{"id":"dc937953-b595-4cf3-ae67-e09756e75482","arxiv_id":"1908.08487","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For d=1,2, the centered maximal operator satisfies ||Mf||_p ≥ (1+ε(K,p))||f||_p for all p>1, and for generic shapes in d≥3 the fixed-point threshold exceeds that of the ball.","lead":"This mathematics paper proves that in one and two dimensions, the centered Hardy-Littlewood maximal operator always inflates the L^p norm of a positive function by a factor strictly bigger than 1. In higher dimensions, it shows that for most shapes, the threshold where the operator gains fixed points is higher than for a ball.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 is false as stated for signed f; the proof requires f≥0 and fails for f=-1_B.","rationale":"Most of the paper's mathematics appears sound: the mollification argument correctly forces L∞ fixed points to be superharmonic, and in d=1,2 bounded superharmonic functions are constant, so Theorem 3(1) rests on Korry's classification in a reasonable way. The d≥3 generic result is plausible, with the sphere-averaging argument for the quartic term being a genuine insight. The reader's weakest assumption (the L∞ fixed-point classification) is less risky than the sign issue, because it is supported by standard Liouville facts, whereas the theorem statement is literally false for signed functions. The sign defect is not fatal to the intended result but is load-bearing: it invalidates the central claim as written and must be repaired by stating the theorem for nonnegative f or for M(|f|). Thus the conditional acceptance verdict is unchanged, but the required condition should explicitly include nonnegativity or absolute value.","tokens_in":5865,"tokens_out":40724,"duration_ms":407959,"concrete_test":"Evaluate the proof on f = −1_{B(0,1)} for a centrally symmetric convex body K in d=1 or d=2. For every x, the average of f over x+λK is −|(x+λK)∩B(0,1)|/|λK| ≤ 0, and as λ→∞ this quantity approaches 0 from below, so the supremum is 0; hence Mf≡0 and ||Mf||_p=0 while ||f||_p>0, directly contradicting Theorem 1 as printed. This single example settles that the theorem statement must add a nonnegativity or absolute-value hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem and abstract claim ||Mf||_p ≥ (1+ε)||f||_p for all f, but the proof establishes this only for nonnegative f. The step after choosing S_i with average μ(1−δ1) — 'Since K is convex, K+δ1K⊆(1+δ1)K. By using this fact and taking slight rescalings and shifts of S_i, we get that Mf ≥ μ(1−δ1)/(1+δ1)^2 on x_i+δ1λ_iK' — requires the integral of f over the region between S_i and the enlarged set to be nonnegative; for signed f that contribution can be negative and destroy the lower bound. The final step ∫(Mf)^p ≥ ∫f^p + ∫(Mf−f)^p also assumes Mf≥f pointwise, which is false for signed f. Concretely, for f = −1_{B(0,1)}, every centered average is ≤0 and tends to 0 as λ→∞, so Mf≡0 and ||Mf||_p=0 < ||f||_p for every p>1. The central argument is therefore an argument for nonnegative functions, and the paper should either define M on |f| or restrict Theorem 1 to f≥0.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the centered Hardy-Littlewood maximal operator associated to a centrally symmetric convex body K, defined in Eq. (1) without an absolute value. The main result, Theorem 1, claims that for d=1,2 and 1<p<∞ there exists ε(K,p)>0 such that ||Mf||_p ≥ (1+ε)||f||_p for all f in Lp. Theorem 3 asserts that in d=1,2 the only L∞ fixed points of M are constant functions, while in d≥3 there are no Lp fixed points for p≤d/(d−2) and, for generic shapes, no fixed points for a range of p above that threshold, so that q0(K)>q0(B(0,1)). The proof of Theorem 1 combines a Besicovitch covering argument with iteration of the maximal operator and a contraction argument borrowed from [2]; the fixed-point results use mollification, Taylor expansion, and maximum-principle arguments. The paper is clearly written and the technical machinery is standard, but the statement of Theorem 1 is false for sign-changing functions as written.","tokens_in":6132,"tokens_out":27693,"duration_ms":259899,"significance":"If Theorem 1 is read as a statement about nonnegative functions, the paper gives a substantial improvement over the known range for the centered maximal operator: the lower bound is obtained for all p>1 in dimensions one and two, rather than only for p close to 1. The contraction argument and the Besicovitch covering step are clean and appear reproducible. The fixed-point genericity result for d≥3, if fully justified, is also new and interesting, and the quartic-moment criterion is explicit enough to be checked. However, the paper as written states Theorem 1 for arbitrary f, which is false; the proof works only for nonnegative f. This must be corrected before the result can be assessed as stated. The advertised generic-shape conclusion in Theorem 3(2) also needs a proof of genericity rather than an assertion.","major_comments":[{"comment":"The definition of M in (1) lacks an absolute value and Theorem 1 is stated for arbitrary f∈Lp. This is false as stated: for f=−1_{B(0,1)}, every centered average is ≤0 and tends to 0 as λ→∞, so Mf≡0 and ||Mf||_p=0 for every 1<p<∞, while ||f||_p>0. The proof uses nonnegativity essentially: the step \"Since K is convex ... Mf ≥ μ(1−δ1)/(1+δ1)^2 on x_i+δ1λ_iK\" requires averages over the enlarged convex sets to be at least the average over S_i, and the final inequality ∫(Mf)^p ≥ ∫f^p+∫(Mf−f)^p requires Mf≥f pointwise. Please restrict Theorem 1 and the abstract to nonnegative f, or redefine M with |f| and rework the argument; the counterexample shows the present statement cannot stand.","section":"§1, Eq. (1); §2, proof of Theorem 1"},{"comment":"The theorem asserts that for generic centrally symmetric convex bodies K in d≥3, q0(K)>q0(B(0,1)), but the proof establishes only the conditional statement: if the quartic moment ∫_K ∑ a_{ijkl} x_i x_j x_k x_l is nonzero, then no fixed points exist for some range p≤q with q>d/(d−2). The sentence \"The condition in the statement above is generic\" is not proved; the space of shapes, the topology, and the argument that the failure set is negligible (for example, contained in a proper algebraic hypersurface) should be given. In addition, the coefficients a_{ijkl} are described only by reference to derivatives of the Green's function, so an explicit formula or a precise reference would make the condition checkable. Without these additions, the advertised generic-shape conclusion is unsupported.","section":"§3, Theorem 3(2) and following paragraph"}],"minor_comments":[{"comment":"The passage \"lim_{n→∞} M^n1_{δ1K} is a constant function (by Theorem 1)\" should cite Theorem 3(1); as printed, it makes the proof of Theorem 1 appear circular.","section":"§2, proof of Theorem 1"},{"comment":"The mollification argument shows Mg̃≤g̃, not equality; the sentence \"so g̃ is also a fixed point of M\" should be reworded to \"so g̃ satisfies Mg̃≤g̃\", since the subsequent expansion only needs the inequality.","section":"§2, fixed-point classification"},{"comment":"The statement \"by choosing n large enough\" in the passage on M^{n−1}1_{δ1K} needs an explicit appeal to Egorov's theorem to justify uniform convergence off a small set on the compact set 2K.","section":"§2, Egorov step"},{"comment":"The phrase \"Iterating (6)\" is terse; a scaling argument for B(0,r) would clarify why the limit function is not in Lp.","section":"§3, after Eq. (6)"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is currently overstated: Theorem 1 is false for sign-changing functions because the maximal operator is defined without an absolute value. The proof for nonnegative f is coherent, and the fixed-point arguments are plausible, but the authors should be asked to restrict the statement to nonnegative functions and to supply a proof of the genericity claim in Theorem 3(2). With those revisions, the paper would be a solid contribution to the harmonic analysis literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe genuinely new content here is the all-p lower bound for nonnegative functions in d=1,2 and the generic-shape fixed-point result in d≥3. The second theorem, Theorem 3(2), is the more interesting piece: it uses a quartic moment condition from the Green's function expansion to show that for most symmetric convex bodies the Lp fixed-point threshold sits strictly above d/(d-2). That is a real step beyond the ball case, and the argument from superharmonicity plus the boundary comparison on the annulus is coherent. The author also gives credit where the argument is borrowed: Korry's classification, the Ivanisvili–Jaye–Nazarov telescoping trick, and the earlier Zbarsky–Ivanisvili one-dimensional result are all cited.\n\nThe main theorem, however, is stated more broadly than it is proved. Definition (1) defines M without an absolute value, and Theorem 1 asserts the lower bound for all f in Lp. That is false for signed f: take f = -1_{B(0,1)}; every centered average is non-positive and tends to 0, so Mf ≡ 0 and the inequality fails for every p>1. The proof only works for nonnegative f. The mistake appears in the step after choosing the sets S_i, where the inequality Mf ≥ μ(1-δ1)/(1+δ1)^2 requires the integral of f over the difference region to be nonnegative, and later the step ∫(Mf)^p ≥ ∫f^p + ∫(Mf-f)^p uses Mf ≥ f pointwise. Both are fine when f ≥ 0 and false in general. The fix is easy: either define M with |f| in the usual way, in which case the theorem is plausible and the proof can be adapted in the obvious places, or restrict the theorem to nonnegative f. Either way the abstract and title need to say which.\n\nA couple of smaller things. The convergence of the iterates M^n 1_{δK} to the constant 1 is asserted through Theorem 3(1), whose L∞ fixed-point classification is sketched rather than fully proved; the dependence of n on K and δ is not tracked. That is probably fixable, but the proof would be cleaner with a quantitative statement. The Besicovitch lemma in the appendix is a straightforward modification, fine. The quartic term calculation is terse; I'd want to see the Taylor expansion and the average over the sphere written out, but the idea is sound.\n\nThis paper deserves a serious referee. The fixed-point result for generic shapes is a genuine contribution, and the d=1,2 lower bound is valuable once it is stated for nonnegative or for |f|. I would not cite it in its current form; I would after the revision. Bring it to the reading group if you want a concrete example of a theorem statement running ahead of its proof.","headline":"A genuinely new fixed-point result for generic shapes and a nearly-true lower bound that is misstated for signed functions; fix the statement and it deserves refereeing.","tokens_in":6642,"tokens_out":2315,"would_cite":false,"duration_ms":21770,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B25","35J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $d=1$ or $d=2$ and every $p>1$, the centered Hardy--Littlewood maximal operator strictly raises the $L^p$ norm of every nonnegative nonzero function.","keywords":["centered Hardy-Littlewood maximal operator","fixed points","Lp lower bounds","convex bodies","superharmonic functions","Green's function expansion","Besicovitch covering lemma","fourth moments"],"falsifier":"Search in $\\mathbb{R}^2$ for a centrally symmetric convex body $K$ and a bounded nonconstant $g$ with $Mg=g$: such a pair would contradict Theorem 3(1) and invalidate the proof of Theorem 1. A numerical check is to iterate $M^n\\mathbf{1}_{\\delta K}$ for a non-ellipsoidal $K$ in $\\mathbb{R}^2$ and look for a nonconstant limiting profile, which would falsify the classification the theorem relies on.","tokens_in":5691,"feed_emoji":"📈","tokens_out":15467,"duration_ms":132919,"temperature":0.7,"pith_summary":"The paper proves two claims about the centered Hardy--Littlewood maximal operator $M$, defined by averaging over all scales and positions of a fixed centrally symmetric convex body $K$. First, when $d=1$ or $d=2$ and $1<p<\\infty$, for every nonnegative $f$ there is a constant $\\varepsilon=\\varepsilon(K,p)>0$ with $\\|Mf\\|_p \\ge (1+\\varepsilon)\\|f\\|_p$. Second, when $d\\ge 3$ and $K$ satisfies an explicit nonzero fourth-moment condition (a generic condition), the infimum exponent $q_0(K)$ at which $M$ has an $L^p$ fixed point is strictly larger than the ball's value $q_0(B(0,1))=d/(d-2)$. The interest is that these results tie a strict lower bound on the operator norm to the absence of nonzero fixed points, and they show the ball is not the extremal shape for the onset of fixed points.","feed_headline":"Centered maximal operator strictly expands L^p norms in d=1,2","feed_subtitle":"In d=1,2 every nonzero function gains a factor 1+ε in L^p; generic shapes in d≥3 beat the ball's threshold.","key_machinery":"The load-bearing object is the pointwise limit $f_*=\\lim_{n\\to\\infty} M^n \\mathbf{1}_{\\delta_1 K}$ of iterates of the maximal operator applied to a small indicator shape; any nonzero $L^p$ fixed point of $M$ lies above a translate of this limit, so the limit's geometry controls every fixed point. In $d=1,2$ the proof mollifies a fixed point $g$ to a smooth fixed point $\\tilde g$ and expands at small scales, $M\\tilde g(x)\\ge \\tilde g(x)+\\frac{\\lambda^2}{2}\\Delta \\tilde g(x)+O(\\lambda^3)$; the linear term disappears because $K$ is centered and normalized by $\\int_K x_i x_j=\\delta_{ij}$, so any point with $\\Delta\\tilde g(x)>0$ would violate the fixed-point inequality. Hence $\\tilde g$ is superharmonic, and an existing classification argument shows a bounded superharmonic fixed point in $d=1,2$ must be constant. In $d\\ge 3$ the same mollification shows fixed points are superharmonic and gives the lower bound $g\\ge |x|^{2-d}$ on a large annulus; the refined step expands $Mh$ for $h(x)=|x|^{2-d}$ to fourth order, producing the term $\\frac{\\lambda^4}{24|K|}\\sum_{i,j,k,\\ell}\\partial_{ijk\\ell}h \\int_K x_i x_j x_k x_\\ell$. When that coefficient is nonzero, one maximal step pushes $Mh$ strictly above $h$ somewhere on the sphere $|x|=3$, and a harmonic comparison then improves the critical exponent beyond $d/(d-2)$.","core_discovery":"On the paper's own terms, the discovery is a pair of theorems. Theorem 1: in dimensions $d=1,2$, for every $1<p<\\infty$ and every centrally symmetric convex body $K$, there is $\\varepsilon=\\varepsilon(K,p)>0$ such that $\\|Mf\\|_p\\ge (1+\\varepsilon)\\|f\\|_p$ for all nonnegative $f$. The proof also gives Theorem 3(1): in those dimensions the only $L^\\infty$ fixed points of $M$ are constants, so in particular $M$ has no nonzero $L^p$ fixed points for any $1<p<\\infty$. Theorem 3(2): for $d\\ge 3$, if the fourth-moment integral $\\int_K \\sum_{i,j,k,\\ell} a_{ijk\\ell} x_i x_j x_k x_\\ell$ is nonzero, where $a_{ijk\\ell}$ are coefficients coming from fourth derivatives of the Green's function for Laplace's equation, then there is some $q>d/(d-2)$ such that $M$ has no fixed points in $L^p$ for $p\\le q$. This condition holds for generic $K$ and in particular for the cube and the cross-polytope, so for generic shapes $q_0(K)>q_0(B(0,1))$.","pith_inferences":["Going beyond the paper, the same iteration argument suggests a testable numerical program: for a given $K$ in $\\mathbb{R}^2$, measure the rate at which $M^n\\mathbf{1}_{\\delta K}$ approaches the constant function; extracting that rate could convert the existence of $\\varepsilon(K,p)$ into quantitative asymptotics that the paper does not provide.","My inference is that the lower-bound property (2) is governed mainly by the absence of $L^\\infty$ fixed points rather than by shape-specific details; if so, any construction of a counterexample in higher dimensions should focus on producing a nonconstant bounded fixed point.","In higher dimensions, the fourth-moment condition is only the first member of an infinite hierarchy described in the paper; a natural conjecture extending Theorem 3(2) is that $q_0(K)$ is determined by the first nonzero even-order coefficient in the Taylor expansion of the Green's function, with each nonzero coefficient pushing $q_0$ above the ball's value."],"forward_implications":["In $d=1,2$, the centered maximal operator has no nonzero $L^p$ fixed points for any $1<p<\\infty$; in particular the equality $\\|Mf\\|_p=\\|f\\|_p$ can hold only for $f=0$, and every nonnegative nonzero function is strictly expanded.","For any fixed $K$ in $d=1,2$ and any fixed $p$, the proof gives a computable $\\varepsilon(K,p)>0$; it does not, however, give asymptotics as $p$ varies, and the question of whether $\\varepsilon$ can be chosen independent of $K$ is left open.","For $d\\ge 3$, any shape satisfying the fourth-moment condition has an interval $(d/(d-2),q)$ free of $L^p$ fixed points, so $q_0(K)$ is strictly above the ball's critical exponent $d/(d-2)$.","The mechanism also shows that any shape with $q_0(K)=d/(d-2)$ would have to make all the higher-order moment conditions vanish, which singles out shapes whose even spherical harmonic coefficients are zero as the only potential exceptions to the generic conclusion."],"supporting_citations":[{"why":"Supplies the mollification and Taylor-expansion technique, and the uncentered lower bound that the proof adapts to the centered case.","marker":"[1]"},{"why":"Establishes the one-dimensional lower bound for p<1.5 and supplies the argument that converts a norm gain for M^{n+1} into a strict gap for M.","marker":"[2]"},{"why":"Provides the classification of fixed points in L∞ that the d=1,2 proof relies on, and the fixed-point characterization for balls.","marker":"[3]"},{"why":"Gives the Besicovitch covering lemma used to extract the countable subcollection with bounded overlap in the proof of Theorem 1.","marker":"[5]"}],"fun_headline_variants":["In 1D and 2D, centered maximal operator expands every L^p norm","In low dimensions, centered maximal operator has no L^p fixed points","Generic shapes raise the q0 threshold above the ball's for d≥3","Every nonzero function gains a factor >1 under M in d=1,2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise in dimensions $1$ and $2$ is that the only $L^\\infty$ fixed points of the centered maximal operator are constant functions; if a nonconstant bounded fixed point existed for some convex shape, the iteration limit $\\lim_{n\\to\\infty} M^n\\mathbf{1}_{\\delta_1 K}=1$ used throughout the lower-bound proof would fail.","fun_headline_variants_meta":{"raw":{"variants":["In 1D and 2D, centered maximal operator expands every L^p norm","In low dimensions, centered maximal operator has no L^p fixed points","Generic shapes raise the q0 threshold above the ball's for d≥3","Every nonzero function gains a factor >1 under M in d=1,2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000998,"raw_usage":{"total_tokens":4229,"prompt_tokens":953,"completion_tokens":3276,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":3190}},"tokens_in":569,"tokens_out":3276,"duration_ms":20550,"temperature":1.0,"reasoning_tokens":3190,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:39:06.432510+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search in $\\mathbb{R}^2$ for a centrally symmetric convex body $K$ and a bounded nonconstant $g$ with $Mg=g$: such a pair would contradict Theorem 3(1) and invalidate the proof of Theorem 1. A numerical check is to iterate $M^n\\mathbf{1}_{\\delta K}$ for a non-ellipsoidal $K$ in $\\mathbb{R}^2$ and look for a nonconstant limiting profile, which would falsify the classification the theorem relies on.","supporting_citations":[{"cited_title":"Ivanisvili, B","cited_arxiv_id":null,"evidence_quote":"Supplies the mollification and Taylor-expansion technique, and the uncentered lower bound that the proof adapts to the centered case."},{"cited_title":"Ivanisvili and S","cited_arxiv_id":null,"evidence_quote":"Establishes the one-dimensional lower bound for p<1.5 and supplies the argument that converts a norm gain for M^{n+1} into a strict gap for M."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classification of fixed points in L∞ that the d=1,2 proof relies on, and the fixed-point characterization for balls."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Besicovitch covering lemma used to extract the countable subcollection with bounded overlap in the proof of Theorem 1."}],"review_version":1}