pith. sign in

arxiv: 2604.08984 · v1 · submitted 2026-04-10 · 🧮 math.CA

Adams type Dunkl Stein-Weiss inequality on Dunkl Morrey spaces on the real line

Sourav Dutta , Saswata Adhikari This is my paper

Pith reviewed 2026-05-10 17:28 UTC · model grok-4.3

classification 🧮 math.CA
keywords Dunkl operatorStein-Weiss inequalityDunkl-Morrey spacesfractional integral operatorweighted boundednessAdams inequalityreal line
0
0 comments X

The pith

The Adams-type Dunkl Stein-Weiss inequality holds for the Dunkl fractional integral operator on Dunkl-Morrey spaces.

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

This paper proves that the Adams-type Stein-Weiss inequality for fractional integrals extends to the Dunkl setting on the real line when the underlying spaces are Dunkl-Morrey spaces. The Dunkl operator modifies the usual derivative with reflection terms, and the result shows the inequality persists under this modification. The authors also prove weighted bounds for the Dunkl fractional maximal function in the same spaces. A sympathetic reader would care because these estimates are fundamental tools in harmonic analysis and the extension shows they remain valid in the presence of reflection symmetries.

Core claim

In this paper, we study the weighted boundedness of the Dunkl fractional integral operator associated with the Dunkl operator on the real line. We obtain the Adams-type Dunkl Stein-Weiss inequality on Dunkl-Morrey spaces. Our result extends the classical Adams type Stein-Weiss inequality on Morrey space result to the Dunkl setting. Furthermore, we establish the weighted boundedness of the Dunkl fractional maximal function on Dunkl Morrey spaces.

What carries the argument

The Dunkl fractional integral operator, built from the Dunkl kernel with reflection invariance, acting on Dunkl-Morrey spaces.

If this is right

  • The weighted boundedness of the Dunkl fractional maximal function holds on Dunkl Morrey spaces.
  • The classical Adams-type Stein-Weiss inequality carries over to the Dunkl framework on the real line.
  • Weighted estimates for the Dunkl fractional integral operator apply directly to functions in these reflection-adjusted Morrey spaces.

Where Pith is reading between the lines

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

  • The same extension technique might apply to Dunkl operators associated with other finite reflection groups beyond the real line.
  • These bounds could yield new a priori estimates for solutions of differential-difference equations involving the Dunkl Laplacian.

Load-bearing premise

The Dunkl fractional integral operator and the Dunkl-Morrey spaces satisfy the structural properties including reflection invariance and weight conditions needed for the Adams-type boundedness to hold.

What would settle it

A concrete counterexample consisting of a specific weight function and a test function in a Dunkl-Morrey space on the real line where the Dunkl fractional integral operator fails to satisfy the predicted Adams-type bound would falsify the claim.

read the original abstract

In this paper, we study the weighted boundedness of the Dunkl fractional integral operator (i.e., Dunkl Stein-Weiss inequality) associated with the Dunkl operator on $\mathbb{R}$. Indeed, we obtain the Adams-type Dunkl Stein-Weiss inequality on Dunkl-Morrey spaces. Our result extends the classical Adams type Stein-Weiss inequality on Morrey space result to the Dunkl setting. Furthermore, we establish the weighted boundedness of the Dunkl fractional maximal function on Dunkl Morrey spaces.

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.

Referee Report

1 major / 2 minor

Summary. The paper claims to study the weighted boundedness of the Dunkl fractional integral operator on the real line and obtain the Adams-type Dunkl Stein-Weiss inequality on Dunkl-Morrey spaces, extending the classical Adams-type Stein-Weiss inequality from Morrey spaces to the Dunkl setting. It further establishes the weighted boundedness of the Dunkl fractional maximal function on Dunkl-Morrey spaces.

Significance. If the central claims hold with rigorous verification of the required measure-theoretic properties, the work would provide a non-trivial extension of classical harmonic analysis inequalities to the Dunkl framework with its reflection-invariant weighted measure. This could be useful for applications involving Dunkl operators, though the abstract supplies no parameter ranges, proof sketches, or explicit checks on covering lemmas.

major comments (1)
  1. The extension of the Adams-type inequality relies on the Dunkl-Morrey spaces inheriting the reflection-invariance and doubling properties needed for the classical weak-type estimate plus covering argument to transfer. The manuscript should explicitly verify these for the weighted measure w_k(x)dx with w_k(x)=|x|^{2k} (k>0), including integrability of the Dunkl kernel near reflection points, as the abstract gives no indication that such a re-verification was performed.
minor comments (2)
  1. The abstract would be strengthened by including the precise ranges of parameters (e.g., for the order of the fractional integral and the Morrey exponent) under which the inequalities hold.
  2. Ensure that all definitions of the Dunkl-Morrey norm and the Dunkl fractional integral operator are stated explicitly in the main text, with clear references to the underlying Dunkl measure and prior literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the underlying measure-theoretic properties. We address the major comment below and will revise the paper accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: The extension of the Adams-type inequality relies on the Dunkl-Morrey spaces inheriting the reflection-invariance and doubling properties needed for the classical weak-type estimate plus covering argument to transfer. The manuscript should explicitly verify these for the weighted measure w_k(x)dx with w_k(x)=|x|^{2k} (k>0), including integrability of the Dunkl kernel near reflection points, as the abstract gives no indication that such a re-verification was performed.

    Authors: We agree that making these verifications explicit will strengthen the presentation. The weighted measure dμ_k(x) = |x|^{2k} dx is reflection-invariant by construction, as the weight w_k is even. The doubling property follows from the fact that μ_k is a power weight with exponent 2k > -1, which is known to be doubling on ℝ (see, e.g., standard results on Muckenhoupt weights in one dimension); we will add a short lemma in the preliminaries section confirming this explicitly for the Dunkl-Morrey spaces. Regarding the Dunkl kernel, its integrability near the reflection point x=0 is ensured by the known analytic properties of the Dunkl kernel (boundedness and smoothness away from the origin combined with the weight), and we will insert an explicit integrability estimate in the proof of the weak-type inequality to show that the classical covering argument transfers directly. These additions will be included in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity; extension rests on independent prior Dunkl theory and classical proofs

full rationale

The paper's central claim is an extension of the classical Adams-type Stein-Weiss inequality to the Dunkl setting on Morrey spaces, together with weighted boundedness of the Dunkl fractional maximal function. No quoted equations or steps reduce the target inequality to a quantity defined by the paper itself, a fitted parameter renamed as a prediction, or a self-citation chain whose cited result is unverified. The derivation invokes structural properties of the Dunkl operator and weighted measure that are taken from established Dunkl theory (reflection invariance, weighted doubling) rather than being smuggled in via ansatz or self-definition. The abstract and description give no indication that the main boundedness result is forced by construction from the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the standard definition and basic boundedness properties of the Dunkl operator together with the classical Adams-type inequality; no free parameters or new entities are mentioned.

axioms (1)
  • domain assumption Standard properties of the Dunkl operator and Dunkl-Morrey spaces on the real line
    The paper invokes the established theory of Dunkl operators to define the fractional integral and the underlying spaces.

pith-pipeline@v0.9.0 · 5382 in / 1371 out tokens · 100534 ms · 2026-05-10T17:28:39.723594+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    Abdelkefi, C., Sifi, M.: Dunkl translation and uncentered maximal operator on the real line. Int. J. Math. Math. Sci.2007, Art. ID 87808, 9 (2007)

    work page 2007

  2. [2]

    R.: A note on Riesz potentials

    Adams, D. R.: A note on Riesz potentials. Duke Math. J.,42, 765-778 (1975)

    work page 1975

  3. [3]

    P., Parui, S.: Existence of extremals of Dunkl-type Sobolev inequal- ity and of Stein–Weiss inequality for Dunkl Riesz potential

    Adhikari, S., Anoop, V. P., Parui, S.: Existence of extremals of Dunkl-type Sobolev inequal- ity and of Stein–Weiss inequality for Dunkl Riesz potential. Complex Analysis and Operator Theory,15(2), (2021)

    work page 2021

  4. [4]

    Indian J

    Adhikari, S., Parui, S.: The boundedness of Dunkl Bessel Riesz operators on Dunkl-type Morrey spaces. Indian J. Pure Appl. Math. 1-14 (2025)

    work page 2025

  5. [5]

    Beckner, W.: Pitt’s inequality with sharp convolution estimates. Proc. Am. Math. Soc.136(5), 1871-1885 (2008)

    work page 2008

  6. [6]

    De Jeu, M.F.E.: The Dunkl transform. Invent. Math.113, 147–162 (1993)

    work page 1993

  7. [7]

    F.: Reflection groups and orthogonal polynomials on the sphere

    Dunkl, C. F.: Reflection groups and orthogonal polynomials on the sphere. Math. Z.197(1), 33-60 (1988)

    work page 1988

  8. [8]

    F.: Differential-difference operators associated to reflection groups

    Dunkl, C. F.: Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc.311, 167-183 (1989)

    work page 1989

  9. [9]

    F.: Operators commuting with Coxeter group actions on polynomials

    Dunkl, C. F.: Operators commuting with Coxeter group actions on polynomials. Invariant theory and tableaux. 107-117 (1990)

    work page 1990

  10. [10]

    F.: Integral kernels with reflection group invariance

    Dunkl, C. F.: Integral kernels with reflection group invariance. Canad. J. Math.43(6), 1213- 1227 (1991)

    work page 1991

  11. [11]

    Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Contemp. Math.138, 123–138 (1992)

    work page 1992

  12. [12]

    L., Lieb, E

    Frank, R. L., Lieb, E. H.: Sharp constants in several inequalities on the Heisenberg group. Ann. Math.176, 349-381 (2012)

    work page 2012

  13. [13]

    B., Stein, E

    Folland, G. B., Stein, E. M.: Estimates for the ∂b complex and analysis on the Heisenberg group. Commun. Pure Appl. Math.27(4), 429-522 (1974)

    work page 1974

  14. [14]

    V., Ivanov, V

    Gorbachev, D. V., Ivanov, V. I., Tikhonov, S. Y.: Riesz potential and maximal function for Dunkl transform. Potential Anal.55(4), 513-538 (2021)

    work page 2021

  15. [15]

    Integral Transforms Spec

    Guliyev, V.S., Mammadov, Y.Y.: (L p, Lq) boundedness of the fractional maximal operator associated with the Dunkl operator on the real line. Integral Transforms Spec. Funct.21(8), 629-639 (2010)

    work page 2010

  16. [16]

    Guliyev, V.S., Mammadov, Y.Y.: On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line. J. Math. Anal. Appl.353(1), 449-459 (2009)

    work page 2009

  17. [17]

    S., Mustafayev, R

    Guliyev, V. S., Mustafayev, R. C., Serbetci, A.: Stein–Weiss inequalities for the fractional integral operators in Carnot groups and applications. Complex Var. Elliptic Equ55(8-10), 847-863 (2010)

    work page 2010

  18. [18]

    Hardy, G. H. and Littlewood, J. E.: Some properties of fractional integrals. I. Math. Zeit.,27, 565–606, (1927)

    work page 1927

  19. [19]

    P.: Two-weight norm, Poincar´ e, Sobolev and Stein–Weiss inequalities on Morrey spaces

    Ho, K. P.: Two-weight norm, Poincar´ e, Sobolev and Stein–Weiss inequalities on Morrey spaces. Publ. Res. Inst. Math. Sci.53(1), 119-139 (2017)

    work page 2017

  20. [20]

    Integral Transforms Spec

    Kassymov, A., Ruzhansky, M., Suragan, D.: Hardy–Littlewood–Sobolev and Stein–Weiss in- equalities on homogeneous Lie groups. Integral Transforms Spec. Funct.30(8), 643-655 (2019)

    work page 2019

  21. [21]

    Forum Math.34(5), 1147-1158 (2022)

    Kassymov, A., Ruzhansky, M., Suragan, D.: Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups. Forum Math.34(5), 1147-1158 (2022)

    work page 2022

  22. [22]

    A., Ruzhansky, M., Suragan, D.: Stein-Weiss-Adams inequality on Morrey spaces

    Kassymov, A., Ragusa, M. A., Ruzhansky, M., Suragan, D.: Stein-Weiss-Adams inequality on Morrey spaces. J. Funct. Anal.285(11), 110-152 (2023)

    work page 2023

  23. [23]

    H.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities

    Lieb, E. H.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math.118, 349-374 (1983)

    work page 1983

  24. [24]

    B.: On the solutions of quasi-linear elliptic partial differential equations

    Morrey, C. B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc.43(1), 126-166 (1938)

    work page 1938

  25. [25]

    A.: Transmutation operators associated with a Dunkl type differential-difference operator on the real line and certain of their applications

    Mourou, M. A.: Transmutation operators associated with a Dunkl type differential-difference operator on the real line and certain of their applications. Integral Transforms Spec. Funct. 12(1), 77-88 (2001)175(1), 75–121 (1995)

    work page 2001

  26. [26]

    Parashar, S., Adhikari, S.: Hardy-Littlewood maximal, generalized Bessel-Riesz and gener- alized fractional integral operators in generalized Morrey andBM O ϕ spaces associated with Dunkl operator on the real line. Fract. Calc. Appl. Anal. (2025). 22 Sourav Dutta, Saswata Adhikari

    work page 2025

  27. [27]

    Peetre, J.: On the theory ofL p spaces. J. Funct. Anal.4, 71–87(1969)

    work page 1969

  28. [28]

    Probability measures on groups and related structures XI

    Rosler, M.: Bessel-type signed hypergroups on R. Probability measures on groups and related structures XI. Proceedings, Oberwolfach, 292-304 (1994)

    work page 1994

  29. [29]

    Duke Math

    R¨ osler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J.98, 445-463 (1999)

    work page 1999

  30. [30]

    R¨ osler, M.: A positive radial product formula for the Dunkl kernel. Trans. Amer. Math. Soc. 355(6), 2413-2438 (2003)

    work page 2003

  31. [31]

    Ruzhansky, M., Yessirkegenov, N.: Hypoelliptic functional inequalities. Math. Zeit.,307(2), (2024)

    work page 2024

  32. [32]

    Samko, S.: Best constant in the weighted Hardy inequality: the spatial and spherical version. Fract. Calc. Appl. Anal.8(1), 39-52 (2005)

    work page 2005

  33. [33]

    Sifi, M., Soltani, F.: Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line. J. Math. Anal. Appl.270(1), 92-106 (2002)

    work page 2002

  34. [34]

    L.: On a theorem in functional analysis (Russian)

    Sobolev, S. L.: On a theorem in functional analysis (Russian). Math. Sbor.4, 471–497, (1938). [English translation in Amer. Math. Soc. Transl.34(2), 39–68, (1963)

    work page 1938

  35. [35]

    Soltani, F.: Lp-Fourier multipliers for the Dunkl operator on the real line. J. Funct. Anal. 209(1), 16-35 (2004)

    work page 2004

  36. [36]

    M.: Fractional integrals on n-dimensional Euclidean spaces

    Stein, E. M.: Fractional integrals on n-dimensional Euclidean spaces. J. Math. Mech.7, 503-514 (1958)

    work page 1958

  37. [37]

    Tanaka H.: Two-weight norm inequalities on Morrey spaces Ann. Acad. Sci. Fenn. Math.40, 773-791 (2015)

    work page 2015

  38. [38]

    and Xu, Y.: Convolution operator and maximal function for the Dunkl trans- form, J

    Thangavelu, S. and Xu, Y.: Convolution operator and maximal function for the Dunkl trans- form, J. Anal. Math.97(1), 25-55 (2005)

    work page 2005

  39. [39]

    Integral Transforms Spec

    Trime’Che, K.: Paley-Wiener theorems for the Dunkl transform and Dunkl translation oper- ators. Integral Transforms Spec. Funct.13(1), 17-38 (2002)

    work page 2002