Fractional Hankel–Sobolev Spaces

• Fractional Hankel–Sobolev spaces are Hilbert spaces defined on (0,∞) using a weighted norm with the fractional Hankel–Bessel transform to capture Sobolev regularity.
• They generalize classical L²-based Sobolev spaces by introducing the parameter α, which interpolates between distinct analytical settings and diagonalizes Bessel operators.
• These spaces exhibit robust properties including continuous embeddings, a well-structured Hilbert scale, and uniform kernel decay estimates that aid in pseudo-differential analysis.

A fractional Hankel–Sobolev space is a Hilbert space of functions on $(0,\infty)$ whose regularity is measured via the fractional Hankel–Bessel transform $\mathcal{H}_\mu^\alpha$, itself a fractionalization of the classical Hankel transform. These spaces, denoted $H^s_{\alpha,\mu}$ with $s\in\mathbb{R}$, $\mu > -\frac12$, and $\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$, arise naturally in the global analysis of pseudo-differential operators associated with Bessel operators when the underlying Fourier analysis is replaced with $\mathcal{H}_\mu^\alpha$ (Pasawan, 6 Jan 2026). Their structure and properties parallel classical $L^2$-based Sobolev spaces but incorporate the spectral geometry associated with the Bessel differential operator $L_{\mu}$. The key novelty is the presence of the fractional Hankel transform parameter $\alpha$, which controls a family of unitary transforms and interpolates between distinct analytical settings.

1. Fractional Hankel–Bessel Transform

Let $\mathcal{H}_\mu^\alpha$0 and $\mathcal{H}_\mu^\alpha$1. The fractional Hankel–Bessel transform of a function $\mathcal{H}_\mu^\alpha$2 is defined by

$\mathcal{H}_\mu^\alpha$3

where $\mathcal{H}_\mu^\alpha$4 is the Bessel function of the first kind and $\mathcal{H}_\mu^\alpha$5 is the oscillatory-Bessel kernel [(Pasawan, 6 Jan 2026), §3]. This transform extends to a unitary operator on $\mathcal{H}_\mu^\alpha$6, with inverse $\mathcal{H}_\mu^\alpha$7: $\mathcal{H}_\mu^\alpha$8 This construction generalizes the classical Hankel transform (recovered when $\mathcal{H}_\mu^\alpha$9), introducing a fractional Fourier-type phase.

2. Definition and Basic Structure of $H^s_{\alpha,\mu}$0

For $H^s_{\alpha,\mu}$1, the fractional Hankel–Sobolev space $H^s_{\alpha,\mu}$2 is defined as

$H^s_{\alpha,\mu}$3

with norm

$H^s_{\alpha,\mu}$4

The “weight function” $H^s_{\alpha,\mu}$5 is fixed, and the $H^s_{\alpha,\mu}$6 factor encodes the Sobolev regularity in analogy with global (Shubin-type) Sobolev spaces [(Pasawan, 6 Jan 2026), §5].

When $H^s_{\alpha,\mu}$7, the space $H^s_{\alpha,\mu}$8 reduces to the classical Hankel–Sobolev space associated with the standard Hankel transform.

3. Spectral Characterization via the Bessel Operator

The Bessel differential operator is given by

$H^s_{\alpha,\mu}$9

which is essentially self-adjoint on $s\in\mathbb{R}$0. The classical Hankel transform diagonalizes $s\in\mathbb{R}$1 via

$s\in\mathbb{R}$2

The same property holds for the fractional transform: $s\in\mathbb{R}$3 corresponds to multiplication by $s\in\mathbb{R}$4 [(Pasawan, 6 Jan 2026), §5].

This yields a functional-calculus interpretation: for $s\in\mathbb{R}$5,

$s\in\mathbb{R}$6

Theorem 5.2 in (Pasawan, 6 Jan 2026) establishes the equivalence of norms: $s\in\mathbb{R}$7 and identifies $s\in\mathbb{R}$8 up to norm equivalence.

4. Functional-Analytic Properties

Fractional Hankel--Sobolev spaces exhibit a robust functional-analytic structure:

• Hilbert Space Structure: $s\in\mathbb{R}$9 is a Hilbert space, inheriting completeness from the closedness of $\mu > -\frac12$0.
• Continuous Embeddings: For $\mu > -\frac12$1, $\mu > -\frac12$2, and the embedding is continuous.
• Hilbert Scale and Interpolation: The family $\mu > -\frac12$3 forms a Hilbert scale. Complex interpolation yields $\mu > -\frac12$4 for $\mu > -\frac12$5.
• Density: $\mu > -\frac12$6 and $\mu > -\frac12$7 are dense in $\mu > -\frac12$8 for all $\mu > -\frac12$9.

These properties follow from the spectral calculus of $\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$0 and general results on Hilbert scales [(Pasawan, 6 Jan 2026), §5].

5. Kernel Estimates and Integral Representations

The oscillatory–Bessel kernel $\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$1 and its integral properties are central to the fractional pseudo-differential analysis.

• By repeated integration by parts in $\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$2 and use of standard Bessel bounds $\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$3, it is demonstrated that for each $\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$4,

$\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$5

• For a symbol $\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$6 in Pasawan’s Shubin-type class $\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$7, the pseudo-differential operator

$\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$8

admits the integral kernel representation

$\alpha \in \mathbb{R}\setminus \pi\mathbb{Z}$9

where

$\mathcal{H}_\mu^\alpha$0

Lemma 4.1 in (Pasawan, 6 Jan 2026) shows that $\mathcal{H}_\mu^\alpha$1 for all $\mathcal{H}_\mu^\alpha$2. For $\mathcal{H}_\mu^\alpha$3, $\mathcal{H}_\mu^\alpha$4 is bounded on all $\mathcal{H}_\mu^\alpha$5 spaces, $\mathcal{H}_\mu^\alpha$6, by Schur’s test.

6. Pseudo-differential Operators and $\mathcal{H}_\mu^\alpha$7-Dependence

The group law $\mathcal{H}_\mu^\alpha$8 underlies the unitarity and spectral properties of $\mathcal{H}_\mu^\alpha$9. The parameter $L^2$0 interpolates continuously between transforms, and the analysis remains uniform in $L^2$1 away from $L^2$2. Integration by parts in $L^2$3 or $L^2$4 in $L^2$5 yields decay estimates with $L^2$6 type factors, demonstrating uniform kernel decay for $L^2$7 bounded away from these exceptional points.

Pseudo-differential operators $L^2$8, conjugated by $L^2$9, are mapped to global Shubin-type operators $L_{\mu}$0 whose $L_{\mu}$1-boundedness properties are governed by standard symbol estimates, allowing direct transfer of Sobolev–boundedness results to the fractional setting [(Pasawan, 6 Jan 2026), Thm 6.2].

7. Interplay with Classical Sobolev and Hankel Spaces

When $L_{\mu}$2, the framework reduces to the classical Hankel–Sobolev analysis as studied in operator theory and harmonic analysis of radial functions. The introduction of the fractional parameter $L_{\mu}$3 generalizes the calculus, enabling new classes of unitary transforms and pseudo-differential operators. This framework is parallel to, and interacts with, the global Weyl–Hörmander and Shubin–Sobolev theory but is distinguished by the geometry of the Bessel operator spectrum (Pasawan, 6 Jan 2026).

A plausible implication is that fractional Hankel–Sobolev spaces provide an adaptable analytic foundation for global analysis on $L_{\mu}$4, particularly for equations and operators exhibiting radial or Bessel-type symmetry, but now with enhanced flexibility dependent on the parameter $L_{\mu}$5.