Gagliardo Seminorms in Fractional Spaces

• Gagliardo seminorms are integral expressions that quantify fractional differences in function values, forming the basis for nonlocal Sobolev spaces.
• They establish a rigorous bridge between nonlocal and local variational frameworks, notably through the BBM theorem and precise asymptotic expansions.
• Extensions such as weighted, periodic, and interpolation-based seminorms enable applications across PDEs, minimal surfaces, and material science.

Gagliardo seminorms are fundamental in the quantitative and qualitative analysis of fractional Sobolev spaces and nonlocal variational problems. They extend the notion of the classical Sobolev seminorm to non-integer differentiation orders, measuring the "fractional" regularity of functions by penalizing differences between values at pairs of points. This framework underlies a vast range of results in analysis, PDEs, geometry, and applied mathematics, including sharp functional inequalities, nonlocal minimal surface theory, variational methods for fractional PDEs, and modern calculus of variations.

1. Definition and Core Properties

Let $\Omega \subset \mathbb{R}^N$ and $s \in (0,1)$, $p \in [1,\infty)$. The (homogeneous) fractional Gagliardo seminorm of $u : \Omega \to \mathbb{R}$ is

$$|u|_{W^{s,p}(\Omega)} = \left( \int_\Omega \int_\Omega \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy \right)^{1/p}.$$

This seminorm, together with the $L^p$-norm, defines the fractional Sobolev space $W^{s,p}(\Omega)$. For $p=2$, the seminorm is closely linked (by an explicit constant via Fourier analysis) to the $L^2$-norm of the fractional Laplacian: $$|u|_{H^s(\mathbb{R}^N)}^2 = C_{N,s} \iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy = \|(-\Delta)^{s/2} u \|_{L^2}^2 \quad [1611.00734].$$ The kernel $|x-y|^{-N-sp}$ ensures scale-invariance, and the seminorm becomes increasingly sensitive to fine-scale oscillations as $s \to 1^-$.

2. Asymptotics and Γ-Limits: Strong and Second-Order Expansions

The regime $s \to 1^-$ connects nonlocal fractional energies to local Sobolev norms. The pioneering Bourgain–Brezis–Mironescu (BBM) theorem asserts that, for $u \in C_c^\infty(\Omega)$, $p > 1$,

$$\lim_{s \to 1^-} (1-s) |u|_{W^{s,p}(\Omega)}^p = K_{N,p} \int_\Omega |\nabla u|^p\,dx.$$

This formula was extended to include weighted settings and domains with minimal regularity (Kijaczko, 2023, Hurri-Syrjänen et al., 2023).

Recent advances have characterized the second-order asymptotics. For $p=2$, with $$G_s(u) := (1-s) \iint_{\Omega \times \Omega} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dx\,dy$$,

$$G_s(u) = G_1(u) - (1-s) G^1(u) + o(1-s), \quad s \to 1^-,$$

where $$G_1(u) = \frac{\omega_N}{2} \int_\Omega |\nabla u|^2 dx$$ and the new nonlocal quadratic form $G^1$ encodes nontrivial higher-order corrections (Kubin et al., 23 Oct 2024). The precise structure of $G^1$ involves both local and nonlocal contributions and admits a characterization via Fourier multipliers.

The associated "rate" energy functionals $G_s^1(u) := \frac{G_1(u) - G_s(u)}{1-s}$ converge in the Mosco sense to $G^1(u)$. This advanced asymptotic analysis enables precise control of nonlocal-to-local transitions and underpins the convergence of the associated $L^2$-gradient flows (Kubin et al., 23 Oct 2024, Luca et al., 2021).

3. Extensions: Weighted, Periodic, and Interpolation-Based Seminorms

The Gagliardo seminorm generalizes in several directions:

• Weighted Fractional Seminorms: For a domain $\Omega$ and weight $\delta(x)^\alpha$ with $\delta(x) = \text{dist}(x, \partial \Omega)$,

$$[u]_{W^{s,p}_\alpha(\Omega)} = \left( \iint_{\Omega \times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,\delta(x)^{\alpha} \delta(y)^{\alpha} dx\,dy \right)^{1/p}$$

recovers local weighted Sobolev norms in the limit $s \to 1^-$ and sets up nonlocal–local identification for power-type weights (Kijaczko, 2023).

• Weighted Inequalities: Results such as

$$\iint_{Q\times Q} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\;w(x)\,dx\,w(y)\,dy \leq C\,[w]_{A_1}\,(1-s)^{-p} \int_Q |\nabla u(x)|^p\,w(x)\,dx$$

extend the BBM paradigm to Muckenhoupt weights (Hurri-Syrjänen et al., 2023).

• Periodic and Symmetrization Structures: For $u$ periodic in a coordinate, the seminorm on the torus,

$$[u]_{W^{s,p}(\mathbb{T}^n)}^p = \int_{\mathbb{T}^n} \int_{\mathbb{T}^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} dx\,dy,$$

satisfies strict Pólya–Szegő type inequalities under periodic and cylindrical rearrangements, with equality cases precisely described for $p\geq 1$ (Csató et al., 22 Nov 2024, Luca et al., 2022).

• Real Interpolation and Moduli of Smoothness: The real interpolation approach yields “correct” fractional variants—so-called Butzer seminorms—with direct connections to fractional Sobolev embeddings, limit formulas, and convergence theorems, surpassing the naively extended Gagliardo seminorm in higher regularity settings (Domínguez et al., 2021).

4. Variational Problems, Minimal Surfaces, and PDEs

Gagliardo seminorms play a central role in modern nonlocal calculus of variations:

• Fractional Minimal Surfaces and Least-Gradient Functions: In the $W^{s,1}$ setting, the seminorm

$$[u]_{W^{s,1}(\Omega)} = \iint_{\Omega \times \Omega} \frac{|u(x)-u(y)|}{|x-y|^{n+s}}dx\,dy$$

serves as a nonlocal total variation, supporting minimization problems that interpolate between classical minimal surfaces and nonlocal geometric functionals. The minimizers—$s$-minimal functions—are shown to be bounded and continuous under mild topological assumptions and regularity of exterior data, highlighting the "fractional counterpart" of the least-gradient problem (Bucur et al., 29 Jul 2024). These results are sharp, with explicit counterexamples demonstrating the necessity of the connectivity and regularity hypotheses.

• Gradient Flows and Heat Flows: The $L^2$-gradient flow of the squared Gagliardo seminorm produces the (nonlocal) fractional heat equation. Convergence properties and stability are fully described as $s\to 0^+$ and $s\to 1^-$, with the flows converging respectively to degenerate ODEs or the classical local heat flow, and finer expansions yielding flows associated to renormalized energies such as the "logarithmic Laplacian" (Luca et al., 2021, Kubin et al., 23 Oct 2024).
• Energy Minimization in Material Science: In applications to lattice dislocation theory, minimization of the $s$-seminorm for piecewise-affine periodic profiles leads rigorously to the van der Merwe energy formula for misfit dislocations, capturing both periodicity of the optimal microstructure and sharp quantitative asymptotics (Luca et al., 2022).

5. Functional Inequalities and Sharp Constants

A significant line of research addresses the Gagliardo–Nirenberg and Sobolev-type inequalities, relating fractional norms and moduli of smoothness. The $L^2$-based inequalities are expressed as

$$\| D^j f \|_{L^r(\mathbb{R}^d)} \leq K_{j,n,\theta}(d) \| f \|_{L^2}^{1-\theta} \| D^n f \|_{L^2}^{\theta},$$

where $D^n f = (-\Delta)^{n/2} f$ and the constants and extremals can be computed exactly in certain endpoints, or bracketed between tight two-sided bounds elsewhere (Morosi et al., 2016). The equivalence of the $L^2$-based Gagliardo seminorm and the Fourier norm is essential in these developments.

Advances in interpolation and real method theory are leveraged to describe sharp embeddings between spaces defined by Gagliardo-type seminorms, their limits as $s \to 1^-$, and endpoint inequalities (Domínguez et al., 2021). The correct scaling and precise identification of constants are crucial for establishing the sharpness of nonlocal–to–local inequalities.

6. Extensions and Further Contexts

• Non-Euclidean and Dunkl Operators: The entire BBM–Maz’ya–Shaposhnikova limiting structure extends to settings as general as those associated with Dunkl operators and finite reflection groups, with dimension-free constants in the limit $s \to 0^+$ and robust analytic techniques not depending on the density of smooth functions (Li et al., 25 Mar 2025).
• Symmetrization and Rearrangement Inequalities: Rigorous symmetrization results for periodic and cylindrical decreasing rearrangements in the nonlocal setting have been developed, with full equality characterizations and proof methodologies extending classical results of Almgren–Lieb and Baernstein–Taylor–Friedberg–Luttinger (Csató et al., 22 Nov 2024). Nonexpansivity properties and their consequences for variational problems are now tightly understood.
• Failure of Naive Extensions: It is established that the classical Gagliardo seminorm, though effective for $0 < s < 1$, fails to control higher smoothness or to yield convergence in certain natural limiting problems; specifically, boundedness of the classical seminorm does not suffice for convergence to higher-order Sobolev spaces—a critical observation for correct function space theory (Domínguez et al., 2021).

7. Summary Table: Key Theorems and Asymptotics

Setting	Main Result	Reference
$s \to 1^-$, $p > 1$	$$(1-s)|u|_{W^{s,p}}^p \to K_{N,p} \int |\nabla u|^p \,dx$$	(Kijaczko, 2023)
Weighted, $s \to 1^-$	$$(1-s)[u]_{W^{s,p}_\alpha}^p \to C_{n,p} \int |\nabla u|^p \delta(x)^{2\alpha}dx$$	(Kijaczko, 2023)
$s \to 0^+$, general	$s |u|_{W^{s,p}}^p \to D_{n,p} \int |u|^p dx$	(Li et al., 25 Mar 2025)
Nonlocal $\to$ Local	$\Gamma$-limits, Mosco-convergence of functionals and flows	(Kubin et al., 23 Oct 2024)
Rearrangement	$[u^*]_{W^{s,p}} \leq [u]_{W^{s,p}}$ (strict with equality cases)	(Csató et al., 22 Nov 2024)

These advances cement the Gagliardo seminorm as a central nonlocal functional, mediating between local and nonlocal analysis, and provide the rigorous underpinnings for modern research in analysis, geometry, and mathematical physics.