Critical Fractional Sobolev Embedding

• Critical fractional Sobolev embedding is a threshold phenomenon governing function space inclusions for fractional smoothness, with optimal target spaces like BMO.
• The embedding uses the nonlocal Gagliardo seminorm to establish optimal inequalities and sharp endpoint conditions in analysis and PDEs.
• Extremal functions and concentration–compactness techniques demonstrate optimal decay rates and stability properties at the critical scaling where sp=n.

The critical fractional Sobolev embedding delineates the threshold phenomena for function space inclusions governed by fractional smoothness. Specifically, it concerns the embedding properties of fractional Sobolev spaces $W^{s,p}(\Omega)$ and their homogeneous counterparts $\dot{W}^{s,p}(\mathbb R^n)$ at the scale where $sp=n$, emphasizing the role of nonlocal seminorms and endpoint spaces such as BMO. These embedding results are sharp and optimal, with significant ramifications for analysis, PDEs, and geometry.

1. Definition and Structure of Fractional Sobolev Spaces

Fractional Sobolev spaces $W^{s,p}(\Omega)$ are constructed via the Gagliardo seminorm: $$[u]_{W^{s,p}(\Omega)} = \left( c_{n,s,p} \iint_{\Omega \times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy \right)^{1/p}$$ where $$c_{n,s,p} = \frac{s\,2^{2s-1}\,\Gamma((ps+p+n-2)/2)}{\pi^{n/2}\,\Gamma(1-s)}$$ is normalized so that, as $s \to 0$, one recovers the $L^p$-norm, and as $s \to 1$, the classical Sobolev seminorm. The full norm is

$$\|u\|_{W^{s,p}(\Omega)} = \left( \|u\|_{L^p(\Omega)}^p + [u]_{W^{s,p}(\Omega)}^p \right)^{1/p}$$

with the homogeneous space $\dot{W}^{s,p}(\mathbb R^n)$ defined as the closure of $C_c^\infty(\mathbb R^n)$ under $[\cdot]_{s,p}$, and explicit constraints $u \in L^{p^*}$, $[u]_{s,p}<\infty$ modulo null sets (Dipierro et al., 19 Nov 2024, Brasco et al., 2015).

2. Critical Embedding Regime: Characterization and Endpoint Spaces

When $sp=n$ and $s \neq p$, the embedding achieves its critical status. Several sharp descriptions hold:

• For any $(\hat{s},\hat{p})$ with $0 < \hat{s} < s$, $1 \leq \hat{p} \leq n/\hat{s}$, or in the degenerate case $\hat{s}=0$, $\hat{p}<\infty$, there exists $C$ such that

$$\|u\|_{W^{\hat{s}, \hat{p}}(\Omega)} \leq C \|u\|_{W^{s, p}(\Omega)}$$

allowing passage to the endpoint exponent $\hat{p} = n/\hat{s}$ (Dipierro et al., 19 Nov 2024).

• Endpoint embedding into BMO:

$$C_1 \|u\|_{BMO} \leq \|u\|_{W^{\hat{s},\hat{p}}(\mathbb R^n)} \leq C_2 \|u\|_{W^{s,p}(\mathbb R^n)}$$

where $\hat{s}\hat{p} = n$ (Dipierro et al., 19 Nov 2024, Domínguez et al., 2021).

• In dimension $n=1$, $sp=1$ reduces to the classical case $W^{1,1}(\Omega) \hookrightarrow L^\infty(\Omega)$.

3. Extremal Functions and Asymptotic Behavior

Extremals for the critical fractional Sobolev inequality, in the Hilbertian case $p=2$, are known explicitly: $$U_{\mu,x_0}(x) = C_{N,s} \left( \frac{\mu}{1 + \mu^2 |x-x_0|^2 } \right)^{(N-2s)/2}$$ and, conjectured for general $p>1$ and $s\in(0,1)$,

$$U_{t,x_0}(x) = C_{N,p,s} t^{(sp-N)/p} (1 + |x-x_0|^{p/(p-1)})^{-(N-sp)/p}$$

These attain equality in the critical embedding and solve the fractional $p$-Laplacian equation

$$(-\Delta)_p^s U = \lambda |U|^{p^*-2} U \quad \text{in } \mathbb R^n$$

with $\lambda = S_{N,s,p}^p$, where $p^* = np/(n-sp)$ is the critical Sobolev exponent (Brasco et al., 2015).

Sharp decay at infinity is characteristically polynomial: $$U(x) \sim U_\infty |x|^{-(N-sp)/(p-1)}, \quad |x|\to\infty$$ with two-sided barriers demonstrating optimality (Brasco et al., 2015).

4. Optimal Embeddings on Domains and Manifolds

On open, bounded domains with Lipschitz boundary, the critical fractional embedding persists provided $0 < \hat{s} \leq s$; for $\hat{s}=0$ embedding into $L^{\hat{p}}$ holds for all finite $\hat{p}$ (Dipierro et al., 19 Nov 2024).

On closed Riemannian manifolds, the heat-kernel-based characterization leads to

$$\|u\|_{L^p(M)}^2 \leq S_{s,n} [u]_{s,2}^2 + A(M) \|u\|_{L^2(M)}^2$$

with $S_{s,n}$ the Euclidean constant and $A(M) = (\mathrm{Vol}(M))^{-2s/n}$ sharp (Tan et al., 21 Dec 2025). In the superquadratic range $p > 2$, no exact $p$-power form of the inequality holds globally; instead, almost-sharp inequalities can be obtained, and further improvements are realized under orthogonality constraints on the test functions (Tan et al., 21 Dec 2025).

5. Interpolation, Concentration-Compactness, and Limiting Phenomena

Sophisticated interpolation techniques, e.g., those of Brezis–Mironescu, construct admissible regions in the $(s,p)$-plane for target norms, leveraging profile decompositions and concentration-compactness principles to analyze defect measures and loss of compactness (Palatucci et al., 2013). Specifically, any noncompactness in maximizing sequences is characterized by atomic measures, with the sharp constant $S^*$ constraining pointwise defects.

Profile decomposition yields a countable sum of dilated and translated bubbles plus remainder vanishing in the critical Lebesgue norm, elucidating the precise structure of compactness breakdown and enabling analysis of subcritical extremal concentration (Palatucci et al., 2013).

On manifold settings, concentration–compactness extends to cover all dimensions and $s$ via localized Euclidean inequalities and partition of unity arguments, with curvature terms entering only in lower-order coefficients (Tan et al., 21 Dec 2025).

6. Endpoint Spaces, Optimality, and Counterexamples

At criticality, the embedding into $L^\infty$ fails:

• Compact embedding into $L^\infty$ or $C^0$ holds iff $sp>n$ (or, generally, for Orlicz $A$, when $\int^{\infty} A(t)/t^{1+n/(n-s)} dt < \infty$) (Alberico et al., 2022).
• For $sp=n$, the best possible embedding is into BMO (or into Orlicz–Lorentz spaces). Attempts to push beyond this regime are thwarted by scaling counterexamples: if $(\hat{s},\hat{p})$ crosses the critical boundary, there exist sequences with bounded $W^{s,p}$ norm and divergent target norm (Dipierro et al., 19 Nov 2024, Domínguez et al., 2021).
• The limiting case $W^{1,n}(\Omega) \hookrightarrow BMO$ (not $L^\infty$) is fully recovered as the integer-order endpoint.

7. Quantitative Stability and Applications

Quantitative stability results establish precise control of the distance to the manifold of bubble extremals by the Sobolev-deficit or Euler–Lagrange error. For $u$ in $\dot H^s(\mathbb R^n)$ with small deficit,

$$\|u - \sum_i U[z_i,\lambda_i]\|_{\dot H^s} \leq C \Phi(T(u))$$

where $\Phi$ is linear except for a logarithmic correction in the critical dimension $n=6s$ (Chen et al., 14 Aug 2024). This underpins uniqueness, compactness, and blow-up analysis for nonlinear PDEs with critical fractional Sobolev structure.

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Table: Endpoint Embeddings for Critical Fractional Sobolev Spaces

Setting	Embedding at $sp = n$	Optimal Target Space
$\mathbb R^n$	$W^{s,p} \hookrightarrow BMO$	BMO or Orlicz–Lorentz
Bounded Lipschitz Domain	$W^{\hat{s},\,\hat{p}}$ for $0<\hat{s}\le s$	$L^{\hat{p}}$ ($\hat{p}<\infty$)
Closed Manifold $(M,g)$	$H^s(M) \hookrightarrow L^{2_s^*}(M)$ (lower-order $L^2$ rem.)	$L^{2_s^*}(M) \oplus L^2(M)$
Orlicz–Sobolev	$V^sL^A \not\hookrightarrow L^\infty$ at edge	Orlicz–Lorentz, but not $L^\infty$

• (Dipierro et al., 19 Nov 2024) Optimal embedding results for fractional Sobolev spaces
• (Tan et al., 21 Dec 2025) Sharp Fractional Sobolev Embeddings on Closed Manifolds
• (Brasco et al., 2015) Optimal decay of extremals for the fractional Sobolev inequality
• (Palatucci et al., 2013) Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces
• (Alberico et al., 2022) Boundedness of functions in fractional Orlicz-Sobolev spaces
• (Domínguez et al., 2021) Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limit formulae for fractional Sobolev spaces via interpolation and extrapolation
• (Chen et al., 14 Aug 2024) Sharp quantitative stability estimates for critical points of fractional Sobolev inequalities