u^2) is not majorized by the L2 norm of ⟨x⟩^3 u^2 without an additional weighted H^1 estimate. The stated hypotheses u ∈ C([0,T]; Z_{2+,2}(R^2)) give ||⟨x⟩^2 u||_{L2} and ||u||_{H^{2+ε}}, but these do not by themselves imply ||⟨x⟩^3 u ∂x1 u||_{L2} < ∞; a family of unit-width bumps centered at |x| ≈ N with amplitude N^{-3} has both data norms uniformly controlled while the weighted derivative product grows like N. The same gap reappears in (6.13) and around (8.1). Since Claim 1 is what forces u0(0)=0 from the second-time hypothesis, Theorem 1.2 and the claimed optimal decay rates rest on this missing product estimate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the initial value problem for the higher-dimensional Benjamin-Ono equation ∂tu − R1∆u + u∂x1u = 0 in Rd, d = 2, 3. It claims local well-posedness in weighted Sobolev spaces Z_{s,r} and ˙Z_{s,r} (Theorem 1.1), and uses a Duhamel/Fourier-weight analysis together with a new Riesz-transform commutator estimate (Proposition 1.2) to prove unique continuation results (Theorems 1.2–1.5). These continuation results are then used to identify optimal spatial L2 decay rates d/2+2 and d/2+3 for arbitrary versus zero-mean data. The weighted local well-posedness proof is an energy method with truncated weights and the operators Γl; the continuation proofs proceed by showing that extra decay at one, two, or three times forces certain Fourier coefficients to vanish.","tokens_in":43530,"tokens_out":15521,"duration_ms":156006,"significance":"If the results are correct, the paper gives sharp spatial decay rates for a genuinely multidimensional analogue of the Benjamin-Ono equation and introduces a commutator estimate of independent harmonic-analysis interest. The proof of Theorem 1.1 in Section 4 is detailed and appears internally consistent, and the paper correctly treats the author's earlier result [14] as an independent black box rather than assuming what it proves. However, the unique continuation part contains a missing product estimate at a load-bearing point: the displayed proof of the key claim in Section 5.1 establishes a bound for u^2 where a bound for u∂x1u is needed. The same gap reappears in the proofs of Theorems 1.3 and 1.5. Until this is repaired, the claimed optimal decay conclusions are not established.","major_comments":[{"comment":"The proof of Claim 1 asserts that u∂x1u ∈ L∞([0,T]; Z_{1,3}(R2)) and then proves only the weighted L2 bound for u^2. Concretely, the displayed argument bounds ||⟨x⟩^3 u^2||_{L2}, whereas the quantity actually needed is ||⟨x⟩^3 u∂x1u||_{L2}; since ∂x1(u^2) = 2u∂x1u, the former does not control the latter without an additional weighted H1 estimate. The stated hypotheses u ∈ C([0,T]; Z_{2+,2}(R2)) do not imply this bound: taking u_N(x) = N^{-2}φ(x − Ne1) with φ a fixed unit-scale bump gives ||u_N||_{H^{2+}} and ||⟨x⟩^2u_N||_{L2} uniformly bounded while ||⟨x⟩^3u_N∂x1u_N||_{L2} grows like N. Since Claim 1 is the mechanism forcing u0(0) = 0, Theorem 1.2 and the claimed optimal decay rate for d = 2 are not established as written.","section":"§5.1, Eqs. (5.11)–(5.12)"},{"comment":"The same gap reappears in the proofs of Theorems 1.3 and 1.5. In (6.13) the assertion u∂x1u ∈ L∞([0,T]; L2(|x|^5dx)) is used for the Duhamel terms in Claim 3, but the text only justifies the corresponding weighted bound for u^2, by the same argument as (5.12). In (6.26) the assertion u∂x1u ∈ L∞([0,T]; ˙Z_{3,9/2}(R3)) is stated as following from 'a similar reasoning to (5.12)', but again only u^2-type estimates are proved. In Section 8, Eq. (8.1) asserts u∂x1u ∈ L∞([0,T]; Z_{d/2+3,d/2+3}(Rd)) from u ∈ C([0,T]; ˙Z_{s,rd}(Rd)) with s ≥ d/2+4; for d = 2 this needs, in particular, |x|^4u∂x1u ∈ L2 and u∂x1u ∈ H^4, neither of which follows from H^4 ∩ L2(|x|^6dx) by the displayed reasoning. Since Theorems 1.3 and 1.5 rely on these assertions, the three-time continuation results and the sharp decay conclusions depending on them are not justified.","section":"§6.1, Eq. (6.13); §6.2, Eq. (6.26); §8, Eq. (8.1)"}],"minor_comments":[{"comment":"In the statement of the d = 3 case of Theorem 1.2 and in the proof of Theorem 1.3, the target spaces are written as Z_{7/2,7/2}(R2) and Z_{9/2,9/2}(R2), respectively; these should be R3.","section":"§5.2 and §6.2"},{"comment":"The conclusion of Theorem 1.5 is stated for ˙Z_{d/2+3,d/2+3}(R2) although the hypotheses are written for d = 2,3; it should presumably be R^d.","section":"Theorem 1.5"},{"comment":"The appendix heading and the closing sentence refer to 'Proposition 1.5' and 'Proposition 9.1', but the result being proved is Proposition 1.2; the labels are inconsistent.","section":"Appendix"},{"comment":"Remark (iv) refers to 'Theorem 1.2 (ii)', but the sharpness statement about the decay rate d/2+3 belongs to Theorem 1.1 (ii).","section":"Remarks after Theorem 1.3"},{"comment":"In several places the variable t is written as r in the supremum, for example 'sup_{r∈[0,T]}' after (4.4); this conflicts with the decay parameter r used throughout the section.","section":"§4, after Eq. (4.4)"},{"comment":"The estimate in (5.21) reduces ∂ξ^m(ξ1u2)φ to weighted u^2 because φ has compact support in frequency, which allows the factor ξ1 to be absorbed; this should be stated explicitly, since it is the reason the d = 3 Claim 2 avoids the difficulty in (5.11).","section":"§5.2, Eq. (5.21)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the author's earlier paper [14] as a black box; this is not circular, but the editor may wish to confirm that [14] is available and accepted. The missing product estimate affects the central continuation theorems, and if it cannot be repaired, the sharp decay claims in the abstract are unsupported. The weighted local well-posedness part, however, appears to be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Riaño's paper has two distinct parts. The first, Theorem 1.1 plus Proposition 1.2, is a genuine contribution: weighted local well-posedness for the higher-dimensional Benjamin-Ono equation with decay r up to d/2+2 (and d/2+3 with zero mean), and a Calderón-type commutator estimate for Riesz transforms that looks correct. The proof of Theorem 1.1 is long but careful, and the use of the author's earlier H^s well-posedness [14] as a black box is fine—it is an independent prior theorem. I would send this part to the journals on its own.\n\nThe problem is in the unique continuation theorems. Theorem 1.2 depends on Claim 1 in Section 5.1, which requires u∂x1u ∈ L∞([0,T]; Z_{1,3}(R^2)). The displayed argument proves (5.12), a bound for ||⟨x⟩^3 u^2||_{L2}. That is a different object. To get the weighted derivative term you need ||⟨x⟩^3 ∂x1(u^2)||_{L2}, or equivalently a weighted H^1 estimate for u^2, and no such estimate is derived. The hypotheses u ∈ Z_{2+,2} do not by themselves give it; interpolation between ⟨x⟩^2 u and H^{2+ε} does not produce a weight on the derivative. The same missing product estimate reappears in (6.13) and (8.1). Without it, the conclusion u0(0) = 0 in Theorem 1.2, and the sharp decay statements that follow, are unsupported. The three-times argument in Theorem 1.3 inherits the same gap.\n\nTo be fair, the stress-test counterexample as written has a scaling that doesn't grow; but the missing inequality is real. The author may be able to prove it with additional effort—for instance, a weighted Sobolev embedding or a different splitting of the Duhamel term—but it is not in the paper. The appendix proof of Proposition 1.2, by contrast, is standard paraproduct business and convincing.\n\nBottom line: this is a paper for dispersive PDE people who care about weighted LWP and unique continuation. Theorem 1.1 and the commutator estimate deserve citation and further use. The continuation part is, at this moment, conditional. Send it to a serious referee; a good referee will identify the same gap. If the author fixes the product estimate, the paper would be a solid contribution. As it stands, the main uniqueness results are not established.\n\nRecommendation: engage with the paper, reject in current form but encourage revision.","headline":"Solid weighted LWP and a new Riesz commutator estimate, but the sharp decay theorems hinge on an unproved weighted product bound.","tokens_in":44033,"tokens_out":7314,"would_cite":true,"duration_ms":68521,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q53","35A01","35B30","42B20","35B60"],"pacs":[],"model":"deepseek-v4-flash","headline":"In d=2,3, weighted local well-posedness holds up to decay d/2+2, or d/2+3 with zero mean; endpoint decay at two or three times forces vanishing.","keywords":["higher-dimensional Benjamin–Ono equation","weighted Sobolev spaces","Riesz transform","commutator estimate","local well-posedness","unique continuation","optimal decay rate","zero mean data"],"falsifier":"In dimension $d=2$, solve the equation numerically from a smooth compactly supported datum in $Z_{2+,2}(\\mathbb{R}^2)\\setminus Z_{3,3}(\\mathbb{R}^2)$ over the existence time and monitor $F(t)=\\|\\langle x\\rangle^3 u(t)\\partial_{x_1}u(t)\\|_{L^2}$. If $F$ becomes unbounded while $\\|\\langle x\\rangle^3 u(t)^2\\|_{L^2}$ stays finite, the Duhamel estimate in Claim 1 of Section 5.1 fails; if $F$ stays bounded, that gap is closed.","tokens_in":42980,"feed_emoji":"📉","tokens_out":15212,"duration_ms":133161,"temperature":0.7,"pith_summary":"The paper studies the initial value problem for the higher-dimensional Benjamin-Ono equation $\\partial_t u - R_1\\Delta u + u\\partial_{x_1}u = 0$ in $\\mathbb{R}^d$, $d\\ge 2$, where $R_1$ is the Riesz transform in the first coordinate. Its main claim is that for $d=2,3$ the problem is locally well-posed in the weighted Sobolev spaces $Z_{s,r}=H^s\\cap L^2(|x|^{2r}dx)$ for every $s$ above the Sobolev threshold and $r<d/2+2$, and in the zero-mean spaces $\\dot Z_{s,r}$ up to $r<d/2+3$. It then proves continuation principles showing these decay exponents are optimal: a solution that reaches decay $d/2+2$ at two different times must have vanishing zero Fourier mode, and one that reaches decay $d/2+3$ at three different times must vanish identically. If correct, this settles the optimal polynomial $L^2$ decay rate for the model and extends to several variables the known one-dimensional Benjamin-Ono phenomenon that polynomial decay is not preserved by the flow beyond a sharp threshold.","feed_headline":"Sharp decay rates pinned down for higher-dimensional Benjamin-Ono","feed_subtitle":"In dimensions 2 and 3, decay beyond d/2+2 is impossible; three times at the endpoint force u≡0.","key_machinery":"The load-bearing object is Proposition 1.2, a new commutator estimate for Riesz transforms: for $1<p<\\infty$ and any multi-index $\\alpha$ with $|\\alpha|\\ge 1$, the difference $R_l(a\\partial^\\alpha f)-aR_l\\partial^\\alpha f$ minus the lower-order Taylor terms with the auxiliary Fourier-multiplier operators $D^\\beta_{R_l}$ is controlled by $\\sum_{|\\beta|=|\\alpha|}\\|\\partial^\\beta a\\|_{L^\\infty}\\|f\\|_{L^p}$. This estimate transfers derivatives onto the weight in the energy method, exchanging polynomial decay for Sobolev regularity. The proof combines it with the $A_2$-weighted boundedness of Riesz transforms uniform in the truncation parameter $N$, an interpolation inequality for $J^a(\\langle x\\rangle^b f)$, and the known $H^s$ local theory. For the continuation theorems, the machinery is a family of frequency-domain operators $F^j_k(t,\\xi,f)$ tracking derivatives of the phase $e^{it\\xi_1|\\xi|}$; the failure of $\\partial^3_{\\xi_k}(\\xi_1|\\xi|)$ or $\\partial^4_{\\xi_k}(\\xi_1|\\xi|)$ to lie in $L^2$ (respectively $H^{1/2}$) near the origin is the mechanism forcing the zero-mode or full-vanishing conclusions.","core_discovery":"The paper's central claim, Theorem 1.1, is local well-posedness in $Z_{s,r}(\\mathbb{R}^d)$ for $d=2,3$, with $s>s_d$ ($s_2=5/3$, $s_3=2$) and $0\\le r<d/2+2$, $r\\le s$; under the zero-mean condition $\\hat u_0(0)=0$, the same result holds in $\\dot Z_{s,r}(\\mathbb{R}^d)$ for $0\\le r<d/2+3$. Theorems 1.2 and 1.3 turn the endpoint weights into rigidity statements: membership in $Z_{d/2+2,d/2+2}(\\mathbb{R}^d)$ at two distinct times forces $\\hat u_0(0)=0$, and membership in $Z_{d/2+3,d/2+3}(\\mathbb{R}^d)$ at three distinct times forces $u\\equiv 0$. The paper reads these as sharpness of Theorem 1.1: nontrivial solutions cannot keep $|x|^{d/2+2}u(t)$ in $L^2$ over a time interval, and nontrivial zero-mean solutions cannot keep $|x|^{d/2+3}u(t)$ in $L^2$, so the decay exponents in the well-posedness theorem are the best possible.","pith_inferences":["Beyond the paper, the commutator estimate of Proposition 1.2 should transfer to any dispersive equation whose dispersion is a Riesz transform of a Laplacian, so a testable extension is weighted well-posedness for Zakharov-Kuznetsov-type variants in dimensions $d\\ge 2$.","The proof isolates the obstruction to extra decay at the zero Fourier frequency, which suggests a quantitative conjecture: the shape of $\\hat u(t,\\xi)$ near $\\xi=0$ controls exactly how much polynomial tail a solution can carry; the explicit time $t^*$ in Theorem 1.5 gives a concrete place to test this numerically.","The missing weighted product bound for $u\\partial_{x_1}u$ in Section 5.1 is the one fragile step in the sharpness argument; if a counterexample with $\\|\\langle x\\rangle^3 u\\partial_{x_1}u\\|_{L^2}=\\infty$ under $u\\in Z_{2+,2}(\\mathbb{R}^2)$ exists, the continuation conclusions may still hold but would need a different Duhamel estimate."],"forward_implications":["For $d=2,3$, initial data in $Z_{s,r}$ with $r<d/2+2$ propagate their polynomial decay, and no nontrivial solution keeps $|x|^{d/2+2}u(t)$ in $L^2$ over a time interval unless $\\hat u_0(0)=0$.","For zero-mean data, decay up to $r<d/2+3$ is propagated; hitting the endpoint at three distinct times forces $u\\equiv 0$, so no nontrivial zero-mean solution sustains $|x|^{d/2+3}$ decay.","A two-time endpoint condition plus the vanishing of $\\int x_1u$ at one of the two times forces $u\\equiv 0$, whereas without that moment condition a nontrivial solution can reach the endpoint exactly once, at the time $t^*=-4\\|u_0\\|_{L^2}^{-2}\\int x_1u_0\\,dx$.","The results generalize the sharp decay theory of the one-dimensional Benjamin-Ono equation to higher dimensions, with the allowable decay exponent increasing by $1/2$ per added dimension and the same two-versus-three time structure."],"supporting_citations":[{"why":"Provides the local well-posedness theory in $H^s(\\mathbb{R}^d)$ and the smooth approximating solutions that justify the weighted energy estimates.","marker":"[14]"},{"why":"Supplies the one-dimensional weighted-space method and sharp decay conclusions that this paper extends to higher dimensions.","marker":"[10]"},{"why":"Gives the dispersion-generalized Benjamin-Ono weighted method whose arguments are adapted in the proof of Theorem 1.1.","marker":"[9]"},{"why":"Contains the classical first commutator estimate that Proposition 1.2 generalizes to Riesz transforms.","marker":"[3]"},{"why":"Provides the weighted-space energy estimate pattern used for Proposition 1.1.","marker":"[6]"},{"why":"Establishes the sharp weighted $A_2$ bound for Riesz transforms that makes the weighted estimates independent of the truncation parameter $N$.","marker":"[19]"},{"why":"Improves the Sobolev local theory in dimension two, setting the regularity threshold $s_2=5/3$.","marker":"[21]"},{"why":"Gives the fractional Sobolev characterization used in interpolation and product estimates.","marker":"[22]"}],"fun_headline_variants":["Higher-D Benjamin-Ono: optimal decay established","New commutator estimate locks sharp BO decay","2D/3D BO: endpoint weights force rigidity","Three critical-time memberships imply zero solution","Decay rate d/2+2 is optimal for Benjamin-Ono"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharp decay conclusions rely on the product $u\\partial_{x_1}u$ lying in the weighted space $Z_{1,3}(\\mathbb{R}^2)$ under the stated solution regularity; the paper proves the bound $\\|\\langle x\\rangle^3 u^2\\|_{L^2}<\\infty$ but not the needed $\\|\\langle x\\rangle^3 u\\partial_{x_1}u\\|_{L^2}<\\infty$, and that missing bound does not obviously follow from $u\\in Z_{2+,2}(\\mathbb{R}^2)$.","fun_headline_variants_meta":{"raw":{"variants":["Higher-D Benjamin-Ono: optimal decay established","New commutator estimate locks sharp BO decay","2D/3D BO: endpoint weights force rigidity","Three critical-time memberships imply zero solution","Decay rate d/2+2 is optimal for Benjamin-Ono"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000339,"raw_usage":{"total_tokens":1852,"prompt_tokens":909,"completion_tokens":943,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":867}},"tokens_in":525,"tokens_out":943,"duration_ms":10093,"temperature":1.0,"reasoning_tokens":867,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:29:09.087795+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In dimension $d=2$, solve the equation numerically from a smooth compactly supported datum in $Z_{2+,2}(\\mathbb{R}^2)\\setminus Z_{3,3}(\\mathbb{R}^2)$ over the existence time and monitor $F(t)=\\|\\langle x\\rangle^3 u(t)\\partial_{x_1}u(t)\\|_{L^2}$. If $F$ becomes unbounded while $\\|\\langle x\\rangle^3 u(t)^2\\|_{L^2}$ stays finite, the Duhamel estimate in Claim 1 of Section 5.1 fails; if $F$ stays bounded, that gap is closed.","supporting_citations":[{"cited_title":"On a higher dimensional version of the Benjamin--Ono equation","cited_arxiv_id":"1901.04817","evidence_quote":"Provides the local well-posedness theory in $H^s(\\mathbb{R}^d)$ and the smooth approximating solutions that justify the weighted energy estimates."},{"cited_title":"The IVP for the Benjamin-Ono equation in weighted Sobolev spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional weighted-space method and sharp decay conclusions that this paper extends to higher dimensions."},{"cited_title":"The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces","cited_arxiv_id":null,"evidence_quote":"Gives the dispersion-generalized Benjamin-Ono weighted method whose arguments are adapted in the proof of Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the classical first commutator estimate that Proposition 1.2 generalizes to Riesz transforms."},{"cited_title":"The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in weighted Sobolev spaces","cited_arxiv_id":null,"evidence_quote":"Provides the weighted-space energy estimate pattern used for Proposition 1.1."},{"cited_title":"The sharp weighted bound for the Riesz transforms","cited_arxiv_id":null,"evidence_quote":"Establishes the sharp weighted $A_2$ bound for Riesz transforms that makes the weighted estimates independent of the truncation parameter $N$."},{"cited_title":"On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations","cited_arxiv_id":"1903.02027","evidence_quote":"Improves the Sobolev local theory in dimension two, setting the regularity threshold $s_2=5/3$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the fractional Sobolev characterization used in interpolation and product estimates."}],"review_version":1}