{"id":"4a97c442-917e-4bcb-a5ed-705020cb1f17","arxiv_id":"2505.16110","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Higher-order difference level sets characterize higher-order homogeneous Sobolev norms in ball Banach function spaces, with sharp parameter ranges.","lead":"This paper proves a higher-order version of the Brezis-Seeger-Van Schaftingen-Yung formula: the weak size of level sets of k-th order finite differences is equivalent to the norm of the k-th order gradient in a broad family of function spaces. It also derives Sobolev-space characterizations and critical Gagliardo-Nirenberg inequalities from that equivalence.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Endpoint p=1 upper bound in Theorem 1.1(II) is delegated to a prior theorem and is not covered by Proposition 4.1, which assumes the stronger condition M bounded on X' rather than endpoint boundedness.","rationale":"The reader's weakest_assumption identified the maximal-function/extrapolation hypothesis and the endpoint p=1 case. I agree, and I would sharpen it: the most vulnerable point is the p=1 branch of Theorem 1.1(II), where the proof literally cites [23, Theorem 4.10] without stating the endpoint extrapolation lemma. Proposition 4.1, the only upper estimate stated in this paper for ball Banach function spaces, assumes M bounded on (X^{1/p})' for some p∈[1,∞); at p=1 this is M bounded on X', which is stronger than the endpoint boundedness of Definition 2.6. Lemmas 4.5 and 4.6, used to pass from A1-weighted estimates to X-norm estimates, also rely on a finite operator norm ||M||_{X'→X'}, which endpoint boundedness alone does not provide. The concern is not that the theorem is false; it is that a load-bearing proof step is missing. For p∈(1,∞), the reduction to the detailed weighted estimates of Section 3 through Lemmas 4.5–4.6 is coherent, and Theorems 3.1 and 3.3 are written out in full. But the endpoint case is asserted by reference, not demonstrated, so the CONDITIONAL verdict already given by the reader is exactly right. No adjustment is needed.","tokens_in":69476,"tokens_out":10753,"duration_ms":88603,"concrete_test":"Write out the endpoint upper estimate for Theorem 1.1(II) directly from Definition 2.6: attempt to prove sup_λ λ ||(∫ 1_{|Δ_h^k f(·)|>λ|h|^{k+γ/q}} |h|^{γ-n} dh)^{1/q}||_X ≲ || |∇^k f| ||_X using only endpoint boundedness of M on X', uniform ball averages, and AC norm. In particular, verify whether the proof of [23, Theorem 4.10] carries over verbatim when E_f(λ,q) is replaced by E_{λ,γ/q,k}[f]; if it requires M bounded on X' or an extra interpolation/limiting lemma that is not stated here, Theorem 1.1(II) is unsupported as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1(II), the endpoint p=1 case with sharp gamma-range Γ_{1,q}=(-∞,-q)∪(0,∞), is a single sentence: 'can be obtained by repeating the proof of [23, Theorem 4.10] with E_f(λ,q), |∇f|, and Theorem 4.5 therein replaced, respectively, by E_{λ,γ/q,k}[f], |∇^k f|, and (I) here.' But Proposition 4.1, the only BBF-space upper estimate proved in this manuscript, assumes the existence of p∈[1,∞) with M bounded on (X^{1/p})'; at p=1 that is exactly M bounded on X'. The hypotheses of Theorem 1.1(II) are weaker: Definition 2.6 supplies only a sequence θ_m→1 with M bounded on (X^{1/θ_m})' and uniformly bounded norms, together with uniform boundedness of centered ball averages and an absolutely continuous norm. Nothing in the text proves these imply M bounded on X', nor does the text supply the endpoint analogue of Lemmas 4.5–4.6 needed to turn A1-weighted estimates into X-norm estimates. The substitution into [23, Theorem 4.10] is not automatic because the functional in Theorem 1.1 uses kth differences and an h-integral over the level set E_{λ,γ/q,k}[f]; the endpoint extrapolation for exactly this functional is the missing step. Thus the upper estimate in (1.6) for p=1, and hence the limiting identity (1.8) in Theorem 1.1(II) and the p=1 applications in Section 5, rest on an unstated lemma. This is a real gap in the written proof, not a disagreement with consensus; it may be closable by supplying the missing endpoint extrapolation, but as printed the argument is incomplete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes sharp higher-order Brezis--Seeger--Van Schaftingen--Yung (BSVY) formulae for kth-order differences in ball Banach function spaces. For a BBF space X satisfying either a convexification/maximal-function condition (Theorem 1.1(I)) or an endpoint maximal-function condition (Theorem 1.1(II)), the authors prove the equivalence (1.6) between a weak-type functional built from the level sets E_{λ,γ/q,k}[f] and the homogeneous Sobolev norm || |∇^k f| ||_X, together with the limiting identity (1.8). The proof proceeds through a higher-order weighted variant of Cohen--Dahmen--Daubechies--DeVore inequalities (Theorem 3.1), a weighted BSVY upper estimate for A1 weights (Theorem 3.3), a sharpness result for the range n(1/p-1/q)<k (Proposition 3.12), and an extrapolation argument to general BBF spaces. Applications include a BSVY characterization of higher-order homogeneous ball Banach Sobolev spaces (Theorem 1.3), critical Gagliardo--Nirenberg and Sobolev-type inequalities (Theorem 1.5), and a catalogue of examples ranging from Lebesgue and weighted Lebesgue spaces to Morrey-type, Herz, mixed-norm, variable, Lorentz, Orlicz, and Orlicz-slice spaces.","tokens_in":69878,"tokens_out":7880,"duration_ms":68413,"significance":"If the endpoint case is completed, this is a substantial contribution. The higher-order weighted estimates and the sparse dyadic characterization in Section 3 are new and of independent interest; the sharp ranges for γ and the range n(1/p-1/q)<k are identified and tested by Proposition 3.12; and the applications to a wide family of concrete BBF spaces are broad. The paper also gives genuine credit to the prior first-order BBF treatments and explicitly records that the higher-order, k≥2 Lebesgue-space case is new. The main concern is the proof of Theorem 1.1(II), where the endpoint p=1 upper estimate is delegated to a prior theorem without supplying the endpoint extrapolation that would be needed; this currently leaves a load-bearing gap in the central claim.","major_comments":[{"comment":"The endpoint case Theorem 1.1(II) is not proved. The text states that (II) 'can be obtained by repeating the proof of [23, Theorem 4.10]' with E_f(λ,q), |∇f|, and Theorem 4.5 there replaced by E_{λ,γ/q,k}[f], |∇^k f|, and (I) here. But Proposition 4.1, the only BBF-space upper estimate proved in this manuscript, assumes the existence of p∈[1,∞) with M bounded on (X^{1/p})'; at p=1 this is exactly M bounded on X'. The hypotheses of Theorem 1.1(II) are weaker: Definition 2.6 only supplies a sequence θ_m→1 with M bounded on (X^{1/θ_m})' and uniformly bounded norms, plus uniform boundedness of centered ball averages and an absolutely continuous norm. Nothing in the text proves that these assumptions imply M bounded on X', and the text explicitly motivates Definition 2.6 by spaces for which M is not known to be bounded on X'. The endpoint analogue of Lemmas 4.5--4.6 and Proposition 4.1 is therefore missing. Since Theorem 1.1(II) supplies the p=1 upper estimate in (1.6), the limiting identity (1.8), and the p=1 applications in Section 5, this is a load-bearing gap rather than a cosmetic omission.","section":"§4.1, proof of Theorem 1.1, final paragraph"},{"comment":"Proposition 4.1 is the key bridge from weighted Lebesgue estimates to general BBF spaces, but its proof is omitted with only the instruction to repeat the proof of [23, (4.10)]. The adaptation is not entirely formal: the functional E_{λ,γ/q,k,ℓ}[f] is defined by a nonlinear level set of a kth-order difference, whereas [23, (4.10)] treats the first-order functional E_f(λ,q). The proof should either be written out, or the authors should identify the specific steps in [23, (4.10)] that are unchanged and the steps that require Theorem 3.3. This is especially important because the endpoint p=1 case of Theorem 1.1 cannot be obtained from Proposition 4.1 as stated.","section":"§4.1, Proposition 4.1"},{"comment":"The proof of Corollary 3.5 asserts that the weighted estimates for A1 weights can be adapted to A_p(R) weights by 'replacing A1, ℓ, and n(1/p-1/q)<ℓ by Ap(R), k, and 1-1/q<k'. However, Theorem 3.3 and Proposition 3.11(ii) are stated and proved only for A1 weights. For υ∈A_p with p>1, the doubling estimate in Lemma 2.13(ii) has the exponent p rather than 1, so the geometric-factor bookkeeping in Proposition 3.11(ii) changes; no A_p version of these weighted estimates is stated or proved in the manuscript. Since Corollary 3.5 is the characterization of A_p weights for n=1 and all p∈[1,∞), the implication (i)⇒(ii) for p>1 is not justified as written.","section":"§3.3, proof of Corollary 3.5, step '(i) implies (ii)'"}],"minor_comments":[{"comment":"The proof says 'we find that (iii) holds', but Theorem 5.12 has only items (i) and (ii); the reference to (iii) should be corrected.","section":"§5.4, proof of Theorem 5.12"},{"comment":"The statement says 'Theorem 1.3 holds with X := LΦ', but in this subsection the relevant space is the Orlicz-slice space (E^r_Φ)_t, not LΦ. This appears to be a copy-and-paste error.","section":"§5.9, Theorem 5.24(ii)"},{"comment":"There is a typo in the sentence defining E_{λ,γ/q,k,ℓ}[f]: 'repalced' should be 'replaced'.","section":"Theorem 3.3"},{"comment":"The phrase 'the the desired inequality' appears in the introduction; it should read 'the desired inequality'.","section":"Introduction"},{"comment":"The proof writes 'We only consider the case q∈[p,∞) because the case q∈(0,p) is quite similar and hence we omit the details here.' Since the q∈(0,p) case uses Proposition 3.11(iii) and (iv) rather than a literal repetition, a short indication of the changes (or a reference to the corresponding lines) would improve readability.","section":"§3.2, proof of Theorem 3.3"},{"comment":"The definition of endpoint boundedness is stated with a sequence θ_m→1, but the definition would be clearer if the uniformity in the limit of the operator norms were explicitly quantified, as part of the inequality lim_m ∥M∥_{(X^{1/θ_m})'→(X^{1/θ_m})'} < ∞ already suggests; the current wording is acceptable but could be tightened.","section":"§2.1, Definition 2.6"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution and the weighted Lebesgue part is detailed and convincing. The main issue is the missing endpoint extrapolation for Theorem 1.1(II); this is a genuine gap in the written proof, but it appears closable by adding a dedicated endpoint lemma. I would encourage the editor to ask for that addition rather than reject the paper on this basis. The paper also leans heavily on a cluster of prior works by the same group; this is a dependency rather than circularity, but the authors should make the precise imported statements explicit in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The higher-order k≥2 part is the real news here, and the weighted machinery behind it—Lemma 3.9's sparse dyadic characterization and the higher-order weighted CDDD variant in Theorem 3.1—is genuinely new and well executed. The paper is honest that k=1 reproduces earlier first-order BSVY results and improves them in places. Even for X=L^p, k≥2 was open, so this is a real advance, not a repackaging.\n\nThe detailed weighted estimates (Theorem 3.3) are written out, and the sharpness construction (Prop 3.12) plus the n=1 A_p characterization (Cor 3.5) give the section real substance. The applications to eight families of spaces are mostly soft consequences of the framework, which is fine.\n\nThe soft spot is the endpoint p=1 case, and it's a genuine proof gap, not a matter of taste. Proposition 4.1—the only BBF-space upper estimate actually proved in the manuscript—assumes some p∈[1,∞) with M bounded on (X^{1/p})'. At p=1, that is M bounded on X'. But Theorem 1.1(II) starts from the weaker endpoint boundedness in Definition 2.6 (a sequence θ_m→1 with M bounded on (X^{1/θ_m})' and uniform norms). The proof of (II) is delegated to a single sentence: repeat [23, Theorem 4.10] with the relevant symbols replaced and '(I) here'. Nothing in the text shows that the endpoint hypotheses imply the proposition's assumption, and no endpoint analogue of Lemmas 4.5–4.6 is supplied to turn A1-weighted estimates into X-norm estimates for this particular functional. So the upper bound in (1.6) for p=1, the limiting identity (1.8), and the p=1 applications in Section 5 rest on an unstated lemma. I believe it is closable—the endpoint boundedness was designed by the same group for exactly this extrapolation—but as printed the argument is incomplete.\n\nThe other delegations are milder. Proposition 4.1 itself is asserted by 'repeating the proof of [23, (4.10)]', which is a key step. Lemmas 4.8–4.10 are only sketched, but the pieces there look standard. I don't see circularity or fitted parameters; the dependency on the group's earlier results is a dependency, not a flaw.\n\nWho this is for: harmonic analysts working on Sobolev norm representations and ball Banach function spaces. It deserves a serious referee; I'd send it to review and ask for the endpoint extrapolation and Proposition 4.1 to be written out. The reader's CONDITIONAL verdict is the right one.","headline":"Genuinely new higher-order BSVY formulae with a real advance in the weighted machinery, but the endpoint p=1 proof is delegated to prior work and lacks the stated extrapolation step.","tokens_in":70445,"tokens_out":3593,"would_cite":true,"duration_ms":29226,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35","26D10","35A23","42B25","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, in any ball Banach function space satisfying a maximal-function hypothesis, the higher-order level-set functional $\\sup_{\\lambda>0} \\lambda \\left\\| \\left( \\int_{\\{ |\\Delta_h^k f(\\cdot)| > \\lambda…","keywords":["BSVY formula","ball Banach function space","higher-order difference","homogeneous Sobolev space","Gagliardo–Nirenberg inequality","Muckenhoupt weight","Hardy–Littlewood maximal operator","extrapolation"],"falsifier":"Take $X := L^1$, $k := 2$, $q := 1$, and $\\gamma := 1$, and let $f(x) := e^{-|x|^2}$. The theorem predicts that $\\sup_{\\lambda>0} \\lambda \\int_{\\mathbb{R}^n} \\int_{|\\Delta_h^2 f(x)| > \\lambda |h|^3} |h|^{1-n}\\,dh\\,dx$ is finite and comparable to $\\|\\,|\\nabla^2 f|\\,\\|_{L^1}$, and that as $\\lambda\\to\\infty$ the normalized expression tends to the sphere integral $\\int_{\\mathbb{R}^n} \\int_{S^{n-1}} |\\sum_{|\\alpha|=2} \\partial^\\alpha f(x)\\,\\xi^\\alpha|\\,d\\mathcal{H}^{n-1}(\\xi)\\,dx$; evaluating this limit numerically for this $f$ gives a concrete check of the limiting identity (1.8).","tokens_in":69272,"feed_emoji":"📏","tokens_out":9775,"duration_ms":82961,"temperature":0.7,"pith_summary":"The paper's central claim is that the weak-type level-set formula known as the BSVY formula admits a higher-order version in ball Banach function spaces, a common umbrella for Lebesgue, weighted, Morrey, Herz, mixed-norm, variable, Lorentz, Orlicz, and Orlicz-slice spaces. For any such space whose $p$-th root is again a ball Banach space with the Hardy–Littlewood maximal operator bounded on its associate space, and for $q$ and $\\gamma$ in sharp ranges, the norm of $\\nabla^k f$ is equivalent to a supremum over $\\lambda$ of $\\lambda$ times the space norm of a level-set integral built from the $k$-th order difference $\\Delta_h^k f$. A companion limiting identity expresses the same norm as a sphere integral of the $k$-th derivatives. The applications are a characterization of higher-order homogeneous ball Banach Sobolev spaces and higher-order fractional Gagliardo–Nirenberg and Sobolev inequalities in critical cases. A sympathetic reader would care because the result unifies and extends all earlier first-order BSVY results and is new even for ordinary $L^q$ spaces when $k \\ge 2$.","feed_headline":"Higher-order BSVY formula proved for ball Banach spaces","feed_subtitle":"For ball Banach spaces, a kth-difference level-set integral recovers the gradient norm.","key_machinery":"The carrying object is the $k$-th order difference $\\Delta_h^k f(x)=\\sum_{j=0}^k (-1)^{k-j}\\binom{k}{j} f(x+jh)$ with its level sets $E_{\\lambda,\\gamma/q,k}[f]=\\{(x,h): |\\Delta_h^k f(x)| > \\lambda |h|^{k+\\gamma/q}\\}$. The machinery has four parts: Lemma 3.9, a sparse characterization of the dyadic cubes $Q$ for which the higher-order local approximation $E_k(f,Q)$ exceeds $\\lambda|Q|^{\\beta+\\ell/n}$; Theorem 3.1, the higher-order weighted inequality that bounds sums over such cubes by the weighted $L^p$ norm of $\\nabla^\\ell f$; Lemma 3.10, a variant higher-order Poincaré inequality that bounds $f(x)-P_B^{(k-1)}(f)(x)$ by nested ball averages of local approximation; and an extrapolation argument (Lemmas 4.5, 4.6, and Proposition 4.1) that converts the weighted estimate into the $X$-norm estimate. The limiting identity additionally uses Proposition 4.3, a subtle limsup bound derived from a Taylor-type expansion of $\\Delta_h^k f$.","core_discovery":"On the paper's own terms, the discovery is Theorem 1.1: with $X$ a ball Banach function space, $k\\in\\mathbb{N}$, $q>0$, and $\\gamma$ in the sharp set $\\Gamma_{p,q}$ together with $n(1/p-1/q)<k$, every locally integrable $f$ with $|\\nabla^k f|\\in X$ satisfies the two-sided estimate in (1.6), with constants independent of $f$; under an absolutely continuous norm the $\\lambda\\to\\infty$ or $\\lambda\\to 0^+$ limit equals $|\\gamma|^{-1/q}$ times the $X$-norm of the $q$-th root sphere average of $|\\sum \\partial^\\alpha f\\, \\xi^\\alpha|$ and is again equivalent to $\\|\\,|\\nabla^k f|\\,\\|_X$. Theorems 1.3 and 1.5 turn this into a characterization of the higher-order homogeneous Sobolev space and into critical fractional Gagliardo–Nirenberg and Sobolev inequalities, including the case where the classical strong-type inequality fails. The proof is built from a sparse dyadic-cube description of level sets of higher-order local approximation, a higher-order weighted inequality extending the classical weighted estimate that Theorem 3.1 generalizes, a variant higher-order Poincaré inequality, and an extrapolation step that transfers the weighted Lebesgue-space estimate to the ball Banach space via an $A_1$ weight constructed from the maximal operator.","pith_inferences":["Editorial inference: The sparse dyadic-cube lemma may be the transportable core; the same lemma could be used to derive higher-order Poincaré-type or trace estimates in any space where the cubes of the level set have the local-approximation structure, even without maximal-operator hypotheses.","Editorial inference: If the endpoint $p=1$ assumptions are the bottleneck, an interesting test is whether the conclusion survives for a ball Banach space where the maximal operator is endpoint bounded on $X'$ but centered ball averages fail to be uniformly bounded; Theorem 1.1(II) currently requires all three conditions, so a counterexample there would not contradict the theorem but would indicate","Editorial inference: The limiting formula suggests a directional integral identity: the $X$-norm of $|\\nabla^k f|$ recovers the $X$-norm of the sphere average of the $k$-th directional derivative, which could be used to characterize functions whose $k$-th derivatives vanish on a given set of directions, a question the paper does not address."],"forward_implications":["For every ball Banach function space covered by the hypotheses, the level-set functional is a genuine equivalent norm on the homogeneous Sobolev space $\\dot W^{k,X}$, so membership, convergence, and boundedness in that space can be tested by weak-type difference quotients.","Theorem 1.3 gives an iff criterion: $f$ lies in $\\dot W^{k,X}$ exactly when $f$ is locally integrable and the displayed supremum is finite, provided $X$ and $X'$ have absolutely continuous norms.","The critical Gagliardo–Nirenberg inequalities of Theorem 1.5 hold in the level-set formulation, including the critical case where the strong $L^q$ inequality fails; replacing the strong norm by the weak $L^q$ quasi-norm restores the inequality.","All statements specialize to weighted Lebesgue, Morrey-type, Herz, mixed-norm, variable Lebesgue, Lorentz, Orlicz, and Orlicz-slice spaces, so the result is a single framework covering many concrete spaces.","Even when $X$ is an ordinary $L^q$ space, the higher-order $k\\ge 2$ statements are new; for $k=1$ they recover the best known BSVY results."],"supporting_citations":[{"why":"The original first-order BSVY formula in Lebesgue spaces that this paper extends to higher order and general spaces.","marker":"[14]"},{"why":"The weighted level-set and local-approximation inequality whose higher-order variant is Theorem 3.1.","marker":"[19]"},{"why":"The prior first-order BSVY framework in ball Banach function spaces that supplies extrapolation tools and density arguments.","marker":"[23]"},{"why":"The first-order BSVY results in ball Banach spaces with Muckenhoupt weights that Theorem 1.1 generalizes and improves.","marker":"[66]"},{"why":"The characterization of homogeneous Sobolev spaces whose ideas replace the unavailable higher-order BBM formula in the proof of Theorem 1.3.","marker":"[38]"},{"why":"The generalized BSVY formulae in ball Banach Sobolev spaces used for the first-order base case and for parameter-range comparisons.","marker":"[107]"},{"why":"The earlier BSVY-type characterization of homogeneous ball Banach Sobolev spaces that supplies Lemma 2.7 and related endpoint tools.","marker":"[108]"},{"why":"The higher-order difference characterization of polynomials that motivates the defect the paper repairs and supports Proposition 2.14.","marker":"[37]"}],"fun_headline_variants":["Higher-order BSVY formulas now hold in ball Banach spaces","Level-set integrals recover gradient norms in ball Banach spaces","Sharp Sobolev inequalities proven for higher-order ball Banach spaces","New proof extends higher-order gradient bounds to ball Banach spaces","Ball Banach spaces admit sharp higher-order gradient formulas"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that some $p$ with $1\\le p<\\infty$ makes $X^{1/p}$ a ball Banach function space and makes the Hardy–Littlewood maximal operator bounded on the associate space $(X^{1/p})'$; in the endpoint case $p=1$ it also requires maximal endpoint boundedness on $X'$, uniform boundedness of centered ball averages on $X$, and an absolutely continuous norm.","fun_headline_variants_meta":{"raw":{"variants":["Higher-order BSVY formulas now hold in ball Banach spaces","Level-set integrals recover gradient norms in ball Banach spaces","Sharp Sobolev inequalities proven for higher-order ball Banach spaces","New proof extends higher-order gradient bounds to ball Banach spaces","Ball Banach spaces admit sharp higher-order gradient formulas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000434,"raw_usage":{"total_tokens":2387,"prompt_tokens":1301,"completion_tokens":1086,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":917,"completion_tokens_details":{"reasoning_tokens":1001}},"tokens_in":917,"tokens_out":1086,"duration_ms":8388,"temperature":1.0,"reasoning_tokens":1001,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:06:03.564920+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $X := L^1$, $k := 2$, $q := 1$, and $\\gamma := 1$, and let $f(x) := e^{-|x|^2}$. The theorem predicts that $\\sup_{\\lambda>0} \\lambda \\int_{\\mathbb{R}^n} \\int_{|\\Delta_h^2 f(x)| > \\lambda |h|^3} |h|^{1-n}\\,dh\\,dx$ is finite and comparable to $\\|\\,|\\nabla^2 f|\\,\\|_{L^1}$, and that as $\\lambda\\to\\infty$ the normalized expression tends to the sphere integral $\\int_{\\mathbb{R}^n} \\int_{S^{n-1}} |\\sum_{|\\alpha|=2} \\partial^\\alpha f(x)\\,\\xi^\\alpha|\\,d\\mathcal{H}^{n-1}(\\xi)\\,dx$; evaluating this limit numerically for this $f$ gives a concrete check of the limiting identity (1.8).","supporting_citations":[{"cited_title":"Sharp Weighted Cohen--Dahmen--Daubechies--DeVore Inequality with Applications to (Weighted) Critical Sobolev Spaces, Gagliardo--Nirenberg Inequalities, and Muckenhoupt Weights","cited_arxiv_id":"2405.19790","evidence_quote":"The first-order BSVY results in ball Banach spaces with Muckenhoupt weights that Theorem 1.1 generalizes and improves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The characterization of homogeneous Sobolev spaces whose ideas replace the unavailable higher-order BBM formula in the proof of Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The generalized BSVY formulae in ball Banach Sobolev spaces used for the first-order base case and for parameter-range comparisons."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The earlier BSVY-type characterization of homogeneous ball Banach Sobolev spaces that supplies Lemma 2.7 and related endpoint tools."},{"cited_title":"Ferreira, C","cited_arxiv_id":null,"evidence_quote":"The higher-order difference characterization of polynomials that motivates the defect the paper repairs and supports Proposition 2.14."}],"review_version":1}