Dunkl-Hausdorff Operator

• The Dunkl-Hausdorff operator is a scaling-averaging operator defined in the Dunkl framework, replacing the Euclidean scaling with the Dunkl effective dimension dₐ = 2α+2.
• It is constructed using the Dunkl measure and translation, leading to boundedness results in Dunkl-type Morrey and Campanato spaces with classical limits at α = -½.
• Its fractional analogue introduces a parameter β, closely mimicking fractional integral operators and linking the study to classical Hardy, Calderón, and Orlicz theories.

The Dunkl-Hausdorff operator is a Hausdorff-type scaling operator adapted to the one-dimensional Dunkl setting associated with the reflection group $\mathbb{Z}_2$. In its basic form, it replaces the Euclidean scaling exponent by the Dunkl “dimension” $d_\alpha=2\alpha+2$, uses the weighted measure $d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$, and is studied on function spaces whose geometry is determined by Dunkl translation rather than ordinary translation. A systematic treatment of the operator and its fractional analogue, including boundedness on Dunkl-type Morrey and Campanato spaces and from $L^p(\mathbb{R},d\mu_\alpha)$ to $L^q(\mathbb{R},d\mu_\alpha)$, was given in "Hausdorff operators and fractional Hausdorff operators in the Dunkl setting" (Parashar et al., 15 Jun 2026).

1. Dunkl framework on the real line

Fix $\alpha\ge -\frac12$. The one-dimensional Dunkl operator associated with $\mathbb{Z}_2$ is

$$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$

When $\alpha=-\frac12$, this reduces to the classical derivative. The operator is a differential-difference operator, and the reflection term $f(x)-f(-x)$ encodes the underlying symmetry (Parashar et al., 15 Jun 2026).

The natural measure is

$d_\alpha=2\alpha+2$0

and the quantity

$d_\alpha=2\alpha+2$1

plays the role of an effective dimension. In particular, $d_\alpha=2\alpha+2$2, so scaling relations in the Dunkl setting are governed by $d_\alpha=2\alpha+2$3 rather than the Euclidean dimension (Parashar et al., 15 Jun 2026).

The Dunkl kernel $d_\alpha=2\alpha+2$4 is the unique solution of

$d_\alpha=2\alpha+2$5

and the Dunkl transform is defined for $d_\alpha=2\alpha+2$6 by

$d_\alpha=2\alpha+2$7

The transform satisfies a Plancherel theorem and an inversion formula. The corresponding Dunkl translation $d_\alpha=2\alpha+2$8 and convolution $d_\alpha=2\alpha+2$9 replace classical translation and convolution; the latter is commutative, associative, and satisfies a Young-type inequality (Parashar et al., 15 Jun 2026).

In this context, “Dunkl-type” means that the ambient measure is $d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$0, balls and averages are defined with the Dunkl weight, and the translation operator entering the function-space norms is $d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$1, not $d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$2 (Parashar et al., 15 Jun 2026).

2. Definition and representations of the operator

The classical Hausdorff operator on $d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$3 is

$d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$4

and on $d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$5 one has the Euclidean variant

$d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$6

In the Dunkl setting on $d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$7, the Dunkl-type Hausdorff operator is defined for $d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$8 by

$d\mu_\alpha(x)=A_\alpha |x|^{2\alpha+1}dx$9

The denominator exponent $L^p(\mathbb{R},d\mu_\alpha)$0 is the Dunkl replacement for the Euclidean dimension (Parashar et al., 15 Jun 2026).

For $L^p(\mathbb{R},d\mu_\alpha)$1, one recovers the classical one-dimensional Hausdorff operator,

$L^p(\mathbb{R},d\mu_\alpha)$2

Thus the Dunkl-Hausdorff operator is a deformation of the classical Hausdorff operator by the Dunkl scaling law (Parashar et al., 15 Jun 2026).

After the change of variables $L^p(\mathbb{R},d\mu_\alpha)$3, the operator admits the equivalent representation

$L^p(\mathbb{R},d\mu_\alpha)$4

This form makes the dependence on the Dunkl measure explicit and is the starting point for the fractional generalization (Parashar et al., 15 Jun 2026).

A key structural fact is the interaction with Dunkl translation. For $L^p(\mathbb{R},d\mu_\alpha)$5, $L^p(\mathbb{R},d\mu_\alpha)$6, and $L^p(\mathbb{R},d\mu_\alpha)$7,

$L^p(\mathbb{R},d\mu_\alpha)$8

This intertwining lemma is central in the Morrey and Campanato estimates because it permits the Hausdorff operator to pass through the translated local averages defining those norms (Parashar et al., 15 Jun 2026).

3. Fractional extension and adapted function spaces

The fractional Dunkl-type Hausdorff operator introduces a parameter $L^p(\mathbb{R},d\mu_\alpha)$9 and is defined by

$L^q(\mathbb{R},d\mu_\alpha)$0

When $L^q(\mathbb{R},d\mu_\alpha)$1, it reduces to $L^q(\mathbb{R},d\mu_\alpha)$2. When $L^q(\mathbb{R},d\mu_\alpha)$3, it becomes the classical fractional Hausdorff operator, modulo normalization factors (Parashar et al., 15 Jun 2026).

The relevant function spaces are Dunkl-type Morrey and Campanato spaces. For $L^q(\mathbb{R},d\mu_\alpha)$4, the Dunkl-type Morrey space $L^q(\mathbb{R},d\mu_\alpha)$5 is defined by the norm

$L^q(\mathbb{R},d\mu_\alpha)$6

Here the Dunkl translation appears inside the local $L^q(\mathbb{R},d\mu_\alpha)$7 average, so the space is not obtained by a formal substitution of weights into the classical Morrey definition (Parashar et al., 15 Jun 2026).

For the Campanato scale, the Dunkl-type Campanato space $L^q(\mathbb{R},d\mu_\alpha)$8 is given by

$L^q(\mathbb{R},d\mu_\alpha)$9

where

$\alpha\ge -\frac12$0

For $\alpha\ge -\frac12$1, this space coincides with $\alpha\ge -\frac12$2 in the Dunkl setting (Parashar et al., 15 Jun 2026).

The same paper emphasizes that Dunkl balls satisfy

$\alpha\ge -\frac12$3

which explains why $\alpha\ge -\frac12$4 controls both the Hausdorff kernel and the local scaling of Morrey and Campanato norms (Parashar et al., 15 Jun 2026).

4. Mapping properties and boundedness theory

A background $\alpha\ge -\frac12$5-result for the non-fractional operator states that if

$\alpha\ge -\frac12$6

then

$\alpha\ge -\frac12$7

The 2026 theory extends this to Dunkl-type Morrey and Campanato spaces (Parashar et al., 15 Jun 2026).

For Morrey spaces, if

$\alpha\ge -\frac12$8

then

$\alpha\ge -\frac12$9

With the same constant $\mathbb{Z}_2$0, one also has

$\mathbb{Z}_2$1

Thus the Dunkl-Hausdorff operator is bounded on both Dunkl-type Morrey and Campanato spaces under the same integrability condition on $\mathbb{Z}_2$2 (Parashar et al., 15 Jun 2026).

For the fractional operator, let

$\mathbb{Z}_2$3

and assume

$\mathbb{Z}_2$4

Then

$\mathbb{Z}_2$5

The relation $\mathbb{Z}_2$6 is the characteristic Sobolev-type scaling for fractional integrals in effective dimension $\mathbb{Z}_2$7 (Parashar et al., 15 Jun 2026).

The fractional Morrey result is more delicate. If $\mathbb{Z}_2$8, $\mathbb{Z}_2$9, $$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$0, $$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$1, and

$$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$2

then boundedness from $$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$3 to $$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$4 holds under the kernel condition

$$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$5

In that case,

$$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$6

The proof uses a decomposition $$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$7, Hölder and Minkowski inequalities, and a dyadic analysis of the tail term (Parashar et al., 15 Jun 2026).

A complementary line of work studies weighted Lebesgue spaces $$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$8. "Boundedness of Dunkl-Hausdorff operator in Lebesgue spaces" characterizes $$\Lambda_\alpha f(x)=\frac{df}{dx}(x)+\frac{2\alpha+1}{x}\,\frac{f(x)-f(-x)}{2},\qquad x\in\mathbb{R}.$$9-boundedness through weight ratios $\alpha=-\frac12$0, and for multiplicative weights $\alpha=-\frac12$1 gives the exact norm

$\alpha=-\frac12$2

Analogous results are proved in two dimensions (Jain et al., 2020).

5. Classical limit and relation to Hardy, Calderón, and Orlicz theory

The classical limit is obtained by setting $\alpha=-\frac12$3. Then $\alpha=-\frac12$4, $\alpha=-\frac12$5, the Dunkl transform becomes the classical Fourier transform, and the Dunkl-Hausdorff operator becomes the usual one-dimensional Hausdorff operator (Parashar et al., 15 Jun 2026).

This limit connects the operator to several standard integral operators. Choosing $\alpha=-\frac12$6 on $\alpha=-\frac12$7 yields the classical Hardy averaging operator, up to natural changes of variables, while $\alpha=-\frac12$8 yields the adjoint Hardy operator; in two variables one similarly recovers the two-dimensional Hardy operator (Jain et al., 2020). The 2025 Orlicz-space study treats

$\alpha=-\frac12$9

on non-negative non-increasing functions and analyzes both the operator and its quasi Dunkl-Hausdorff adjoint in weighted Orlicz spaces (Madan et al., 17 Aug 2025).

Under a structural condition on $f(x)-f(-x)$0,

$f(x)-f(-x)$1

the Orlicz theory shows that $f(x)-f(-x)$2 is comparable to a sum of Hardy-type terms,

$f(x)-f(-x)$3

This places the Dunkl-Hausdorff operator within the broader Hardy-Calderón framework while retaining the Dunkl parameter in the kernel (Madan et al., 17 Aug 2025).

The 2026 paper also notes that even in the classical Euclidean case the boundedness of the fractional Hausdorff operator on Morrey spaces had not been fully studied. A plausible implication is that the Dunkl results are informative not only as deformations of Euclidean theory but also as a route back to unresolved classical mapping questions through the limit $f(x)-f(-x)$4 (Parashar et al., 15 Jun 2026).

6. Conceptual role, methods, and adjacent constructions

Conceptually, the Dunkl-Hausdorff operator is a scaling-averaging operator that respects the Dunkl measure, the effective dimension $f(x)-f(-x)$5, and the reflection symmetry $f(x)-f(-x)$6. Its fractional version behaves like a fractional integral operator of order $f(x)-f(-x)$7 in Dunkl dimension $f(x)-f(-x)$8, with the same kind of $f(x)-f(-x)$9-$d_\alpha=2\alpha+2$00 scaling that appears in Riesz-potential theory (Parashar et al., 15 Jun 2026).

The proofs of boundedness rely on several specifically Dunkl-analytic ingredients: boundedness of the translation operator $d_\alpha=2\alpha+2$01, the intertwining lemma for $d_\alpha=2\alpha+2$02 and $d_\alpha=2\alpha+2$03, support control for translated characteristic functions, integral comparisons between intervals and Dunkl balls, Minkowski and Hölder inequalities, and a Young inequality on the multiplicative group $d_\alpha=2\alpha+2$04 with Haar measure $d_\alpha=2\alpha+2$05 (Parashar et al., 15 Jun 2026).

The present terminology should not be conflated with every Dunkl-type averaging construction. Some related papers provide surrounding operator calculus without explicitly introducing a Dunkl-Hausdorff operator in the harmonic-analysis sense. "A Dunkl Analogue of Operators Including Two-variable Hermite polynomials" constructs positive linear approximation operators with Dunkl kernels and reflection-adapted nodes, but does not define a Dunkl-Hausdorff operator (Aktaş et al., 2017). Likewise, "On The Dunkl Intertwining Opereator" gives an integral representation for $d_\alpha=2\alpha+2$06 with a kernel $d_\alpha=2\alpha+2$07 and positivity-preserving properties, which is structurally relevant to Dunkl-type averaging, but is not the same scaling operator (Maslouhi, 2016).

Natural extensions explicitly identified in the 2026 theory include higher-dimensional Dunkl settings attached to reflection groups such as $d_\alpha=2\alpha+2$08 and $d_\alpha=2\alpha+2$09, more general symbols $d_\alpha=2\alpha+2$10, and further function spaces such as Dunkl Hardy spaces, Besov and Triebel-Lizorkin spaces, and variable-exponent spaces. Commutators with $d_\alpha=2\alpha+2$11 functions are another stated direction. This suggests that the Dunkl-Hausdorff operator is best understood not as an isolated object but as part of a broader program transferring classical harmonic-analysis operators into the Dunkl framework (Parashar et al., 15 Jun 2026).