Fractional Bessel–Sobolev Spaces

• Fractional Bessel–Sobolev spaces are fractional-order Sobolev spaces defined via Bessel potentials and spectral multipliers, unifying Euclidean and non-Euclidean frameworks.
• They offer a robust tool for interpolation, embedding, and mapping properties, bridging classical Sobolev theory with transform-adapted and weighted analyses.
• These spaces underpin various analytical applications, including nonlocal PDEs, pseudo-differential calculus, and the study of boundary integral equations.

Fractional Bessel–Sobolev spaces are fractional-order Sobolev spaces defined spectrally through Bessel potentials, and more generally through operator or transform-adapted multiplier calculi. In the Euclidean setting they are the spaces

$$H^{s,p}(\mathbb{R}^n):=\{f\in \mathcal{S}'(\mathbb{R}^n):(I-\Delta)^{s/2}f\in L^p(\mathbb{R}^n)\}, \qquad \|f\|_{H^{s,p}}=\|(I-\Delta)^{s/2}f\|_{L^p},$$

equivalently the image of $L^p$ under the Bessel potential $J_s=(I-\Delta)^{-s/2}$ (Bellido et al., 6 Mar 2025). The same structural principle persists in non-Euclidean and weighted settings: fractional Hankel–Bessel spaces $H^s_{\alpha,\nu}$ are defined on the transform side by the fractional Hankel–Bessel transform (Pasawan, 6 Jan 2026), while weighted Bessel geometries on $\mathbb{R}^n_+$ use the Bessel–Laplace operator $\Delta_a$ and Bessel translation rather than the Euclidean Laplacian and ordinary differences (Chegaar et al., 3 Sep 2025).

1. Euclidean Bessel potential spaces

In the classical Euclidean theory, the Bessel potential operator of order $s$ is

$J^{s}:=(I-\Delta)^{-s/2},$

whose Fourier symbol is $(1+|\xi|^2)^{-s/2}$. A widely used equivalent convention replaces this symbol by $(1+4\pi^2|\xi|^2)^{-s/2}$; the two choices differ only by constants determined by the Fourier convention (Bellido et al., 6 Mar 2025). For $L^p$0, there is a Bessel kernel $L^p$1 such that

$L^p$2

and $L^p$3 acts boundedly on $L^p$4, $L^p$5. Accordingly, $L^p$6 if and only if $L^p$7 for some $L^p$8, with $L^p$9 (Bellido et al., 6 Mar 2025).

The Hilbert case is completely explicit. For $J_s=(I-\Delta)^{-s/2}$0,

$J_s=(I-\Delta)^{-s/2}$1

and $J_s=(I-\Delta)^{-s/2}$2 for $J_s=(I-\Delta)^{-s/2}$3 (Maione, 18 Sep 2025). Moreover, for $J_s=(I-\Delta)^{-s/2}$4, $J_s=(I-\Delta)^{-s/2}$5 with equivalence of norms (Bellido et al., 6 Mar 2025). Density and duality are part of the basic structure: $J_s=(I-\Delta)^{-s/2}$6 is dense in $J_s=(I-\Delta)^{-s/2}$7, and

$J_s=(I-\Delta)^{-s/2}$8

for $J_s=(I-\Delta)^{-s/2}$9 (Maione, 18 Sep 2025).

A distinct but related $H^s_{\alpha,\nu}$0-based convention appears in one-dimensional approximation theory, where

$H^s_{\alpha,\nu}$1

with norm $H^s_{\alpha,\nu}$2, where $H^s_{\alpha,\nu}$3 is the strong Riesz derivative. In that framework, $H^s_{\alpha,\nu}$4 for integer $H^s_{\alpha,\nu}$5, and $H^s_{\alpha,\nu}$6 (Butzer et al., 2016). This is a homogeneous formulation rather than the inhomogeneous $H^s_{\alpha,\nu}$7-based norm, but it fits the same general Bessel–Sobolev paradigm.

2. Interpolation scale, embeddings, and relation to $H^s_{\alpha,\nu}$8

A central structural fact is that Euclidean Bessel potential spaces are interpolation spaces. If $H^s_{\alpha,\nu}$9, $\mathbb{R}^n_+$0, $\mathbb{R}^n_+$1, and $\mathbb{R}^n_+$2, then

$\mathbb{R}^n_+$3

with equivalence of norms (Bellido et al., 6 Mar 2025). More generally, for $\mathbb{R}^n_+$4, $\mathbb{R}^n_+$5, and $\mathbb{R}^n_+$6,

$\mathbb{R}^n_+$7

In the Hilbert scale $\mathbb{R}^n_+$8, real and complex interpolation coincide (Bellido et al., 6 Mar 2025).

This immediately separates Bessel potential spaces from Gagliardo–Slobodeckij spaces. For $\mathbb{R}^n_+$9,

$\Delta_a$0

is a real interpolation space, whereas $\Delta_a$1 is the corresponding complex interpolation space (Maione, 18 Sep 2025). The two scales coincide only in the Hilbert case. Specifically,

$\Delta_a$2

while for $\Delta_a$3,

$\Delta_a$4

and for $\Delta_a$5,

$\Delta_a$6

with strict inclusions unless $\Delta_a$7 (Bellido et al., 6 Mar 2025). A common misconception is therefore that “fractional Sobolev,” “Bessel potential,” and “Slobodeckij” are interchangeable at arbitrary $\Delta_a$8; the cited results show that this is false outside the Hilbert scale.

The Euclidean embedding theory is also standard and sharp. For $\Delta_a$9, $s$0, and $s$1,

$s$2

If $s$3, then $s$4 for every $s$5, and more precisely $s$6. If $s$7, then

$s$8

and more generally into Morrey–Campanato spaces (Bellido et al., 6 Mar 2025). There are also scale-to-scale embeddings: $s$9 (Bellido et al., 6 Mar 2025).

On bounded Lipschitz domains, compactness complements continuity. If $J^{s}:=(I-\Delta)^{-s/2},$0 is bounded Lipschitz and $J^{s}:=(I-\Delta)^{-s/2},$1, then the embeddings into Lebesgue or Hölder targets are compact in the subcritical, critical, and supercritical regimes, and moreover

$J^{s}:=(I-\Delta)^{-s/2},$2

(Bellido et al., 2 Jun 2025). These compact embeddings are established by three distinct arguments: complex interpolation, translation estimates combined with Fréchet–Kolmogorov–Riesz, and reduction to the corresponding compactness theorem for Gagliardo spaces (Bellido et al., 2 Jun 2025).

3. Fractional Hankel–Bessel and transform-side Sobolev spaces

A transform-adapted version of the theory replaces the Euclidean Fourier transform by the fractional Hankel–Bessel transform. Fix $J^{s}:=(I-\Delta)^{-s/2},$3 and $J^{s}:=(I-\Delta)^{-s/2},$4. On a Schwartz-type space $J^{s}:=(I-\Delta)^{-s/2},$5,

$J^{s}:=(I-\Delta)^{-s/2},$6

Its kernel is

$J^{s}:=(I-\Delta)^{-s/2},$7

At $J^{s}:=(I-\Delta)^{-s/2},$8, one recovers the classical Hankel transform of order $J^{s}:=(I-\Delta)^{-s/2},$9 (Pasawan, 6 Jan 2026).

The transform $(1+|\xi|^2)^{-s/2}$0 extends uniquely to a unitary operator on $(1+|\xi|^2)^{-s/2}$1, with inverse $(1+|\xi|^2)^{-s/2}$2, and satisfies the Plancherel identity

$(1+|\xi|^2)^{-s/2}$3

The parameter $(1+|\xi|^2)^{-s/2}$4 inserts a fractional Fourier-type phase into the Hankel kernel: the oscillatory factor is rotated by $(1+|\xi|^2)^{-s/2}$5, and the Bessel argument is scaled by $(1+|\xi|^2)^{-s/2}$6 (Pasawan, 6 Jan 2026).

The associated fractional Hankel–Sobolev spaces are

$(1+|\xi|^2)^{-s/2}$7

with norm

$(1+|\xi|^2)^{-s/2}$8

They are Hilbert spaces, reduce to the classical Hankel–Sobolev spaces when $(1+|\xi|^2)^{-s/2}$9, satisfy the monotone embedding

$(1+4\pi^2|\xi|^2)^{-s/2}$0

and enjoy density of $(1+4\pi^2|\xi|^2)^{-s/2}$1 together with the duality identification

$(1+4\pi^2|\xi|^2)^{-s/2}$2

through the $(1+4\pi^2|\xi|^2)^{-s/2}$3-pairing and unitarity of $(1+4\pi^2|\xi|^2)^{-s/2}$4 (Pasawan, 6 Jan 2026).

The definition is spectral. If $(1+4\pi^2|\xi|^2)^{-s/2}$5 is the model self-adjoint operator determined by

$(1+4\pi^2|\xi|^2)^{-s/2}$6

then $(1+4\pi^2|\xi|^2)^{-s/2}$7 is the domain of $(1+4\pi^2|\xi|^2)^{-s/2}$8, equipped with the graph norm (Pasawan, 6 Jan 2026). This parallels the classical Fourier characterization of Euclidean $(1+4\pi^2|\xi|^2)^{-s/2}$9, but the analysis proceeds entirely through the transform and multiplier side rather than through an explicit differential expression for $L^p$00.

These spaces form the natural background for a pseudo-differential calculus adapted to $L^p$01. For $L^p$02, a symbol $L^p$03 belongs to the global Shubin-type class $L^p$04 if

$L^p$05

The associated operator is

$L^p$06

Kernel estimates imply $L^p$07-boundedness for order-zero symbols and, more generally,

$L^p$08

continuously for every $L^p$09 and $L^p$10 (Pasawan, 6 Jan 2026). In this sense, the fractional Bessel–Sobolev scale is not only a regularity scale but also the exact domain/range scale for a transform-dependent pseudo-differential calculus.

4. Weighted Bessel geometry on $L^p$11

A different meaning of “fractional Bessel–Sobolev” arises in Bessel geometry on the positive orthant. Here the ambient space is

$L^p$12

equipped with the weighted measure $L^p$13, where $L^p$14, $L^p$15, and $L^p$16. The underlying operator is the Bessel–Laplace operator

$L^p$17

together with the Bessel translation $L^p$18 and the associated Bessel convolution and Hankel transform $L^p$19 (Chegaar et al., 3 Sep 2025).

The integer-order Sobolev scale is

$L^p$20

where $L^p$21 denotes the weighted space with norm

$L^p$22

The fractional theory is built not from Euclidean increments but from spherical Bessel differences and the corresponding modulus of smoothness. For $L^p$23,

$L^p$24

with

$L^p$25

where $L^p$26 is the spherical Gagliardo-type seminorm induced by Bessel translation (Chegaar et al., 3 Sep 2025).

A foundational theorem identifies the Bessel $L^p$27-functional with the spherical modulus of smoothness: $L^p$28 for $L^p$29, $L^p$30, and $L^p$31 (Chegaar et al., 3 Sep 2025). This yields the interpolation-theoretic description of the fractional spaces and shows that the relevant modulus is adapted to the Bessel geometry rather than to Euclidean translation.

The corresponding Bessel $L^p$32-potential spaces are defined through the Bessel kernel $L^p$33 by

$L^p$34

For integer orders,

$L^p$35

with equivalent norms, while for $L^p$36 and any $L^p$37,

$L^p$38

(Chegaar et al., 3 Sep 2025). The paper also emphasizes that $L^p$39 does not coincide with the weighted classical Sobolev space defined through ordinary derivatives $L^p$40: the two scales are genuinely different because the Bessel operator, Bessel translation, and weight are built into the norm (Chegaar et al., 3 Sep 2025).

This Bessel geometry also supports a potential theory. For operators

$L^p$41

removability of compact sets is characterized by the vanishing of Bessel capacities $L^p$42; in the single-term case $L^p$43, a compact set $L^p$44 is removable if and only if $L^p$45 (Chegaar et al., 3 Sep 2025).

5. Weighted, Orlicz, and geometric generalizations

The Euclidean $L^p$46 theory has several substantial extensions in which the Bessel potential viewpoint survives intact.

In the weighted $L^p$47 setting, for $L^p$48 and $L^p$49, the Bessel potential space on $L^p$50 is

$L^p$51

with $L^p$52. The spaces built from the Riesz fractional gradient,

$L^p$53

satisfy

$L^p$54

with equivalence of norms (García-Sáez, 10 Dec 2025). On bounded Lipschitz domains,

$L^p$55

there is a weighted Poincaré inequality,

$L^p$56

and the embedding

$L^p$57

is compact (García-Sáez, 10 Dec 2025).

In generalized Orlicz and Musielak–Orlicz settings, the same equivalence persists. If $L^p$58 satisfies $L^p$59, $L^p$60, $L^p$61, $L^p$62, and $L^p$63, then the fractional Sobolev–Orlicz space defined through the Riesz fractional gradient,

$L^p$64

coincides with the generalized Bessel potential space

$L^p$65

and

$L^p$66

for $L^p$67 (Campos et al., 16 Jun 2026). A closely related framework, denoted $L^p$68, is built on the norm $L^p$69 and likewise extends classical Lions–Calderón/Bessel potential spaces to generalized $L^p$70-growth (Campos, 2024).

Geometric irregularity changes the theory substantially. On non-Lipschitz subsets of $L^p$71, extrinsic Bessel potential spaces defined by restriction or support,

$L^p$72

retain clean duality and restriction properties, but intrinsic Slobodeckij spaces may fail to coincide with these extrinsic spaces (Chandler-Wilde et al., 2016). On quasicircles in the plane, the unit-circle identity

$L^p$73

holds for $L^p$74, whereas for $L^p$75 the corresponding paper explicitly states that $L^p$76 is not equal to the Bessel potential image $L^p$77 and therefore uses Besov terminology rather than Bessel-potential terminology (Wei et al., 4 Jan 2026). This distinction is a genuine structural feature, not a notational preference.

A noncommutative analogue appears on Carnot groups. If $L^p$78 is the positive sub-Laplacian, then

$L^p$79

and Poisson square functions give lower bounds for $L^p$80, implying in particular that the fractional Sobolev norm controls the Poisson-defined Besov norm with $L^p$81 (Maalaoui et al., 2020). This is a spectral Bessel–Sobolev theory with the sub-Laplacian in place of the Euclidean Laplacian.

6. Analytical roles, applications, and terminological scope

Fractional Bessel–Sobolev spaces are used not merely as definitions of regularity but as functional settings for operators and PDEs.

One line of application concerns operator theory. In the fractional Hankel–Bessel calculus, the mapping theorem

$L^p$82

for $L^p$83 is the analogue of classical Sobolev mapping for pseudo-differential operators (Pasawan, 6 Jan 2026). On Carnot groups, Poisson-square-function bounds for $L^p$84 are used to establish commutator estimates for fractional powers of the sub-Laplacian (Maalaoui et al., 2020).

Another line concerns nonlocal PDEs driven by the Riesz fractional gradient. In the Euclidean $L^p$85 setting, the identification of $L^p$86 with the closure of $L^p$87 under $L^p$88 links Bessel potential spaces directly to equations involving $L^p$89, $L^p$90, and $L^p$91 (Bellido et al., 6 Mar 2025). In the generalized Orlicz framework, the spaces $L^p$92 support existence, uniqueness, and continuous dependence results for quasilinear equations of the form

$L^p$93

with $L^p$94 on $L^p$95 (Campos, 2024). In the weighted $L^p$96 setting, the same role is played by $L^p$97, which yields existence and uniqueness for degenerate fractional elliptic problems

$L^p$98

and, in the Hilbert case, for linear equations with degenerate elliptic matrix coefficients (García-Sáez, 10 Dec 2025).

Compact embeddings are essential in variational arguments. For ordinary Bessel potential spaces on bounded Lipschitz domains, compactness into $L^p$99, into lower-order Bessel spaces, and into Hölder spaces in the supercritical regime is used to control nonlinear terms and to pass from weak to strong convergence (Bellido et al., 2 Jun 2025). A closely related interpolation inequality also enters spectral theory for mixed local–nonlocal operators: for

$J_s=(I-\Delta)^{-s/2}$00

on a bounded $J_s=(I-\Delta)^{-s/2}$01 domain with Dirichlet exterior condition, the cited note proves a discrete spectrum with compact resolvent and a min–max characterization of eigenvalues, using the compact embedding of the energy space into $J_s=(I-\Delta)^{-s/2}$02 (Maione, 18 Sep 2025).

Potential theory provides a further role. In the Bessel geometry of $J_s=(I-\Delta)^{-s/2}$03, Bessel capacities determine removable sets for fractional elliptic operators built from $J_s=(I-\Delta)^{-s/2}$04 (Chegaar et al., 3 Sep 2025). On non-Lipschitz and fractal subsets of Euclidean space, the extrinsic $J_s=(I-\Delta)^{-s/2}$05-scale supplies precise well-posedness criteria for boundary integral equations in terms of $J_s=(I-\Delta)^{-s/2}$06-nullity and support conditions, rather than in terms of smooth boundaries (Chandler-Wilde et al., 2016).

Terminologically, “fractional Bessel–Sobolev space” therefore does not refer to a single model. It may designate the classical Euclidean spaces $J_s=(I-\Delta)^{-s/2}$07, the transform-defined spaces $J_s=(I-\Delta)^{-s/2}$08, the Bessel-geometry spaces $J_s=(I-\Delta)^{-s/2}$09, weighted $J_s=(I-\Delta)^{-s/2}$10 analogues, Musielak–Orlicz analogues, or spectral spaces associated with non-Euclidean operators such as sub-Laplacians. What unifies these constructions is the use of a Bessel potential, a spectral multiplier, or a transform-side weight as the primary definition of fractional regularity; the comparison with difference-quotient or Gagliardo-type norms is then a theorem, not a definition.