Fractional Sobolev Spaces

• Fractional Sobolev spaces are function spaces that extend classical Sobolev spaces to non-integer orders, capturing intermediate differentiability and nonlocal effects.
• They are defined through various equivalent methods including integral, difference-quotient, Fourier, and interpolation techniques on diverse domains.
• These spaces underpin the analysis of nonlocal PDEs, embedding theorems, and trace results, impacting variational methods and regularity theory.

Fractional Sobolev spaces, also known as Sobolev–Slobodeckij or Gagliardo spaces, generalize the classical Sobolev spaces to incorporate non-integer orders of smoothness. For a real order $0 < s < 1$ and $1 \leq p < \infty$, these spaces rigorously quantify intermediate differentiability between $L^p$-integrability and full $W^{1,p}$ regularity, capturing both regularity and nonlocal effects. Definitions extend across Euclidean domains, manifolds, weighted or Orlicz scales, and even non-Euclidean settings, and admit multiple equivalent formulations via integral, difference-quotient, interpolation, and Fourier methods.

1. Core Definitions and Equivalent Representations

The canonical fractional Sobolev space $W^{s,p}(\Omega)$, for an open $\Omega \subset \mathbb{R}^n$, is defined as the set

$$W^{s,p}(\Omega) := \left\{ u \in L^p(\Omega) : [u]_{W^{s,p}(\Omega)} < \infty \right\}$$

where the (Gagliardo–Slobodeckij) seminorm is

$$[u]_{W^{s,p}(\Omega)}^p = \iint_{\Omega \times \Omega} \frac{|u(x) - u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy.$$

This is complemented with the full norm $$\|u\|_{W^{s,p}(\Omega)} = \|u\|_{L^p(\Omega)} + [u]_{W^{s,p}(\Omega)}$$ (Nezza et al., 2011).

Alternative characterizations include:

• Difference-quotient: For bounded Lipschitz $\Omega$, $u \in W^{s,p}(\Omega)$ if $$\liminf_{h\to 0} \int_{\Omega_h} \frac{|u(x+h)-u(x)|^p}{|h|^{sp}}\,dx < \infty$$, with $\Omega_h = \{x \in \Omega : x+h \in \Omega\}$ (Nezza et al., 2011).
• Fourier (Bessel-potential for $p=2$): For $u \in L^2(\mathbb{R}^n)$,

$$H^s(\mathbb{R}^n) = \left\{ u : \int_{\mathbb{R}^n} (1+|\xi|^2)^{s} |\widehat{u}(\xi)|^2\,d\xi < \infty \right\}.$$

• Real Interpolation: $$W^{s,p}(\Omega) \cong (L^p(\Omega), W^{1,p}(\Omega))_{s,p}$$, with norm equivalence (Maione, 18 Sep 2025).

For homogeneous spaces, the completion of $C_c^\infty(\mathbb{R}^n)$ under the seminorm $\big[\cdot\big]_{W^{s,p}(\mathbb{R}^n)}$ defines $\mathcal{D}^{s,p}(\mathbb{R}^n)$ (Brasco et al., 2020).

2. Embeddings, Trace Theorems, and Density

Fractional Sobolev spaces exhibit a fine embedding structure determined by $sp$ relative to $n$:

• Subcritical regime ($sp < n$):

$$W^{s,p}(\Omega) \hookrightarrow L^q(\Omega), \quad \forall q \in [p, p^*],~~p^* = \frac{np}{n-sp}$$

with compactness for $q < p^*$ (Nezza et al., 2011, Leoni, 2023, Maione, 18 Sep 2025).

• Critical regime ($sp=n$): Compact embedding fails, but there is dense embedding into $L^q$ for all $q < \infty$, and a limiting Orlicz–Trudinger type result holds (Nezza et al., 2011).
• Supercritical regime ($sp>n$): One obtains embedding into Hölder spaces $C^{0,\alpha}(\overline{\Omega})$ with $\alpha = s - \frac{n}{p}$ (Nezza et al., 2011).

For bounded Lipschitz domains, linear continuous extension operators $E:W^{s,p}(\Omega)\to W^{s,p}(\mathbb{R}^n)$ exist (Nezza et al., 2011, Leoni, 2023). Density of $C_c^\infty(\Omega)$ in $W^{s,p}(\Omega)$ holds for such domains, but fails in domains with fractal or outward-cusp boundaries (Nezza et al., 2011).

Trace theory identifies fractional traces on $W^{s,p}(\Omega)$ for $s>1/p$, mapping continuously and surjectively onto $W^{s-1/p,p}(\partial\Omega)$, a Besov or Slobodeckij space on the boundary (Nezza et al., 2011).

3. Interpolation Theory and the Gagliardo–Nirenberg Framework

Fractional Sobolev spaces can be realized precisely as (real) interpolation spaces:

$$W^{s,p}(\Omega) = (L^p(\Omega), W^{1,p}(\Omega))_{s,p},$$

with norm equivalence and precise functional-analytic structure (Maione, 18 Sep 2025, Brasco et al., 2018). The Peetre $K$-functional for $u \in L^p(\Omega) + W^{1,p}(\Omega)$,

$$K(t,u; L^p, W^{1,p}) = \inf_{u=g+h} \{ \|g\|_{L^p} + t\|\nabla h\|_{L^p} \}$$

produces a norm

$$\|u\|_{(L^p, W^{1,p})_{s,p}} = \left( \int_0^\infty [K(t,u)/t^s]^p\,\frac{dt}{t} \right)^{1/p}$$

equivalent to the fractional Gagliardo seminorm (Maione, 18 Sep 2025, Brasco et al., 2018).

In the Hilbertian case ($p=2$), complex interpolation produces the Bessel-potential (Fourier-based) Sobolev scale, which coincides with $W^{s,2}$ (Maione, 18 Sep 2025). For $p \neq 2$, the complex method yields a strictly different scale (Bessel-potential/Besov) (Maione, 18 Sep 2025).

4. Comparison of Constructions and Domain Pathologies

Several mechanisms produce different but related fractional spaces:

• Slobodeckij/Gagliardo spaces (integral definition): Fundamental for general open sets, but may depend sensitively on domain geometry.
• Interpolation spaces: $(L^p, W^{1,p})_{s,p}$ involves $K$-functionals and always embeds into or equals the above, but may not coincide if the domain is rough or has vanishing capacity sets (Brasco et al., 2018).
• On $\mathbb{R}^n$ or bounded convex/Lipschitz domains, both constructions give equivalent spaces and norms (Brasco et al., 2018, Maione, 18 Sep 2025).
• On domains with “cracks” or fractal boundaries the spaces may differ: certain pathologies (lack of continuity of zero-extension, dependence on interpolation capacity) arise, notably in the real interpolation spaces (Brasco et al., 2018).

Weighted and Orlicz fractional Sobolev spaces ($W^{s,p;w,v}$, $W^{s,M}$) further generalize the construction by allowing power or generalized function weights, with comparability and density criteria governed by boundary geometry (Assouad codimension) and $\Delta_2$-conditions on the $N$-function (Kijaczko, 2021, Azroul et al., 2018).

5. Connections to Fractional Operators, BV Theory, and Further Generalizations

There is deep interplay between fractional Sobolev spaces and nonlocal/fractional operators:

• Fractional Laplacian: The form $$\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x-y|^{n+2s}}dx\,dy$$ induces the spectral and variational definition of the fractional Laplacian $(-\Delta)^s$ (Leoni, 2023).
• Riemann–Liouville and Caputo derivatives: In one dimension, fractional Sobolev spaces may be equivalently defined using weak Riemann–Liouville derivatives. The embedding $SBV([a,b])\subset W^{s,1}_{RL}([a,b])$ for all $s\in(0,1)$ holds, while a full characterization of $BV\subset W^{s,1}_{RL}$ for arbitrary singular measures remains open (Bergounioux et al., 2016, Carbotti et al., 2020).
• Finer scales: Generalizations include variable-exponent/fractional Sobolev spaces on manifolds using kernel functions and nonconstant exponents (Aberqi et al., 2023), as well as spaces based on ultradistributions and Fourier growth (Amaonyeiro et al., 3 Oct 2024). Interpolation-based scales also allow direct connection to mixed local-nonlocal operator theory and generalized nonlocal equations (Maione, 18 Sep 2025).

6. Gagliardo–Nirenberg, Embedding, and Compactness Theorems

Gagliardo–Nirenberg type interpolation, Poincaré, and inequality results structure the analysis:

• Fractional Gagliardo–Nirenberg: For $u\in W^{s_0,p}\cap W^{s_1,p}$, $$|u|_{W^{s,p}}\leq |u|_{W^{s_0,p}}^{1-\theta} |u|_{W^{s_1,p}}^{\theta}$$, with $s=(1-\theta)s_0+\theta s_1$ (Leoni, 2023).
• Fractional Sobolev inequality: There exists $C>0$ with

$$\|u\|_{L^{p^*}(\Omega)} \leq C\, [u]_{W^{s,p}(\Omega)}$$

for $sp<n$ (Brasco et al., 2020, 1110.04012).

• Morrey–Campanato estimates: For $sp > n$, seminorm control yields Hölder continuity of $u$ (Brasco et al., 2020).

7. Applications and Contemporary Directions

Fractional Sobolev spaces underpin the rigorous analysis of nonlocal PDEs, calculus of variations, and regularity theory. Current research elucidates the behavior on metric spaces, weighted spaces, fractals, and manifolds (Aberqi et al., 2023, Kijaczko, 2021), and establishes their role as natural domains and energy spaces for fractional and mixed local-nonlocal operators (Maione, 18 Sep 2025), as well as advancing the theory of embeddings, traces, and nonlocal equations involving variable exponents and measure data. Contemporary questions include the optimal functional-analytic framework for variational and PDE techniques in these generalized settings and the delicate identification of function spaces under domain singularities and capacity-theoretic phenomena (Brasco et al., 2018, Amaonyeiro et al., 3 Oct 2024).