{"id":"1fe9ba61-f132-432a-81b3-c3c3218a217d","arxiv_id":"2506.14631","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For fractional Kolmogorov semigroups, the rho-variation operators with rho>2 are bounded on L^p exactly for p>1∨d/beta, while the 2-variation operator is never bounded from L^p to weak L^p.","lead":"This paper proves sharp boundedness results for variation, oscillation, and jump operators attached to semigroups generated by fractional Kolmogorov operators, which are fractional Laplacians with a singular drift toward the origin. The results pin down the exact L^p range where these operators behave well and show that the 2-variation operator always fails.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sharp p-range and negative-κ scope are inherited from [11, Prop. 2.9], but the manuscript neither states that bound's exact hypotheses nor verifies it for β>d (Ψ(β)<0); the Section 4 'since β<d' line shows the inherited estimates need checking.","rationale":"Every claimed L^p result is a corollary of the boundedness of V_ρ(T-W), whose proof is a Schur test on kernels built from the imported pointwise estimates. The sharp threshold p>1∨d/β is not derived from a new mechanism in this paper; it is exactly the integrability condition imposed by the factor (1+t^{1/α}/|y|)^{d-β}. Thus the central claim is only as strong as that citation. The paper contains a real internal sign error in Proposition 4.1's proof ('since β<d' under the assumption β≥d), which corroborates that the inherited estimate is being used in a regime where its behavior changes; this is not a mere cosmetic typo because the global kernel bound in that proof is false when β<d for small y. I do not see an internal contradiction in the main L^p argument if the imported bound holds on the full stated range; in particular, the endpoint failure construction in Section 4 is salvageable (the displayed exponent d−β in the pairing should evidently be β−d), and the V_2 counterexample follows from standard reduction once V_2(T-W) is bounded. The reader's weakest_assumption partially matches this concern; I disagree with the reader's assertion that κ≤0 is necessarily excluded, since β>d with Ψ(β)<0 may well cover it, but that only reinforces that the exact range of [11, Prop. 2.9] must be verified.","tokens_in":26792,"tokens_out":26066,"duration_ms":255184,"concrete_test":"For d=2, α=3/2, β=5/2 (so β>d and Ψ(β)<0), re-derive [11, Proposition 2.9] and evaluate the Schur constants of the global kernel G_ℓ from (3.4). If the derivation yields a divergent L^1_x or L^1_y bound for G_ℓ, or if [11, Prop. 2.9] is only proven for β with Ψ(β)>0, then the p>1 range and V_2 transfer for negative κ are unsupported; if both checks pass, the concern is settled.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The decisive assumption is the pointwise kernel estimate displayed after Theorem 1.1: |T_{t,ℓ}^α(x,y)| ≤ C t^{-d/α} (t^{1/α}/(t^{1/α}+|x-y|))^{d+α-ε} (1+t^{1/α}/|y|)^{d-β}, imported from [11, Proposition 2.9] and used throughout Sections 3, 4, 7 and 8. The factor (1+t^{1/α}/|y|)^{d-β} is what fixes every threshold 1∨d/β and the endpoint statements. The manuscript applies it for all β∈((d+α)/2,d+α), including β>d where Ψ(β)<0, and the abstract advertises κ∈R; however, the exact hypotheses of [11, Proposition 2.9] are never stated, and the proof of Proposition 4.1 contains the line 'since β<d we deduce' inside a lemma that assumes β∈[d,d+α). That phrase points to exactly the dangerous regime: if β<d the factor is unbounded as y→0, while if β≥d it is harmless. The paper needs to resolve whether the imported bound is available on the whole claimed range, in particular for β>d; otherwise Theorems 1.2, 1.4 and 1.5(b) for negative κ are conditional on an unverified input. The Schur-test step in Section 3 is also deferred to '[11, pp. 33-35]', so the key boundedness of T_ℓ is not independently checkable from the manuscript.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"Abstract; Theorems 1.2–1.5: The paper studies ρ-variation, oscillation, and jump operators associated with the semigroup generated by the fractional Kolmogorov operator Λ_κ = (-Δ)^{α/2} + κ|x|^{-α} x·∇ on R^d, with α∈(1,2), α<(d+2)/2 and κ<κ_c. The main result (Theorem 1.2) asserts that, for β∈((d+α)/2,d+α), κ=Ψ(β), ℓ∈N∪{0} and ρ>2, V_ρ({t^ℓ∂_t^ℓT_t^α}) and the corresponding oscillation operator are bounded on L^p for 1∨d/β<p<∞, with weak (1,1) endpoint when d≤β and failure of L^{d/β}→L^{d/β,∞} when d>β. The paper also establishes weighted L^p inequalities (Theorem 1.4), jump inequalities (Theorem 1.3), and the failure of V_2({T_t^α}) from L^p to L^{p,∞} (Theorem 1.5). The proof compares T_t^α with the fractional heat semigroup W_t^α, using pointwise kernel bounds and a Schur test imported from [11], local/global decompositions, abstract weighted results from [4], and a one-dimensional construction from [15] for the V_2 failure. The exposition is detailed, but the precise hypotheses of several imported estimates are not stated and the scope of κ in the abstract is wider than the hypotheses of the theorems.","tokens_in":27176,"tokens_out":17413,"duration_ms":155068,"significance":"I found the main theorems plausible and, conditional on the imported estimates, the proofs are coherent; the paper gives a sharp L^p range, an endpoint, and a negative V_2 result, which would be a useful advance for nonlocal nonsymmetric semigroups. I also found no circular dependence: [5] is used only for standard fractional heat-kernel estimates, and [11] supplies the kernel bounds. The main reservations are verification and scope: the kernel bound after Theorem 1.1 is load-bearing for Theorems 1.2–1.5, but its hypotheses are not quoted; the Schur-test step that yields the threshold is deferred; and Proposition 4.1 contains an apparent contradiction in the β range. These issues are local and fixable, so I do not recommend rejection, but they must be resolved before the claims are fully established.","major_comments":[{"comment":"The abstract announces the operator for κ∈R, while every theorem is stated only for κ=Ψ(β) with β∈((d+α)/2,d+α). The range of Ψ on this interval is never identified, so the relation between these statements is not established. This is not just a wording issue: for d=1 (an allowed case, α∈(1,3/2)), inspection of (1.1) at β→(1+α)^- gives a positive limit, and since Ψ is strictly decreasing the range is contained in (0,∞); hence negative couplings are not covered by Theorem 1.2. Please state the exact range of Ψ and restrict the abstract and theorems to the couplings actually covered, or supply an additional argument for the remaining values of κ.","section":"Abstract; Theorem 1.2; Eq. (1.1)"},{"comment":"The pointwise bound displayed after Theorem 1.1 is quoted from [11, Proposition 2.9] and is used throughout Sections 3, 4, 7 and 8, but its precise hypotheses are never stated. In particular, the manuscript applies it for all β∈((d+α)/2,d+α), including β>d, where the factor (1+t^{1/α}/|y|)^{d-β} changes from singular (β<d) to bounded (β>d). Please quote the exact statement from [11], including the allowed range of β and the relation to κ, and verify that it covers the full parameter range used here. The final boundedness of the operator T_ℓ in Section 3 is then completed by 'applying the Schur test as in [11, pp. 33-35]'; since this is the step that produces the threshold 1∨d/β, that imported result should also be stated explicitly or reproduced.","section":"After Theorem 1.1; Sections 3, 4, 7, 8"},{"comment":"Proposition 4.1 assumes β∈[d,d+α), but the proof of the estimate for T_{ℓ,glob} contains the line 'since β<d we deduce'. These two statements are contradictory. The endpoint weak-type claim depends on whether (1+t^{1/α}/|y|)^{d-β} is bounded (β≥d) or not (β<d), so this step must be corrected and the argument made unambiguous.","section":"Section 4, Proposition 4.1"}],"minor_comments":[{"comment":"The statements write κ=Φ(β), whereas the function is defined as Ψ in (1.1); please unify the notation.","section":"Propositions 3.1 and 4.1"},{"comment":"The condition '1<α∧(d+2)/2' is malformed; the intended statement is probably α∈(1,2∧(d+2)/2) and 1∨d/β<p<∞.","section":"Theorem 1.5(b)"},{"comment":"Lemma 8.3 is stated without proof, referring only to [15, Proposition 8.3]; since this lemma is the basis of Theorem 1.5(a), a brief explanation of the adaptation would improve readability.","section":"Lemma 8.3"},{"comment":"Please proofread the displayed exponent comparisons: the expression with exponent d−α+1−ϵ2 appears inconsistent with the surrounding homogeneous estimates and should be checked against the intended computation.","section":"Section 3, around (3.1)–(3.2)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper relies centrally on the unpublished arXiv preprint [11] (Bui–Duong–Merz) for the kernel bound and the Schur test, and on [5] for standard estimates. I recommend asking the authors to make the imported hypotheses explicit and, if [11] is not yet published, to provide self-contained proofs of the few estimates on which the main theorems rest. The abstract/Theorem 1.2 mismatch on the range of κ should also be resolved in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper is serious and mostly sound. It is the first treatment of ρ-variation, oscillation, and jump operators for the semigroup generated by the fractional Kolmogorov operator Λκ = (−Δ)^{α/2} + (κ/|x|^α)x·∇, a nonlocal and nonsymmetric setting that the harmonic-analysis literature has avoided. The main technical work is in comparing T_t^α to the fractional heat semigroup W_t^α: Propositions 3.1 and 4.1 establish control of V_ρ of the difference on L^p and L^1→L^{1,∞}, and those estimates are substantial. The sharp range p>1∨d/β and the endpoint negative result for V_2 are genuinely new. I found no circularity; the arguments reduce to kernel bounds in [11], and the self-citation to [5] is for standard fractional heat kernel facts. If the imported kernel bound holds as stated, the main theorems follow.\n\nSoft spots, in order of importance. First, the manuscript never states the exact hypotheses of [11, Proposition 2.9], the pointwise bound that carries the whole range. That is a real problem in a paper whose threshold depends on β. The stress-test note is right: the proof of Proposition 4.1 contains 'since β<d' inside a statement that assumes β∈[d,d+α). I think that specific line is a typo—for β≥d the factor (1+t^{1/α}/|y|)^{d−β} is ≤1 and the estimate is actually easier—but the paper should say so. The sharper question is whether [11] supplies the bound for all β∈((d+α)/2,d+α), including β>d. If it does, the abstract's κ∈R is defensible because Ψ(β)<0 there; if it does not, the negative-κ endpoint statements are conditional. The paper needs to either verify that range or state the theorem only for β≤d. Second, the scope mismatch: the abstract advertises κ∈R, the theorems state κ=Ψ(β) with β in an interval, and Φ/Ψ are used interchangeably. Clean that up. Third, the Schur test in Section 3 is deferred to [11, pp. 33–35]; acceptable, but a referee should confirm the local/global decomposition really matches the one in [11].\n\nWho this is for: harmonic analysts working on variational inequalities for semigroups. The paper deserves a serious referee, and my guess is that the referee will ask for the imported bound's hypotheses to be stated and checked, not for a new proof.","headline":"Serious, proof-heavy paper that likely delivers sharp L^p variational inequalities for fractional Kolmogorov semigroups; it deserves a referee, though one imported kernel bound's range needs checking and the notation needs cleanup.","tokens_in":27705,"tokens_out":3490,"would_cite":true,"duration_ms":35124,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B20","42B25","42B35","47D03"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes sharp L^p variation bounds for the fractional Kolmogorov semigroup in the exact range $1\\vee d/\\beta<p<\\infty$, with endpoint failure when $d>\\beta$ and no L^p bound for the 2-variation operator.","keywords":["Kolmogorov operator","fractional Laplacian","ρ-variation","oscillation operator","jump operator","variational inequalities","semigroup","critical coupling constant"],"falsifier":"Test the kernel estimate for a concrete $\\beta\\in((d+\\alpha)/2,d+\\alpha)$: compute or bound $T_{t,\\ell}^\\alpha(x,y)$ near $y=0$ with $x-y$ large, and check whether the factor $(1+t^{1/\\alpha}/|y|)^{d-\\beta}$ appears with the stated exponent; a single violation would change the Schur-test integrability in Sections 3\\textendash 4 and invalidate the $p>d/\\beta$ threshold. Alternatively, exhibit a function in $L^{d/\\beta}(\\mathbb{R}^d)$ for which the maximal operator $T_*^\\alpha(f)$ is not in weak $L^{d/\\beta}$, which would overturn the claimed endpoint failure range. For $\\kappa\\le0$, run the same construction with the drift reversed; if the comparison operator $V_2(\\{T_t^\\alpha-W_t^\\alpha\\})$ fails to be bounded on $L^p$ for $p>1\\vee d/\\beta$, the abstract's $\\kappa\\in\\mathbb{R}$ claim is false as stated.","tokens_in":26627,"feed_emoji":"🧮","tokens_out":12271,"duration_ms":102690,"temperature":0.7,"pith_summary":"The paper studies the fractional Kolmogorov operator $\\Lambda_\\kappa = (-\\Delta)^{\\alpha/2} + \\frac{\\kappa}{|x|^\\alpha} x\\cdot\\nabla$ on $\\mathbb{R}^d$, a fractional Laplacian carrying a highly singular drift that points away from the origin. It establishes that the $\\rho$-variation operators $V_\\rho(\\{t^\\ell \\partial_t^\\ell T_t^\\alpha\\})$ of the semigroup generated by $-\\Lambda_\\kappa$ are bounded on $L^p(\\mathbb{R}^d)$ exactly when $1\\vee d/\\beta < p < \\infty$, where $\\beta$ is tied to the coupling constant $\\kappa$ through an explicit function $\\Psi(\\beta)$, and that these bounds are sharp: at the endpoint $p=d/\\beta$ (when $d>\\beta$) the weak-type inequality fails, and the 2-variation operator is never bounded for any $1<p<\\infty$. A sympathetic reader would care because variation inequalities of this kind control pointwise convergence of the semigroup, and the paper locates the exact threshold for that convergence in the presence of a nonlocal, nonsymmetric drift.","feed_headline":"Singular-drift semigroup gets sharp L^p variation bounds","feed_subtitle":"The ρ-variation operator is bounded exactly when p exceeds d/β; at ρ=2 it fails for every p.","key_machinery":"The load-bearing mechanism is a pointwise estimate for the integral kernel $T_{t,\\ell}^\\alpha(x,y)$ of $t^\\ell\\partial_t^\\ell T_t^\\alpha$, taken from [11, Proposition 2.9]: under $\\kappa=\\Psi(\\beta)$, $|T_{t,\\ell}^\\alpha(x,y)| \\le C t^{-d/\\alpha} (t^{1/\\alpha}/(t^{1/\\alpha}+|x-y|))^{d+\\alpha-\\varepsilon} (1+t^{1/\\alpha}/|y|)^{d-\\beta}$ for every $\\varepsilon>0$. The extra factor $(1+t^{1/\\alpha}/|y|)^{d-\\beta}$ is the signature of the drift: it softens the singularity near $y=0$ when $\\beta<d$ and provides the integrability that forces the threshold $p>d/\\beta$. On top of this kernel bound, the paper builds a local/global decomposition of the time-space region, uses the Schur test to control the local part, and patches the global part by comparison with the fractional heat kernel $W_t^\\alpha$, whose variation operators are handled by vector-valued Calder\\'on\\textendash Zygmund theory.","core_discovery":"The central discovery is that the $L^p$ theory for the fractional Kolmogorov semigroup mirrors the heat-semigroup theory up to a single exponent: for $\\alpha\\in(1,2\\wedge(d+2)/2)$, $\\beta\\in((d+\\alpha)/2,d+\\alpha)$, $\\kappa=\\Psi(\\beta)$, $\\ell\\in\\mathbb{N}\\cup\\{0\\}$, and $\\rho>2$, the operators $V_\\rho(\\{t^\\ell\\partial_t^\\ell T_t^\\alpha\\})$ and the associated oscillation operators are bounded on $L^p(\\mathbb{R}^d)$ whenever $1\\vee d/\\beta<p<\\infty$, and from $L^1(\\mathbb{R}^d)$ to weak $L^1$ when $d\\le\\beta$. The same range is shown to be optimal, since the $\\rho$-variation operator fails to be bounded from $L^{d/\\beta}(\\mathbb{R}^d)$ to weak $L^{d/\\beta}(\\mathbb{R}^d)$ when $d>\\beta$, and $V_2(\\{T_t^\\alpha\\})$ is not bounded from $L^p$ to weak $L^p$ for any $1<p<\\infty$. The proof goes through the difference with the fractional heat semigroup $W_t^\\alpha$: the variation operator of $T_t^\\alpha-W_t^\\alpha$ is shown to be bounded in the same range, so the fractional Kolmogorov semigroup inherits the good variation theory of the fractional heat semigroup, with the sharp threshold determined by the drift-induced decay in the kernel.","pith_inferences":["The paper's theorems only cover $\\kappa=\\Psi(\\beta)>0$ in the range $\\beta\\in((d+\\alpha)/2,d+\\alpha)$, which corresponds to positive (attractive) couplings below the critical value; despite the abstract's $\\kappa\\in\\mathbb{R}$, the repulsive case $\\kappa\\le0$ is not treated by any theorem, and it is unclear whether the same kernel bound or the same $p$-threshold holds there.","The threshold $p>d/\\beta$ has the form of a Hardy-type critical exponent: on the function $|x|^{\\beta-d}$ the drift term becomes borderline, which suggests the range $1\\vee d/\\beta<p<\\infty$ is the natural Sobolev-critical range for this operator, and the same threshold may appear for other nonlocal Kolmogorov-type operators with scaling-critical drifts.","The manuscript uses $\\kappa=\\Phi(\\beta)$ in the statements of Propositions 3.1 and 4.1 while the theorems use $\\kappa=\\Psi(\\beta)$; if $\\Phi\\neq\\Psi$, the proofs as written may contain a typo that should be resolved before the propositions are quoted, though the argument itself only needs the kernel bound valid for $\\kappa=\\Psi(\\beta)$.","A testable extension: the comparison operator $V_\\rho(\\{T_t^\\alpha-W_t^\\alpha\\})$ is the only part of the proof that uses the sharp $\\beta$-range; one could try to push the same strategy to $\\kappa$ near $\\kappa_c$ from below, where $\\beta$ approaches $(d+\\alpha)/2$ and the endpoint $p=d/\\beta$ approaches $2d/(d+\\alpha)$, to see whether the threshold formula remains valid at the critical coupling."],"forward_implications":["For every $f\\in L^p(\\mathbb{R}^d)$ with $p>1\\vee d/\\beta$, the semigroup averages $T_t^\\alpha f$ converge to $f$ as $t\\to0^+$ almost everywhere, and the convergence is quantitative: the number of times the trajectory crosses a vertical window of height $\\lambda$ is controlled by a $\\lambda$-jump bound.","The sharp range transfers to weighted spaces: $V_\\rho$ and the oscillation operator are bounded on $L^p(w)$ for every Muckenhoupt weight $w\\in A_{p/(1\\vee d/\\beta)}$, so the unweighted threshold is also the weight threshold.","The endpoint failure at $p=d/\\beta$ (when $d>\\beta$) is a genuine obstruction, not an artifact of the proof: the maximal operator of the semigroup already fails weak-type there, so no $L^{d/\\beta}\\to L^{d/\\beta,\\infty}$ bound for $V_\\rho$ can hold.","The 2-variation operator is intrinsically different: no $L^p\\to\\text{weak-}L^p$ bound exists for any $1<p<\\infty$, so the usual restriction $\\rho>2$ in variational inequalities is necessary for this semigroup just as it is for the fractional heat semigroup.","The same conclusions hold for oscillation operators with respect to any decreasing sequence $t_j\\to0$, so the results are stable under the choice of the discrete sampling."],"supporting_citations":[{"why":"Supplies the pointwise kernel estimate $|T_{t,\\ell}^\\alpha(x,y)| \\le C t^{-d/\\alpha}(\\cdots)(1+t^{1/\\alpha}/|y|)^{d-\\beta}$ and the $L^p$ theory for $T_t^\\alpha$ that anchors the threshold $1\\vee d/\\beta$.","marker":"[11]"},{"why":"Provides the strong $q$-variation inequalities for analytic semigroups that yield the $L^2$ boundedness of $V_\\rho$ and oscillation operators.","marker":"[32]"},{"why":"Gives the kernel estimates and vector-valued Calder\\'on\\textendash Zygmund framework for the fractional heat semigroup $W_t^\\alpha$ used in Proposition 2.1 and the comparison arguments.","marker":"[5]"},{"why":"The construction of the Rademacher-based test functions $g_N$ and the $\\sim_2$ relation used to disprove $L^p\\to L^{p,\\infty}$ boundedness of $V_2$.","marker":"[15]"},{"why":"Establishes the heat-kernel bounds and the desingularizing-weight framework for the fractional Kolmogorov operator that underpin the drift-induced decay factor.","marker":"[30]"},{"why":"Provides the abstract theorem (Theorem 6.6) on weighted boundedness from which the Muckenhoupt weight $A_{p/(1\\vee d/\\beta)}$ conclusion is drawn.","marker":"[4]"},{"why":"The weighted proof scheme and local/global decomposition for variation operators of Schr\\\"odinger-type semigroups that the paper adapts.","marker":"[43]"},{"why":"Yields the bound relating the $\\lambda$-jump operator to the $\\rho$-variation operator, used to obtain Theorem 1.3.","marker":"[24]"}],"fun_headline_variants":["Sharp L^p variation bounds for fractional Kolmogorov semigroups","Critical exponent found for variation operators on Kolmogorov semigroups","Fractional Kolmogorov semigroup: variation bounded iff p > d/β","ρ-variation fails at ρ=2 for all L^p on Kolmogorov semigroup"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $L^p$ theorems rest on a pointwise kernel bound imported from an earlier paper that fixes the admissible $\\beta$-range and thus the threshold $p>1\\vee d/\\beta$; if that bound failed, or failed to extend to the repulsive couplings $\\kappa\\le0$ promised by the abstract, the results would not cover the claimed scope.","fun_headline_variants_meta":{"raw":{"variants":["Sharp L^p variation bounds for fractional Kolmogorov semigroups","Critical exponent found for variation operators on Kolmogorov semigroups","Fractional Kolmogorov semigroup: variation bounded iff p > d/β","ρ-variation fails at ρ=2 for all L^p on Kolmogorov semigroup"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000262,"raw_usage":{"total_tokens":1696,"prompt_tokens":1147,"completion_tokens":549,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":763,"completion_tokens_details":{"reasoning_tokens":463}},"tokens_in":763,"tokens_out":549,"duration_ms":5607,"temperature":1.0,"reasoning_tokens":463,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:50:27.748699+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the kernel estimate for a concrete $\\beta\\in((d+\\alpha)/2,d+\\alpha)$: compute or bound $T_{t,\\ell}^\\alpha(x,y)$ near $y=0$ with $x-y$ large, and check whether the factor $(1+t^{1/\\alpha}/|y|)^{d-\\beta}$ appears with the stated exponent; a single violation would change the Schur-test integrability in Sections 3\\textendash 4 and invalidate the $p>d/\\beta$ threshold. Alternatively, exhibit a function in $L^{d/\\beta}(\\mathbb{R}^d)$ for which the maximal operator $T_*^\\alpha(f)$ is not in weak $L^{d/\\beta}$, which would overturn the claimed endpoint failure range. For $\\kappa\\le0$, run the same construction with the drift reversed; if the comparison operator $V_2(\\{T_t^\\alpha-W_t^\\alpha\\})$ fails to be bounded on $L^p$ for $p>1\\vee d/\\beta$, the abstract's $\\kappa\\in\\mathbb{R}$ claim is false as stated.","supporting_citations":[{"cited_title":"Strong q -variation inequalities for analytic semigroups","cited_arxiv_id":null,"evidence_quote":"Provides the strong $q$-variation inequalities for analytic semigroups that yield the $L^2$ boundedness of $V_\\rho$ and oscillation operators."},{"cited_title":"Variation operators associated with semigroups generated by Hardy operators involving fractional Laplacians in a half space","cited_arxiv_id":"2310.03540","evidence_quote":"Gives the kernel estimates and vector-valued Calder\\'on\\textendash Zygmund framework for the fractional heat semigroup $W_t^\\alpha$ used in Proposition 2.1 and the comparison arguments."},{"cited_title":"Variational inequalities for the O rnstein- U hlenbeck semigroup: the higher-dimensional case","cited_arxiv_id":null,"evidence_quote":"The construction of the Rademacher-based test functions $g_N$ and the $\\sim_2$ relation used to disprove $L^p\\to L^{p,\\infty}$ boundedness of $V_2$."},{"cited_title":"A., and Szczypkowski, K","cited_arxiv_id":null,"evidence_quote":"Establishes the heat-kernel bounds and the desingularizing-weight framework for the fractional Kolmogorov operator that underpin the drift-induced decay factor."},{"cited_title":"New abstract H ardy spaces","cited_arxiv_id":null,"evidence_quote":"Provides the abstract theorem (Theorem 6.6) on weighted boundedness from which the Muckenhoupt weight $A_{p/(1\\vee d/\\beta)}$ conclusion is drawn."},{"cited_title":"Variation and oscillation operators associated to semigroup generated by S chr\\\" o dinger operator with fractional power","cited_arxiv_id":null,"evidence_quote":"The weighted proof scheme and local/global decomposition for variation operators of Schr\\\"odinger-type semigroups that the paper adapts."},{"cited_title":"L., Seeger, A., and Wright, J","cited_arxiv_id":null,"evidence_quote":"Yields the bound relating the $\\lambda$-jump operator to the $\\rho$-variation operator, used to obtain Theorem 1.3."}],"review_version":1}