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 $\mu > -\frac{1}{2}$ and $\alpha \in \mathbb{R} \setminus \pi\mathbb{Z}$. The fractional Hankel–Bessel transform of a function $f \in S(0,\infty)$ is defined by

$$(\mathcal{H}^{\alpha}_{\mu} f)(x) = \int_0^\infty (xy)^{1/2} J_\mu(xy \sin\alpha) \, e^{i(x^2 + y^2)\cos\alpha/2} f(y)\,dy,$$

where $J_\mu$ is the Bessel function of the first kind and $$K^{\alpha}_\mu(x,y) := (xy)^{1/2} J_\mu(xy\sin\alpha) e^{i(x^2 + y^2)\cos\alpha/2}$$ is the oscillatory-Bessel kernel [(Pasawan, 6 Jan 2026), §3]. This transform extends to a unitary operator on $L^2(0,\infty)$, with inverse $$(\mathcal{H}_\mu^{\alpha})^{-1} = \mathcal{H}_\mu^{-\alpha}$$: $$\| \mathcal{H}_\mu^{\alpha} f \|_{L^2(0,\infty)} = \|f\|_{L^2(0,\infty)}, \qquad \mathcal{H}_\mu^{\alpha} \circ \mathcal{H}_\mu^{-\alpha} = \mathrm{Id}.$$ This construction generalizes the classical Hankel transform (recovered when $\alpha = \pi/2$), introducing a fractional Fourier-type phase.

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

For $s \in \mathbb{R}$, the fractional Hankel–Sobolev space $H^s_{\alpha,\mu}$ is defined as

$$H^s_{\alpha,\mu} := \big\{ f \in L^2(0,\infty) : (1 + x^2)^{s/2} (\mathcal{H}_\mu^{\alpha}f)(x) \in L^2(0,\infty) \big\},$$

with norm

$$\|f\|_{H^s_{\alpha,\mu}} = \big\| (1 + x^2)^{s/2} (\mathcal{H}_\mu^{\alpha}f)(x) \big\|_{L^2(0,\infty)}.$$

The “weight function” $w(x)=1$ is fixed, and the $(1+x^2)^s$ factor encodes the Sobolev regularity in analogy with global (Shubin-type) Sobolev spaces [(Pasawan, 6 Jan 2026), §5].

When $\alpha = \pi/2$, the space $H^s_{\alpha,\mu}$ 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

$$L_\mu := -\frac{d^2}{dx^2} - \frac{2\mu+1}{x} \frac{d}{dx},$$

which is essentially self-adjoint on $L^2(0,\infty)$. The classical Hankel transform diagonalizes $L_\mu$ via

$$\mathcal{H}_\mu \big( L_\mu f \big)(\xi) = \xi^2 (\mathcal{H}_\mu f)(\xi).$$

The same property holds for the fractional transform: $$\mathcal{H}_\mu^{\alpha} \circ L_\mu \circ (\mathcal{H}_\mu^{\alpha})^{-1}$$ corresponds to multiplication by $x^2$ [(Pasawan, 6 Jan 2026), §5].

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

$$(1+L_\mu)^{s/2} = (\mathcal{H}_\mu^{\alpha})^{-1} (1 + x^2)^{s/2} \mathcal{H}_\mu^{\alpha}.$$

Theorem 5.2 in (Pasawan, 6 Jan 2026) establishes the equivalence of norms: $$C_1\| (1+L_\mu)^{s/2} f \|_{L^2(0,\infty)} \leq \|f\|_{H^s_{\alpha,\mu}} \leq C_2 \| (1+L_\mu)^{s/2} f \|_{L^2(0,\infty)},$$ and identifies $H^s_{\alpha,\mu} = \mathrm{Dom}((1+L_\mu)^{s/2})$ up to norm equivalence.

4. Functional-Analytic Properties

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

• Hilbert Space Structure: $H^{s}_{\alpha,\mu}$ is a Hilbert space, inheriting completeness from the closedness of $(1+L_\mu)^{s/2}$.
• Continuous Embeddings: For $s_1 > s_2$, $$H^{s_1}_{\alpha,\mu} \hookrightarrow H^{s_2}_{\alpha,\mu}$$, and the embedding is continuous.
• Hilbert Scale and Interpolation: The family $\{H^s_{\alpha,\mu}\}_{s\in\mathbb{R}}$ forms a Hilbert scale. Complex interpolation yields $$[H^{s_0}_{\alpha,\mu}, H^{s_1}_{\alpha,\mu}]_\theta = H^{(1-\theta)s_0+\theta s_1}_{\alpha,\mu}$$ for $0<\theta<1$.
• Density: $S(0,\infty)$ and $C_c^\infty(0,\infty)$ are dense in $H^s_{\alpha,\mu}$ for all $s$.

These properties follow from the spectral calculus of $L_\mu$ and general results on Hilbert scales [(Pasawan, 6 Jan 2026), §5].

5. Kernel Estimates and Integral Representations

The oscillatory–Bessel kernel $K^{\alpha}_\mu$ and its integral properties are central to the fractional pseudo-differential analysis.

• By repeated integration by parts in $x$ and use of standard Bessel bounds $|J_\mu(z)| \leq C\langle z\rangle^{-1/2}$, it is demonstrated that for each $N\geq 0$,

$$|K_{\mu}^{\alpha}(x,y)| \leq C_N \langle x \rangle^{-N} \langle y \rangle^{-N}, \qquad \langle x \rangle = (1+x^2)^{1/2}.$$

• For a symbol $\sigma(x, y)$ in Pasawan’s Shubin-type class $S_{FH}^m$, the pseudo-differential operator

$$A_{\sigma,\alpha}f(x) = \int_{0}^{\infty} K_{\mu}^{\alpha}(x,y)\, \sigma(x, y)\, (\mathcal{H}_{\mu}^{\alpha}f)(y) dy$$

admits the integral kernel representation

$$A_{\sigma,\alpha}f(x) = \int_{0}^{\infty} M_{\sigma,\alpha}(x, z)\, f(z)\, dz,$$

where

$$M_{\sigma,\alpha}(x, z) = \int_{0}^{\infty} K_{\mu}^{\alpha}(x,y) \sigma(x, y) K_{\mu}^{\alpha}(y, z) dy.$$

Lemma 4.1 in (Pasawan, 6 Jan 2026) shows that $$|M_{\sigma,\alpha}(x, z)| \leq C_N \langle x \rangle^{-N} \langle z \rangle^{m}$$ for all $N\geq 0$. For $m=0$, $A_{\sigma,\alpha}$ is bounded on all $L^p$ spaces, $1 \leq p \leq \infty$, by Schur’s test.

6. Pseudo-differential Operators and $\alpha$-Dependence

The group law $$\mathcal{H}_{\mu}^{\alpha} \circ \mathcal{H}_\mu^{\beta} = \mathcal{H}_\mu^{\alpha+\beta}$$ underlies the unitarity and spectral properties of $\mathcal{H}_\mu^\alpha$. The parameter $\alpha$ interpolates continuously between transforms, and the analysis remains uniform in $\alpha$ away from $\cos\alpha=0$. Integration by parts in $x$ or $y$ in $K^{\alpha}_\mu(x, y)$ yields decay estimates with $(i x\cos\alpha)^{-1}$ type factors, demonstrating uniform kernel decay for $\alpha$ bounded away from these exceptional points.

Pseudo-differential operators $A_{\sigma, \alpha}$, conjugated by $\mathcal{H}_{\mu}^{\alpha}$, are mapped to global Shubin-type operators $T_{\sigma}$ whose $L^2$-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 $\alpha = \pi/2$, 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 $\alpha$ 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 $(0,\infty)$, particularly for equations and operators exhibiting radial or Bessel-type symmetry, but now with enhanced flexibility dependent on the parameter $\alpha$.