Fractional Hausdorff Operators

• Fractional Hausdorff operators are dilation-invariant integral operators modified by a fractional parameter, extending the classical averaging mechanism to include power-type kernels.
• They employ precise scaling laws and Mellin-type integrability conditions to ensure boundedness in weighted Lebesgue, Hardy, and analytic spaces.
• These operators generalize fractional integrals in various settings, including Dunkl and p-adic frameworks, linking classical fractional calculus with modern harmonic analysis.

Fractional Hausdorff operators are dilation-invariant integral operators in which the classical Hausdorff averaging mechanism is modified by a fractional homogeneity parameter, by power-type kernel factors, or by ambient scaling parameters attached to the underlying function space. Representative models include the Euclidean fractional operator

$$\mathcal{H}_{\psi,\beta}f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$

the one-dimensional weighted variants studied on Lebesgue and Hardy scales, and analytic upper-half-plane operators obtained by fractionalizing the kernel in

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$

Across these formulations, the common structure is an average over dilations or multiplicative translations, with boundedness governed by Mellin-type moments of the kernel and by scaling relations between source and target spaces (Parashar et al., 15 Jun 2026, Yu et al., 26 Sep 2025, Hung et al., 7 May 2025).

1. Definitions and principal models

The classical Hausdorff operator on the upper half-plane and on the real line is defined by

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$

with measurable symbol $\varphi:(0,\infty)\to\mathbb C$. In both forms, the operator is a weighted average of dilations of the function $f$, with dilation factor $t$ and weight $\varphi(t)/t$ (Hung et al., 7 May 2025). On holomorphic Hardy spaces of the upper half-plane, the same formula underlies the complete norm theory for nonnegative kernels (Hung et al., 2017).

A direct fractionalization introduces a parameter $\beta$ into the homogeneity of the denominator. In the classical $n$-dimensional form,

$$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$

and for $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$0 this reduces to the usual Hausdorff operator (Parashar et al., 15 Jun 2026). In one dimension, Yu and Li study the corresponding fractional variants $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$1 for $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$2, with $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$3 entering simultaneously into the kernel homogeneity, the scaling relation between $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$4 and $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$5, and the weighted Hardy-space balance conditions (Yu et al., 26 Sep 2025).

A second parameterization does not alter the denominator but instead absorbs the fractional order into the kernel. Mirotin’s general Hausdorff-type framework covers operators of the form

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$6

and the prototypical fractional Euclidean example

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$7

appears as a one-variable kernel Hausdorff-type operator with $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$8 (Mirotin, 17 Jun 2025). A closely related abstract formulation,

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$9

treats fractional behavior through non-integer powers of the modulus $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$0 or of the Lipschitz constant $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$1 of the automorphism family (Mirotin, 2023).

The terminology is therefore not completely uniform. Some works define fractional Hausdorff operators explicitly through a parameter $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$2; others identify fractionality through power weights in the kernel; and some papers on analytic Hausdorff operators do not define a separate fractional operator but provide the framework from which such operators arise naturally (Yu et al., 26 Sep 2025, Mirotin, 17 Jun 2025, Hung et al., 7 May 2025).

2. Scaling laws, Mellin structure, and integrability conditions

A central feature of the subject is that boundedness is controlled by precise scaling identities. In the one-dimensional Euclidean Lebesgue theory of Yu and Li, the basic relation is

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$3

while in power-weighted Hardy spaces it becomes

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$4

These formulas are the fractional analogues of the non-fractional Hausdorff scaling constraint and are necessary in the Hardy setting (Yu et al., 26 Sep 2025).

In analytic spaces on the upper half-plane, the decisive quantity is a Mellin-type moment of the symbol. For weighted Bergman spaces $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$5 and power weighted Hardy spaces $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$6, the key condition is

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$7

For the Dirichlet space $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$8, the relevant hypothesis is instead

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt,\qquad H_\varphi(f^*)(x)=\int_0^\infty f^*\!\left(\frac{x}{t}\right)\frac{\varphi(t)}{t}\,dt,$$9

These are explicitly described as Mellin-type integrability conditions, and they are precisely where fractional exponents enter the theory (Hung et al., 7 May 2025).

The same Mellin structure appears in abstract $\varphi:(0,\infty)\to\mathbb C$0 theory. In the two-variable kernel formalism, if $\varphi:(0,\infty)\to\mathbb C$1 on $\varphi:(0,\infty)\to\mathbb C$2, then the measure-distortion factor is $\varphi:(0,\infty)\to\mathbb C$3. Specializing the general $\varphi:(0,\infty)\to\mathbb C$4 criterion yields

$\varphi:(0,\infty)\to\mathbb C$5

as a sufficient condition for boundedness of the fractional Hausdorff operator $\varphi:(0,\infty)\to\mathbb C$6 on $\varphi:(0,\infty)\to\mathbb C$7, with the operator norm bounded by the same integral (Mirotin, 17 Jun 2025).

Non-Euclidean analogues preserve the same structure after replacing the Euclidean dimension by an effective or homogeneous dimension. In the Dunkl setting, $\varphi:(0,\infty)\to\mathbb C$8 plays that role, and the fractional $\varphi:(0,\infty)\to\mathbb C$9-$f$0 scaling is

$f$1

The boundedness condition on the kernel $f$2 is expressed by finiteness of

$f$3

which is the Dunkl analogue of a weighted Mellin moment (Parashar et al., 15 Jun 2026).

A recurrent misconception is that “fractional” refers only to the explicit denominator $f$4. The literature shows a broader picture: fractionality can be encoded by $f$5, by kernel factors such as $f$6, by Mellin moments $f$7, or by effective dimensions such as $f$8 and modulus powers $f$9 (Yu et al., 26 Sep 2025, Mirotin, 17 Jun 2025, Parashar et al., 15 Jun 2026).

3. Weighted Lebesgue and Hardy theories

The sharpest one-dimensional results presently available are due to Yu and Li. On weighted Lebesgue spaces $t$0, they study even weights $t$1 that are either increasing or decreasing on $t$2. For increasing weights, boundedness

$t$3

is characterized by a two-weight Hardy-type condition $t$4, under the hypotheses

$t$5

together with local and large-scale conditions on the kernel $t$6. For decreasing weights, the analogous characterization uses a condition $t$7 and requires

$t$8

The lower bounds on $t$9 and $\varphi(t)/t$0 are explicitly identified as new constraints that do not appear in the non-fractional case $\varphi(t)/t$1 (Yu et al., 26 Sep 2025).

On power-weighted Hardy spaces $\varphi(t)/t$2, the same paper establishes that the relation

$\varphi(t)/t$3

is necessary for boundedness

$\varphi(t)/t$4

Sufficient conditions are obtained in two distinct ways. One route uses the radial maximal function characterization of weighted Hardy spaces together with weighted Campanato–Morrey duality, producing weak-type $\varphi(t)/t$5 estimates and, after Marcinkiewicz interpolation, strong $\varphi(t)/t$6 bounds. The second route uses the Hilbert transform characterization of $\varphi(t)/t$7, the identity $\varphi(t)/t$8, and interpolation to obtain a parallel strong-type theorem under a compact-support assumption on $\varphi(t)/t$9 (Yu et al., 26 Sep 2025).

For non-fractional holomorphic Hardy spaces on the upper half-plane, the operator norm is exactly determined by a Mellin moment of the kernel: $\beta$0 and

$\beta$1

This exact norm identity supplies a model case for fractional-type kernels of one-sided power form, even though the paper itself studies the nonnegative Hausdorff operator rather than a separately named fractional variant (Hung et al., 2017).

The methodological core of this part of the theory combines group convolution and Young’s inequality on the multiplicative group $\beta$2, two-weight fractional Hardy inequalities, maximal-function and Hilbert-transform characterizations of weighted Hardy spaces, and interpolation. These tools explain why fractional Hausdorff operators sit simultaneously in dilation analysis, weighted harmonic analysis, and Hardy-space theory (Yu et al., 26 Sep 2025).

4. Analytic function spaces and boundary-value compatibility

The upper-half-plane analytic theory extends Hausdorff operators beyond real-variable spaces. For $\beta$3, $\beta$4, and measurable $\beta$5, the operator

$\beta$6

is bounded on the weighted Bergman space $\beta$7 whenever

$\beta$8

and then

$\beta$9

For nonnegative kernels, the same condition is also necessary. The operator norm is comparable to $n$0, with constants depending only on $n$1. Exactly the same scalar condition is necessary and sufficient for boundedness on the power weighted Hardy space $n$2 under $n$3, again with norm comparability (Hung et al., 7 May 2025).

The Dirichlet space requires a different threshold: $n$4 is sufficient for boundedness on $n$5, and the operator norm is controlled by the same logarithmic integral. The paper explains that this logarithmic condition appears naturally because $n$6 is one derivative more singular than Bergman space (Hung et al., 7 May 2025).

A structurally important identity is

$n$7

used repeatedly in Hardy and Dirichlet estimates. Equally important is boundary compatibility: if $n$8 has boundary values $n$9, then under $$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$0,

$$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$1

Consequently, the real Hausdorff operator is bounded on $$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$2, with norm comparable to $$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$3, and it commutes with the Hilbert transform whenever $$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$4 (Hung et al., 7 May 2025).

This analytic theory is also where several fractional themes become explicit. The paper notes that the Riemann–Liouville fractional integral operators and the Weyl fractional integral operators can be derived from the Hausdorff operators, and it proposes a natural fractionalization

$$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$5

A plausible implication, stated in the paper as a suggestion rather than a theorem, is that boundedness on $$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$6 or $$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$7 should then be governed by

$$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$8

In that sense, the analytic upper-half-plane theory provides a template for fractional Hausdorff operators even when it does not isolate them as a separate class (Hung et al., 7 May 2025).

5. Generalized, logarithmic, and non-Euclidean extensions

A major line of recent work replaces the fixed Euclidean dilation model by very general automorphism families. In Mirotin’s two-variable-kernel theory, the operator

$$\mathcal{H}_{\psi,\beta} f(x)=\int_{\mathbb{R}^n}\frac{\psi(x/y)}{|y|^{n-\beta}}\,f(y)\,dy,\qquad 0\le \beta<n,$$9

acts from $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$00 to $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$01 under a weak agreement condition

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$02

If a mixed norm of the kernel is finite, then $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$03 is bounded $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$04. For one-variable kernels on $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$05 with $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$06, this recovers the fractional criterion

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$07

for boundedness of $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$08 on $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$09 (Mirotin, 17 Jun 2025).

A related categorical framework defines

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$10

for automorphisms in abstract categories. On $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$11, boundedness follows from

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$12

and on atomic Hardy spaces of homogeneous type it is controlled by

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$13

Because the theory allows arbitrary real powers of $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$14 and $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$15, it is directly applicable to fractional kernel choices (Mirotin, 2023).

Non-Euclidean harmonic analysis supplies further concrete realizations. In the Dunkl setting, the fractional Dunkl-type Hausdorff operator is

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$16

When $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$17, this reduces to the classical one-dimensional fractional Hausdorff operator. The operator is bounded from $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$18 to $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$19 under

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$20

and finiteness of $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$21, and it also admits a Morrey-space theory under structural hypotheses on the kernel (Parashar et al., 15 Jun 2026).

The p-adic theory takes a different but parallel form: $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$22 Weak-type estimates are proved on weighted p-adic weak Lebesgue spaces, strong-type estimates on weighted p-adic Lorentz spaces follow by Marcinkiewicz interpolation, and commutators with Lipschitz symbols are controlled by additional kernel integrability conditions (Sarfraz et al., 2019).

There is also a logarithmic or Mellin-side extension. Hadamard fractional integrals and derivatives of variable order are described as fractional operators of Hausdorff type adapted to multiplicative, logarithmic structures. Their kernels depend on $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$23 and the measure $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$24, placing them conceptually within the Hausdorff–Mellin family rather than the additive Riesz family (Almeida et al., 2014). Likewise, in the Opdam–Cherednik setting, the Hausdorff operator

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$25

includes the Riemann–Liouville fractional derivative as a special case for a specific power-type kernel $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$26, even though the paper does not introduce a separate “fractional Hausdorff operator” by name (Mondal et al., 2021).

6. Classical descendants, methods, and conceptual issues

A defining feature of the subject is that many named operators appear as special cases of the Hausdorff mechanism. The analytic upper-half-plane literature explicitly lists the classical Hardy operator, its adjoint operator, the Cesàro type operators, the Erdélyi–Kober fractional integral operators, the de La Vallée-Poussin type operators, the Picar and Bessel operators, the Stieltjes type operators, and also states that the Riemann–Liouville fractional integral operators and the Weyl fractional integral operators can be derived from the Hausdorff operators (Hung et al., 7 May 2025). In Yu and Li’s one-dimensional fractional theory, appropriate choices of $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$27 recover the fractional Hardy operator, the fractional adjoint Hardy operator, and the fractional Hardy–Littlewood–Pólya operator (Yu et al., 26 Sep 2025).

A particularly explicit bridge to generalized fractional calculus is provided by multiple Erdélyi–Kober operators involving Fox’s $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$28-function. The operators

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$29

are Hausdorff-type operators with kernels $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$30 given by Fox $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$31-functions. Under delta-neutral and parameter positivity conditions, they are bounded on $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$32, with operator norms controlled by the explicit integrals

$$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$33

The paper also emphasizes that general Hausdorff-operator results on $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$34 cannot be applied directly unless certain positivity conditions are imposed, which is one of the clearest technical caveats in the area (Chen et al., 20 Jul 2025).

Methodologically, the literature is highly coherent despite the diversity of settings. Recurrent tools include pointwise growth estimates on Bergman and Hardy spaces, test functions for lower bounds, Mellin-transform or multiplicative-group viewpoints, group convolution and Young’s inequality, two-weight fractional Hardy inequalities, radial maximal function and Hilbert transform characterizations of Hardy spaces, boundary-value transfer from analytic to real-variable settings, dyadic decompositions, and interpolation. These methods explain why the same scaling exponents reappear in Euclidean, analytic, Dunkl, p-adic, and abstract automorphism-based theories (Hung et al., 7 May 2025, Yu et al., 26 Sep 2025, Parashar et al., 15 Jun 2026).

The most important conceptual clarification is that fractional Hausdorff operators do not form a single rigidly standardized class. The current literature supports several structurally compatible viewpoints: a direct fractional homogeneity parameter $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$35, a power-modified kernel $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$36, an analytic Mellin-type fractionalization $$\mathscr H_\varphi f(z)=\int_0^\infty f\!\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}\,dt.$$37, and non-Euclidean versions in which the ambient dimension is replaced by a homogeneous dimension or by an automorphism modulus. What remains stable across these viewpoints is the Hausdorff principle itself: averaging over dilations or automorphisms, with mapping properties determined by the interaction between kernel moments and the scaling geometry of the underlying space (Mirotin, 17 Jun 2025, Mirotin, 2023, Parashar et al., 15 Jun 2026).