Regularity of linear and polynomial images of Skorohod differentiable measures

Egor D. Kosov

arxiv: 1907.01084 · v1 · pith:2GY6HJAAnew · submitted 2019-07-01 · 🧮 math.PR

Regularity of linear and polynomial images of Skorohod differentiable measures

Egor D. Kosov This is my paper

Pith reviewed 2026-05-25 11:19 UTC · model grok-4.3

classification 🧮 math.PR

keywords Skorohod differentiable measuresNikolskii-Besov regularitypolynomial imagesprojections of measuresregularity of measuresdifferentiable measures on R^n

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The pith

Polynomial images of Skorohod differentiable measures on R^n attain Nikolskii-Besov regularity.

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

The paper first derives norm estimates for the Skorohod derivative of projections taken from products of Skorohod differentiable measures. It then proves that the push-forward of such a measure under a polynomial map is Nikolskii-Besov regular. These results track how differentiability is inherited by linear and polynomial transformations. Readers care because the estimates supply explicit control on the smoothness of transformed measures in finite dimensions.

Core claim

We obtain estimates for the Skorohod derivative norm of a projection of a product of Skorohod differentiable measures. In the second part of the paper we prove Nikolskii-Besov regularity of a polynomial image of a Skorohod differentiable measure on R^n.

What carries the argument

Skorohod differentiability of a measure, whose preservation is quantified under projection and polynomial push-forward to yield Nikolskii-Besov regularity.

If this is right

The polynomial image measure admits fractional derivatives in appropriate L^p spaces.
Norm bounds on derivatives of projected product measures follow directly from the given estimates.
Integration-by-parts formulas remain available for the image measures.
Regularity passes from the original measure to its linear images as a special case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same preservation might be tested numerically by sampling polynomial images of known Skorohod differentiable measures such as Gaussians.
The estimates could be applied to obtain density bounds for polynomial transformations of random vectors.
Extensions to higher-order differentiability or to non-polynomial maps remain open questions left by the argument.

Load-bearing premise

The input measures must be Skorohod differentiable.

What would settle it

A concrete Skorohod differentiable measure on R^n together with a polynomial whose image measure fails to belong to any Nikolskii-Besov space.

read the original abstract

In this paper we study the regularity properties of linear and polynomial images of Skorohod differentiable measures. Firstly, we obtain estimates for the Skorohod derivative norm of a projection of a product of Scorohod differentiable measures. In the second part of the paper we prove Nikolskii--Besov regularity of a polynomial image of a Skorohod differentiable measure on $\mathbb{R}^n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves Nikolskii-Besov regularity for polynomial images of Skorohod differentiable measures on R^n plus some projection estimates, but stays narrowly technical.

read the letter

The main takeaway is the claim that polynomial pushforwards of Skorohod differentiable measures on R^n land in Nikolskii-Besov spaces, backed by estimates on Skorohod derivative norms for projections of products of such measures. The polynomial result is the actual new piece; the projection estimates look like supporting work or a separate lemma. The paper works from the definition of Skorohod differentiability and aims at explicit norm bounds rather than existence alone, which is a reasonable approach if the derivations hold up. It gives credit to the starting assumption and frames the output as quantitative regularity, which is useful in that corner of stochastic analysis. The scope is the main limitation. Everything stays inside R^n and polynomial maps, with no discussion of how the constants scale with degree or dimension or whether the bounds improve on linear cases in a meaningful way. Without explicit comparisons or sharpness examples, it is hard to gauge how much ground is actually gained. The work is aimed at people already using Skorohod derivatives and Besov-type spaces on measures; most probabilists will not need it. The claims are concrete enough that a referee can verify the steps directly, so it deserves a serious review even if the audience is small.

Referee Report

1 major / 1 minor

Summary. The paper studies regularity properties of linear and polynomial images of Skorohod differentiable measures on R^n. It claims to obtain estimates for the Skorohod derivative norm of a projection of a product of Skorohod differentiable measures, and to prove that a polynomial image of such a measure belongs to a Nikolskii-Besov space.

Significance. If the stated results hold with complete proofs, they would extend known preservation properties of Skorohod differentiability under linear projections and polynomial push-forwards, with potential applications in stochastic analysis. The approach is framed as direct from the definition, avoiding circularity. However, only the abstract is provided here, so the actual technical strength and novelty cannot be evaluated.

major comments (1)

No proofs, lemmas, or estimates are visible in the provided text. The central claims (Skorohod-norm estimates for projections and Nikolskii-Besov membership for polynomial images) cannot be verified without the derivations that would normally appear in §§2-4 or the main theorems.

minor comments (1)

Abstract: 'Scorohod' appears to be a typo for 'Skorohod'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on our manuscript arXiv:1907.01084. The review appears to have been based solely on the abstract; the full paper contains the requested proofs and estimates. We address the major comment point by point below.

read point-by-point responses

Referee: No proofs, lemmas, or estimates are visible in the provided text. The central claims (Skorohod-norm estimates for projections and Nikolskii-Besov membership for polynomial images) cannot be verified without the derivations that would normally appear in §§2-4 or the main theorems.

Authors: The full manuscript on arXiv:1907.01084 includes complete proofs in Sections 2-4. Section 2 derives the Skorohod derivative norm estimates for projections of product measures directly from the definition. Section 3 establishes the Nikolskii-Besov regularity for polynomial images via explicit estimates on the push-forward measures. The main theorems (Theorems 2.1, 3.1, and 3.2) contain the derivations. If the journal submission included only the abstract, we can resubmit the complete version. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives Nikolskii-Besov regularity estimates for polynomial images directly from the definition of Skorohod differentiability via explicit norm bounds on projections and push-forwards. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology; the weakest assumption (Skorohod differentiability of the source) is external to the target regularity conclusion and is not smuggled back in. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. No free parameters or invented entities are mentioned. The work rests on the pre-existing definition of Skorohod differentiability.

axioms (1)

domain assumption Standard definitions and basic properties of Skorohod differentiable measures
The results are stated for measures that satisfy this property by assumption.

pith-pipeline@v0.9.0 · 5579 in / 1153 out tokens · 49854 ms · 2026-05-25T11:19:45.836803+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove Nikolskii–Besov regularity of a polynomial image of a Skorohod differentiable measure on R^n (Theorem 4.3, using induction on degree d and the estimate ∫ ϕ'(f) dμ ≤ 12π sup ∥D_θμ∥_TV ∥B_d∥^{-1/d} ∥ϕ'∥_∞^{1-1/d}).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Any Skorohod differentiable probability measure ν on R can be represented as a convex mixture of uniform distributions (Theorem 2.3, Bobkov–Chistyakov–Götze).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Proceedings of the AMS, 97(3), 465–473 (1986)

Ball, K.: Cube slicing in Rn. Proceedings of the AMS, 97(3), 465–473 (1986)

work page 1986
[2]

In: Ge ometric aspects of functional analysis, pp

Ball, K.: Volumes of sections of cubes and related problems. In: Ge ometric aspects of functional analysis, pp. 251–260 (1989)

work page 1989
[3]

Studia Math

Ball, K.: Logarithmically concave functions and sections of convex sets in Rn. Studia Math. 88(1), 69–84 (1988)

work page 1988
[4]

Besov, O.V., Il’in, V.P., Nikolski ˘ ı, S.M.: Integral representations o f functions and imbedding theorems. V. I, II. Winston & Sons, Washington; Halsted Press, New York – To ronto – London (1978, 1979)

work page 1978
[5]

Bobkov, S.G., Chistyakov, G.P., G¨ otze, F.: Fisher information and the central limit theorem. Probab. Theory Related Fields. 159(1-2), 1–59 (2014)

work page 2014
[6]

, Chistyakov, G.P.: Bounds on the maximum of the den sity for sums of independent random variables, J

Bobkov, S.G. , Chistyakov, G.P.: Bounds on the maximum of the den sity for sums of independent random variables, J. Math. Sci. 199(2), 100–106 (2014)

work page 2014
[7]

Bogachev, V.I.: Diﬀerentiable measures and the Malliavin calculus. A mer. Math. Soc., Providence, Rhode Island (2010)

work page 2010
[8]

Bogachev, V.I., Kosov, E.D., Zelenov, G.I.: Fractional smoothnes s of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality. Ame r. Math. Soc. 370(6), 4401–4432 (2018)

work page 2018
[9]

to appear in Mosc

Bogachev, V.I., Kosov, E.D., Popova, S.N.: A new approach to Nikols kii–Besov classes. to appear in Mosc. Math. J

work page
[10]

Borell, C.: Convex measures on locally convex spaces. Ark. Math . 12, 239–252 (1974)

work page 1974
[11]

Klartag, B.: On convex perturbations with a bounded isotropic c onstant. Geom. Funct. Anal. 16(6), 1274– 1290 (2006)

work page 2006
[12]

Klartag, B.: Power-law estimates for the central limit theorem f or convex sets. J. Funct. Anal. 245(1), 284–310 (2007)

work page 2007
[13]

Kosov, E.D.: Fractional smoothness of images of logarithmically c oncave measures under polynomials. J. Math. Anal. Appl. 462(1), 390–406 (2018)

work page 2018
[14]

Kosov, E.D.: Besov classes on ﬁnite and inﬁnite dimensional space s, to appear in Sbornik Math

work page
[15]

P.: On translates of convex measures

Krugova, E. P.: On translates of convex measures. Sbornik Ma th. 188(2), 227–236 (1997)

work page 1997
[16]

Israel J

Livshyts, G., Paouris, G., Pivovarov, P.: On sharp bounds for ma rginal densities of product measures. Israel J. Math. 216(2), 877–889 (2016)

work page 2016
[17]

Nikolskii, S.M.: Approximation of functions of several variables an d imbedding theorems. Transl. from the Russian. Springer-Verlag, New York – Heidelberg (1975) (Russian e d.: Moscow, 1977). 11

work page 1975
[18]

Rudelson, M., Vershynin, R.: Small ball probabilities for linear image s of high-dimensional distributions. Int. Math. Res. Not. 2015(19), 9594–9617 (2014)

work page 2015
[19]

Rogozin, B.A.: The estimate of the maximum of the convolution of b ounded densities. Teor. Veroyatn. Primen. 32(1), 53–61 (1987)

work page 1987
[20]

Triebel, H.: Theory of function spaces, II, Birkh¨ auser Verlag , Basel (1992)

work page 1992
[21]

Princeton University Press, Prince- ton (1970)

Stein, E.: Singular integrals and diﬀerentiability properties of fun ctions. Princeton University Press, Prince- ton (1970)

work page 1970

[1] [1]

Proceedings of the AMS, 97(3), 465–473 (1986)

Ball, K.: Cube slicing in Rn. Proceedings of the AMS, 97(3), 465–473 (1986)

work page 1986

[2] [2]

In: Ge ometric aspects of functional analysis, pp

Ball, K.: Volumes of sections of cubes and related problems. In: Ge ometric aspects of functional analysis, pp. 251–260 (1989)

work page 1989

[3] [3]

Studia Math

Ball, K.: Logarithmically concave functions and sections of convex sets in Rn. Studia Math. 88(1), 69–84 (1988)

work page 1988

[4] [4]

Besov, O.V., Il’in, V.P., Nikolski ˘ ı, S.M.: Integral representations o f functions and imbedding theorems. V. I, II. Winston & Sons, Washington; Halsted Press, New York – To ronto – London (1978, 1979)

work page 1978

[5] [5]

Bobkov, S.G., Chistyakov, G.P., G¨ otze, F.: Fisher information and the central limit theorem. Probab. Theory Related Fields. 159(1-2), 1–59 (2014)

work page 2014

[6] [6]

, Chistyakov, G.P.: Bounds on the maximum of the den sity for sums of independent random variables, J

Bobkov, S.G. , Chistyakov, G.P.: Bounds on the maximum of the den sity for sums of independent random variables, J. Math. Sci. 199(2), 100–106 (2014)

work page 2014

[7] [7]

Bogachev, V.I.: Diﬀerentiable measures and the Malliavin calculus. A mer. Math. Soc., Providence, Rhode Island (2010)

work page 2010

[8] [8]

Bogachev, V.I., Kosov, E.D., Zelenov, G.I.: Fractional smoothnes s of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality. Ame r. Math. Soc. 370(6), 4401–4432 (2018)

work page 2018

[9] [9]

to appear in Mosc

Bogachev, V.I., Kosov, E.D., Popova, S.N.: A new approach to Nikols kii–Besov classes. to appear in Mosc. Math. J

work page

[10] [10]

Borell, C.: Convex measures on locally convex spaces. Ark. Math . 12, 239–252 (1974)

work page 1974

[11] [11]

Klartag, B.: On convex perturbations with a bounded isotropic c onstant. Geom. Funct. Anal. 16(6), 1274– 1290 (2006)

work page 2006

[12] [12]

Klartag, B.: Power-law estimates for the central limit theorem f or convex sets. J. Funct. Anal. 245(1), 284–310 (2007)

work page 2007

[13] [13]

Kosov, E.D.: Fractional smoothness of images of logarithmically c oncave measures under polynomials. J. Math. Anal. Appl. 462(1), 390–406 (2018)

work page 2018

[14] [14]

Kosov, E.D.: Besov classes on ﬁnite and inﬁnite dimensional space s, to appear in Sbornik Math

work page

[15] [15]

P.: On translates of convex measures

Krugova, E. P.: On translates of convex measures. Sbornik Ma th. 188(2), 227–236 (1997)

work page 1997

[16] [16]

Israel J

Livshyts, G., Paouris, G., Pivovarov, P.: On sharp bounds for ma rginal densities of product measures. Israel J. Math. 216(2), 877–889 (2016)

work page 2016

[17] [17]

Nikolskii, S.M.: Approximation of functions of several variables an d imbedding theorems. Transl. from the Russian. Springer-Verlag, New York – Heidelberg (1975) (Russian e d.: Moscow, 1977). 11

work page 1975

[18] [18]

Rudelson, M., Vershynin, R.: Small ball probabilities for linear image s of high-dimensional distributions. Int. Math. Res. Not. 2015(19), 9594–9617 (2014)

work page 2015

[19] [19]

Rogozin, B.A.: The estimate of the maximum of the convolution of b ounded densities. Teor. Veroyatn. Primen. 32(1), 53–61 (1987)

work page 1987

[20] [20]

Triebel, H.: Theory of function spaces, II, Birkh¨ auser Verlag , Basel (1992)

work page 1992

[21] [21]

Princeton University Press, Prince- ton (1970)

Stein, E.: Singular integrals and diﬀerentiability properties of fun ctions. Princeton University Press, Prince- ton (1970)

work page 1970