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Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.14631 v1 pith:FF4BIJ6F submitted 2025-06-17 math.AP

Jorge J. Betancor , Estefanía Dalmasso , Pablo Quijano This is my paper · ORCID
classification math.AP MSC 42B2042B2542B3547D03
keywords KolmogorovoperatorfractionalLaplacianρ-variationoscillationjumpvariationalinequalitiessemigroupcriticalcouplingconstant
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Abstract; Theorem 1.2; Eq. (1.1)] 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.
  2. [After Theorem 1.1; Sections 3, 4, 7, 8] 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.
  3. [Section 4, Proposition 4.1] 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.
minor comments (4)
  1. [Propositions 3.1 and 4.1] The statements write κ=Φ(β), whereas the function is defined as Ψ in (1.1); please unify the notation.
  2. [Theorem 1.5(b)] 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<∞.
  3. [Lemma 8.3] 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.
  4. [Section 3, around (3.1)–(3.2)] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the L^p variational bounds are derived from independent kernel estimates in [11] (Bui–Duong–Merz); the sole self-citation [5] is used for standard fractional heat-kernel estimates and is not load-bearing.

full rationale

The derivation chain is self-contained in the required sense. Theorem 1.1 (L^2 variational and oscillation bounds) is imported from Le Merdy–Xu [32], an independent source. The L^p extension proceeds by comparing T_t^alpha with the fractional heat semigroup W_t^alpha: Proposition 3.1 bounds V_rho({t^k ∂_t^k(T_t^alpha−W_t^alpha)}) via the pointwise kernel estimate imported from [11, Proposition 2.9] and the Schur test 'as in [11, pp. 33–35]'. This is a genuine external input, not a restatement of the paper's conclusion: [11] is an independent paper by Bui, Duong and Merz proving Sobolev-norm equivalence and kernel bounds, not variational inequalities. The threshold 1∨d/β is derived in the proof (the condition βp−d>0 appears explicitly), not assumed as the theorem's conclusion, and β is a reparameterization κ=Ψ(β) from [19]/[29], not a fitted constant. The endpoint and failure results are likewise built on independent inputs: Proposition 2.1 uses Calderón–Zygmund theory for W_t^alpha, and Theorem 1.5 uses the construction of [15] plus Proposition 3.1. The only self-citation is [5], used in Section 2 for kernel-size and derivative estimates of the fractional heat semigroup; the same estimates are available from [11, (2.13) and Lemma 2.11] and are standard, so the self-citation is not load-bearing. Two non-circular caveats should be weighed as correctness/scope risks rather than circularity: (i) the exact hypotheses of [11, Proposition 2.9] are never stated, while it is applied for β∈((d+α)/2,d+α), including values β>d where Ψ(β)<0; and (ii) Section 4's proof of Proposition 4.1 contains the line 'since β<d' inside a lemma assuming β∈[d,d+α), which appears to be a typo (the estimate requires β≥d). The abstract advertises κ∈R, but the theorems only treat κ=Ψ(β) for β in the stated interval. These gaps concern whether the imported bound is available on the full claimed range, not whether any theorem is defined in terms of its own conclusion.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entities or empirical parameters. Its free-parameter count is zero: beta is a reparametrization of the coupling constant kappa through kappa = Psi(beta), not an extra degree of freedom. The load-bearing input is the external kernel estimate from [11], which effectively supplies the range of beta and hence the L^p thresholds. The paper's own contribution is the difference estimates and the V_2 counterexample adaptation.

assumptions (7)
  • domain assumption The operator -Lambda_kappa generates a contractive holomorphic C_0-semigroup {T_t^alpha} in L^2(R^d) for kappa < kappa_c.
    Adopted from [28, Section 7] and [29, 30]; the paper relies on this for L^2 boundedness via [32] and for the a.e. limit arguments in Section 4.
  • domain assumption Kernel bounds from [11, Proposition 2.9]: |T_{t,ell}^alpha(x,y)| ≤ C t^{-d/alpha} (t^{1/alpha}/(t^{1/alpha}+|x-y|))^{d+alpha-epsilon} (1+t^{1/alpha}/|y|)^{d-beta} for kappa = Psi(beta).
    This is the central external input; it fixes the admissible beta range and the L^p threshold 1∨d/beta. The paper does not prove it.
  • standard math Fractional heat kernel derivative estimates from [5, Proposition 2.2] and [11, (2.13)].
    Standard estimates for W_t^alpha used in Proposition 2.1 and the comparison kernels.
  • standard math Strong q-variation inequalities for analytic semigroups in L^2 by Le Merdy and Xu [32, Corollary 4.5].
    Gives Theorem 1.1, the L^2 baseline.
  • standard math Weighted extrapolation theorem of Bernicot and Zhao [4, Theorem 6.6].
    Used in Section 7 to derive weighted inequalities from the off-diagonal estimates in Lemma 7.1.
  • standard math Jump inequality bound lambda[J(...)]^{1/rho} ≤ 2^{1+1/rho} V_rho(...) from [24, p. 6712].
    Used in Theorem 1.3 to convert variation bounds into jump bounds.
  • standard math Schur test and vector-valued Calderón-Zygmund theory.
    Background tools used in Propositions 2.1, 3.1, 4.1 and Lemma 7.1.

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Pith. "Pith review of Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators." pith.science (2026). https://pith.science/paper/FF4BIJ6F

@misc{pith2026250614631,
  author       = {Pith},
  title        = {Pith review of: Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FF4BIJ6F}},
  note         = {Machine review of arXiv:2506.14631}
}
abstract

In this paper we consider fractional Kolmogorov operators defined, in $\mathbb{R}^d$, by \[\Lambda_\kappa=(-\Delta)^{\alpha/2}+\frac{\kappa}{|x|^\alpha} x\cdot \nabla,\] with $\alpha\in (1,2)$, $\alpha<(d+2)/2$ and $\kappa\in \mathbb{R}$. The operator $\Lambda_\alpha$ generates a holomorphic semigroup $\{T_t^\alpha\}_{t>0}$ in $L^2(\mathbb{R}^d)$ provided that $\kappa<\kappa_c$ where $\kappa_c$ is a critical coupling constant. We establish $L^p$-boundedness properties for the variation operators $V_\rho\left(\{t^\ell\partial_t^\ell T_t^\alpha\}_{t>0}\right)$ with $\rho> 2$, $\ell\in \mathbb{N}$ and $1\vee \frac{d}{\beta}<p<\infty$, where $\beta$ depends on $\kappa$. We also study the behavior of these variation operators in the endpoint $L^{1\vee \frac{d}{\beta}}(\mathbb{R}^d)$ and we prove that $V_2(\{T_t^\alpha\}_{t>0})$ is not bounded from $L^p(\mathbb{R}^d)$ to $L^{p,\infty}(\mathbb{R}^d)$ for any $1< p<\infty$.

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Works this paper leans on

49 extracted references · 47 canonical work pages

  1. [11]

    A., Duong, X

    Bui, T. A., Duong, X. T., and Merz, K. Equivalence of S obolev norms for K olmogorov operators with scaling-critical drift, 2024. arXiv:2410.00191

    arXiv 2024

  2. [5]

    Variation operators associated with semigroups generated by Hardy operators involving fractional Laplacians in a half space

    Betancor, J. J., Dalmasso, E., and Quijano, P. Variation operators associated with semigroups generated by H ardy operators involving fractional L aplacians in a half space, 2023. arXiv:2310.03540

    work page Pith review arXiv 2023

  3. [4]

    New abstract H ardy spaces

    Bernicot, F., and Zhao, J. New abstract H ardy spaces. J. Funct. Anal. 255 , 7 (2008), 1761--1796

    work page 2008

  4. [15]

    Variational inequalities for the O rnstein- U hlenbeck semigroup: the higher-dimensional case

    Casarino, V., Ciatti, P., and Sj\" o gren, P. Variational inequalities for the O rnstein- U hlenbeck semigroup: the higher-dimensional case. J. Geom. Anal. 35 , 1 (2025), Paper No. 32, 33

    work page 2025

  5. [1]

    Almeida, V., and Betancor, J. J. Variation and oscillation for harmonic operators in the inverse G aussian setting. Commun. Pure Appl. Anal. 21 , 2 (2022), 419--470

    work page 2022

  6. [2]

    J., and Rodr\' guez-Mesa, L

    Almeida, V., Betancor, J. J., and Rodr\' guez-Mesa, L. Variation operators associated with the semigroups generated by S chr\" o dinger operators with inverse square potentials. Anal. Appl. (Singap.) 21 , 2 (2023), 353--383

    work page 2023

  7. [3]

    Variation bounds for spherical averages

    Beltran, D., Oberlin, R., Roncal, L., Seeger, A., and Stovall, B. Variation bounds for spherical averages. Math. Ann. 382 , 1-2 (2022), 459--512

    work page 2022

  8. [6]

    On the maximal ergodic theorem for certain subsets of the integers

    Bourgain, J. On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61 , 1 (1988), 39--72

    work page 1988

Show all 49 references
  1. [7]

    On the pointwise ergodic theorem on L^p for arithmetic sets

    Bourgain, J. On the pointwise ergodic theorem on L^p for arithmetic sets. Israel J. Math. 61 , 1 (1988), 73--84

    1988

  2. [8]

    Pointwise ergodic theorems for arithmetic sets

    Bourgain, J. Pointwise ergodic theorems for arithmetic sets. Inst. Hautes \' E tudes Sci. Publ. Math. , 69 (1989), 5--45. With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein

    1989

  3. [9]

    M., and Wr\' o bel, B

    Bourgain, J., Mirek, M., Stein, E. M., and Wr\' o bel, B. a. Dimension-free estimates for discrete H ardy- L ittlewood averaging operators over the cubes in Z^d . Amer. J. Math. 141 , 4 (2019), 857--905

    2019

  4. [10]

    A., and D'Ancona, P

    Bui, T. A., and D'Ancona, P. Generalized H ardy operators. Nonlinearity 36 , 1 (2023), 171--198

    2023

  5. [12]

    A., and Merz, K

    Bui, T. A., and Merz, K. Equivalence of S obolev norms in L ebesgue spaces for H ardy operators in a half-space, 2023. arXiv:2309.02928

    2023 arXiv

  6. [13]

    T., Jones, R

    Campbell, J. T., Jones, R. L., Reinhold, K., and Wierdl, M. Oscillation and variation for the H ilbert transform. Duke Math. J. 105 , 1 (2000), 59--83

    2000

  7. [14]

    T., Jones, R

    Campbell, J. T., Jones, R. L., Reinhold, K., and Wierdl, M. Oscillation and variation for singular integrals in higher dimensions. Trans. Amer. Math. Soc. 355 , 5 (2003), 2115--2137

    2003

  8. [16]

    A., Men\' a rguez, T., Torrea, J

    Crescimbeni, R., Mac\' as, R. A., Men\' a rguez, T., Torrea, J. L., and Viviani, B. The -variation as an operator between maximal operators and singular integrals. J. Evol. Equ. 9 , 1 (2009), 81--102

    2009

  9. [17]

    Q., and Uraltsev, G

    Di Plinio, F., Do, Y. Q., and Uraltsev, G. N. Positive sparse domination of variational C arleson operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 , 4 (2018), 1443--1458

    2018

  10. [18]

    Weighted bounds for variational F ourier series

    Do, Y., and Lacey, M. Weighted bounds for variational F ourier series. Studia Math. 211 , 2 (2012), 153--190

    2012

  11. [19]

    L., Merz, K., and Siedentop, H

    Frank, R. L., Merz, K., and Siedentop, H. Equivalence of S obolev norms involving generalized H ardy operators. Int. Math. Res. Not. IMRN , 3 (2021), 2284--2303

    2021

  12. [20]

    A., and Torrea, J

    Gillespie, T. A., and Torrea, J. L. Dimension free estimates for the oscillation of R iesz transforms. Israel J. Math. 141\/ (2004), 125--144

    2004

  13. [21]

    D., Magyar, A., Mirek, M., and Szarek, T

    Ionescu, A. D., Magyar, A., Mirek, M., and Szarek, T. Z. Polynomial averages and pointwise ergodic theorems on nilpotent groups. Invent. Math. 231 , 3 (2023), 1023--1140

    2023

  14. [22]

    L., Kaufman, R., Rosenblatt, J

    Jones, R. L., Kaufman, R., Rosenblatt, J. M., and Wierdl, M. Oscillation in ergodic theory. Ergodic Theory Dynam. Systems 18 , 4 (1998), 889--935

    1998

  15. [23]

    L., and Reinhold, K

    Jones, R. L., and Reinhold, K. Oscillation and variation inequalities for convolution powers. Ergodic Theory Dynam. Systems 21 , 6 (2001), 1809--1829

    2001

  16. [24]

    L., Seeger, A., and Wright, J

    Jones, R. L., Seeger, A., and Wright, J. Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc. 360 , 12 (2008), 6711--6742

    2008

  17. [25]

    L., and Wang, G

    Jones, R. L., and Wang, G. Variation inequalities for the F ej\' e r and P oisson kernels. Trans. Amer. Math. Soc. 356 , 11 (2004), 4493--4518

    2004

  18. [26]

    Sobolev spaces adapted to the S chr\" o dinger operator with inverse-square potential

    Killip, R., Miao, C., Visan, M., Zhang, J., and Zheng, J. Sobolev spaces adapted to the S chr\" o dinger operator with inverse-square potential. Math. Z. 288 , 3-4 (2018), 1273--1298

    2018

  19. [27]

    R., and Sem\" e nov, Y

    Kinzebulatov, D., Madou, K. R., and Sem\" e nov, Y. A. On the supercritical fractional diffusion equation with H ardy-type drift. J. Anal. Math. 152 , 2 (2024), 401--420

    2024

  20. [28]

    Kinzebulatov, D., and Sem\" e nov, Y. A. On the theory of the K olmogorov operator in the spaces L^p and C_ . Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 21\/ (2020), 1573--1647

    2020

  21. [29]

    Kinzebulatov, D., and Sem\" e nov, Y. A. Fractional K olmogorov operator and desingularizing weights. Publ. Res. Inst. Math. Sci. 59 , 2 (2023), 339--391

    2023

  22. [30]

    A., and Szczypkowski, K

    Kinzebulatov, D., Sem\" e nov, Y. A., and Szczypkowski, K. Heat kernel of fractional L aplacian with H ardy drift via desingularizing weights. J. Lond. Math. Soc. (2) 104 , 4 (2021), 1861--1900

    2021

  23. [31]

    Pointwise ergodic theorems for non-conventional bilinear polynomial averages

    Krause, B., Mirek, M., and Tao, T. Pointwise ergodic theorems for non-conventional bilinear polynomial averages. Ann. of Math. (2) 195 , 3 (2022), 997--1109

    2022

  24. [32]

    Strong q -variation inequalities for analytic semigroups

    Le Merdy, C., and Xu, Q. Strong q -variation inequalities for analytic semigroups. Ann. Inst. Fourier (Grenoble) 62 , 6 (2012), 2069--2097 (2013)

    2012

  25. [33]

    La variation d'ordre p des semi-martingales

    L\' e pingle, D. La variation d'ordre p des semi-martingales. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 , 4 (1976), 295--316

    1976

  26. [34]

    L., and Xu, Q

    Ma, T., Torrea, J. L., and Xu, Q. Weighted variation inequalities for differential operators and singular integrals. J. Funct. Anal. 268 , 2 (2015), 376--416

    2015

  27. [35]

    L., and Xu, Q

    Ma, T., Torrea, J. L., and Xu, Q. Weighted variation inequalities for differential operators and singular integrals in higher dimensions. Sci. China Math. 60 , 8 (2017), 1419--1442

    2017

  28. [36]

    Variation and oscillation for singular integrals with odd kernel on L ipschitz graphs

    Mas, A., and Tolsa, X. Variation and oscillation for singular integrals with odd kernel on L ipschitz graphs. Proc. Lond. Math. Soc. (3) 105 , 1 (2012), 49--86

    2012

  29. [37]

    Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes

    Mehlhop, N., and S omian, W. Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes. Math. Ann. 388 , 3 (2024), 2807--2842

    2024

  30. [38]

    On scales of S obolev spaces associated to generalized H ardy operators

    Merz, K. On scales of S obolev spaces associated to generalized H ardy operators. Math. Z. 299 , 1-2 (2021), 101--121

    2021

  31. [39]

    Mirek, M., S omian, W., and Szarek, T. Z. Some remarks on oscillation inequalities. Ergodic Theory Dynam. Systems 43 , 10 (2023), 3383--3412

    2023

  32. [40]

    M., and Trojan, B

    Mirek, M., Stein, E. M., and Trojan, B. ^p( Z^d) -estimates for discrete operators of R adon type: variational estimates. Invent. Math. 209 , 3 (2017), 665--748

    2017

  33. [41]

    M., and Zorin-Kranich, P

    Mirek, M., Stein, E. M., and Zorin-Kranich, P. A bootstrapping approach to jump inequalities and their applications. Anal. PDE 13 , 2 (2020), 527--558

    2020

  34. [42]

    Variational estimates for averages and truncated singular integrals along the prime numbers

    Mirek, M., Trojan, B., and Zorin-Kranich, P. Variational estimates for averages and truncated singular integrals along the prime numbers. Trans. Amer. Math. Soc. 369 , 8 (2017), 5403--5423

    2017

  35. [43]

    Variation and oscillation operators associated to semigroup generated by S chr\" o dinger operator with fractional power

    Nader, G. Variation and oscillation operators associated to semigroup generated by S chr\" o dinger operator with fractional power. J. Math. Anal. Appl. 523 , 1 (2023), Paper No. 127000, 26

    2023

  36. [44]

    A variation norm C arleson theorem

    Oberlin, R., Seeger, A., Tao, T., Thiele, C., and Wright, J. A variation norm C arleson theorem. J. Eur. Math. Soc. (JEMS) 14 , 2 (2012), 421--464

    2012

  37. [45]

    Pisier, G., and Xu, Q. H. The strong p -variation of martingales and orthogonal series. Probab. Theory Related Fields 77 , 4 (1988), 497--514

    1988

  38. [46]

    The p -variation of partial sum processes and the empirical process

    Qian, J. The p -variation of partial sum processes and the empirical process. Ann. Probab. 26 , 3 (1998), 1370--1383

    1998

  39. [47]

    Stein, E. M. Topics in harmonic analysis related to the L ittlewood- P aley theory . Annals of Mathematics Studies, No. 63. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970

    1970

  40. [48]

    Functional analysis , sixth ed., vol

    Yosida, K. Functional analysis , sixth ed., vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin-New York, 1980

    1980

  41. [49]

    Variational truncations of singular integrals on spaces of homogeneous type, 2020

    Zorin-Kranich, P. Variational truncations of singular integrals on spaces of homogeneous type, 2020. arXiv:2009.04541

    2020 arXiv

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