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

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.

• The paper establishes boundedness of Dunkl fractional integral operators on one-dimensional Dunkl Morrey spaces with explicit parameter ranges.
• It introduces Adams-type Stein-Weiss inequalities that adapt classical weighted potential results to the Dunkl framework with reflection-induced weight structures.
• The methodology employs kernel decomposition and refined maximal operator estimates, offering new tools for analyzing PDEs with Dunkl symmetry.

Adams-Type Stein-Weiss Inequalities for Dunkl Morrey Spaces on the Real Line

Introduction and Motivation

The paper "Adams type Dunkl Stein-Weiss inequality on Dunkl Morrey spaces on the real line" (2604.08984) addresses the extension of classical weighted fractional integral inequalities—originally formulated in the Euclidean setting (Stein-Weiss and Adams inequalities)—to the setting of Dunkl analysis on the real line, specifically within Dunkl Morrey spaces. These generalizations are non-trivial due to the presence of the Dunkl operator, which intertwines differential-difference structures related to reflection groups and induces additional complications in harmonic analysis analogous to, but more subtle than, those encountered in standard Fourier analysis.

Classical Stein-Weiss inequalities provide weighted $L^p$-$L^q$ bounds for Riesz potentials, and their analogues in Morrey spaces (Adams type) are foundational in fine regularity theory of PDEs and potential analysis. The Dunkl setting, powered by the Dunkl operator $\Lambda_\nu$ and the associated transform, generalizes much of the harmonic analysis over $\mathbb{R}^n$, but introduces weighted measures and reflection group symmetries, complicating kernel estimates, maximal function controls, and translation structures.

Preliminaries: Dunkl Theory and Function Spaces

The Dunkl operator $\Lambda_\nu$ on $\mathbb{R}$, parametrized by $\nu \ge -\frac{1}{2}$, fuses differentiation with a reflection-induced term, generating a non-trivial generalization of the classical derivative. The operator's associated weight measure

$$d\mu_\nu(x) = (2^{\nu+1} \Gamma(\nu+1))^{-1} |x|^{2\nu+1} dx$$

dictates the natural setting for $L^p$ and Morrey-type spaces in Dunkl analysis.

Central analytic constructs include:

• Dunkl Transform $\mathcal{F}_\nu$: Generalizes the Fourier transform, involving the Dunkl kernel $L^q$0 (related to Bessel functions) and providing a unitary map on $L^q$1.
• Convolution and Translation: Defined via the Dunkl translation operator $L^q$2, essential for formulating integral operators and proving kernel estimates.
• Dunkl Morrey Spaces $L^q$3: Analogs of classical Morrey spaces using the Dunkl translation, allowing localization and scaling behaviors sensitive to Dunkl structure.

These tools permit the definition of the Dunkl fractional integral operator $L^q$4 (Dunkl–Riesz potential) and the Dunkl (fractional) maximal operator $L^q$5, generalizing the classical Riesz potential and Hardy-Littlewood maximal function, respectively.

Main Results: Adams-Type Stein-Weiss Inequality in Dunkl Morrey Spaces

The core achievements of the paper are:

1. Boundedness of Dunkl Fractional Integral Operators on Dunkl Morrey Spaces

Let $L^q$6, $L^q$7 with $L^q$8, and $L^q$9 satisfying

$\Lambda_\nu$0

Then

$\Lambda_\nu$1

is a bounded linear operator. The proof combines precise kernel decomposition, sharp use of Dunkl maximal operator bounds, and the peculiar scaling of Dunkl convolution kernels. Critical is the interpolation between localized ($\Lambda_\nu$2) and global ($\Lambda_\nu$3) control via Morrey scaling, tailored to the Dunkl dilation structure.

2. Adams-Type Dunkl Stein-Weiss Inequality

For $\Lambda_\nu$4, $\Lambda_\nu$5, and $\Lambda_\nu$6, with

$\Lambda_\nu$7

and $\Lambda_\nu$8, the following holds: $\Lambda_\nu$9 This generalizes both the Stein-Weiss weighted inequalities and the Adams-type inequalities for Morrey spaces to the Dunkl setting, respecting the reflection-induced weight structure and Dunkl-specific dilation invariants.

A methodical proof strategy decomposes the fractional integral kernel into three regions, handles localized and far-field interactions using pointwise kernel bounds, and utilizes weighted Hardy inequalities adapted to the Dunkl framework. The underlying structure is sufficiently robust to yield, as a consequence, strong-type estimates for the Dunkl fractional maximal function in the same weighted Morrey context.

Theoretical and Practical Implications

Theoretically, these results extend the reach of functional inequalities central to analysis and PDE theory (such as the Hardy-Littlewood-Sobolev and Stein-Weiss inequalities) into a broad family of weight-modified and reflection-invariant function spaces. The proper embedding of Dunkl Morrey spaces within this framework offers new tools for analyzing regularity and potential estimates in Dunkl-type PDEs and makes precise the quantitative interplay between weights, scaling, and symmetry encoded by the Dunkl operator.

Practically, such inequalities may find application in the regularity theory of equations invariant under reflection groups or featuring singular potential terms reflecting the Dunkl structure, such as certain nonlocal or nonclassical elliptic and parabolic PDEs, quantum models with symmetry, and analysis on special classes of Lie groups or hypergroups related to Dunkl theory.

Notably, the results are contingent on the precise range of parameters, and the proofs show nontrivial adaptations required to move classical potential and maximal function theory to the Dunkl context, particularly in weighted, quasi-local spaces. The required sharp kernel estimate techniques and intricate interplay of weights and scaling underscore the technical depth of the analysis.

Outlook and Future Developments

These advances prompt several future research avenues:

• Extending these weighted inequalities to higher dimensions, more general reflection group settings, or other Dunkl-type operators.
• Investigating further sharp constant questions and extremal functions, as well as endpoint cases within Dunkl Morrey and related Besov or Triebel-Lizorkin spaces.
• Applications to the regularity and boundedness of solutions to nonlocal Dunkl-type PDEs and potential analysis with reflection symmetry.
• Exploring analogs in discrete Dunkl frameworks and potential cross-connections with special functions and representation theory.

The robust extension of weighted potential theory to the Dunkl and reflection-invariant world opens up a wide research landscape at the intersection of harmonic analysis, special function theory, and partial differential equations.

Conclusion

The paper decisively establishes Adams-type Stein-Weiss inequalities for Dunkl fractional integral operators on one-dimensional Dunkl Morrey spaces, providing a comprehensive generalization of classical potential and maximal inequalities to settings governed by Dunkl symmetry. The results are technically strong, demonstrate precise control over kernel singularity and weight interplay, and generate new tools for analysis in non-Euclidean, reflection-invariant environments (2604.08984).