Rellich–Kondrachov Compact Embedding Theorem

• Rellich–Kondrachov-like compact embedding theorems are a family of results establishing when bounded Sobolev-type spaces precompactly embed into lower Lebesgue spaces.
• They extend classical embeddings to degenerate, non-Euclidean, and fractal settings via Poincaré-type inequalities and covering arguments.
• These results underpin applications in PDEs, spectral theory, and variational problems by ensuring compactness even in non-standard geometric contexts.

The Rellich–Kondrachov-like compact embedding theorem encompasses a family of powerful results that assert, under suitable geometric and analytic conditions, the precompactness of bounded sets in generalized Sobolev-type spaces within lower Lebesgue spaces. These results extend the classical compact embedding theorems from bounded Euclidean domains and uniformly elliptic operators to a range of degenerate, non-Euclidean, and highly structured settings, such as Hörmander (subelliptic) vector fields, fractal domains, discrete lattices, noncompact symmetric spaces, and function spaces with group symmetry or holomorphic structure.

1. Classical and Abstract Rellich–Kondrachov Theorems

The prototypical Rellich–Kondrachov theorem establishes that for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ and %%%%1%%%%, the embedding

$$W^{1,p}(\Omega) \hookrightarrow L^q(\Omega) \quad \text{is compact for all}\; 1\leq q < p^* = \frac{np}{n-p}.$$

Any bounded sequence in $W^{1,p}(\Omega)$ has a strongly convergent subsequence in $L^q(\Omega)$.

This result generalizes through an abstract formalism, as in (Chua et al., 2011), where the authors show that if

• An "abstract Sobolev" norm controls function oscillations via Poincaré-type inequalities on a suitable cover,
• The cover has bounded overlap and local balls capture the measure structure, then boundedness in the norm implies strong precompactness in a lower $L^q$ space. This framework accommodates degenerate settings, weighted spaces, and even spaces where the "gradient" is measured in a nonstandard way.

For example, in degenerate Sobolev spaces $W^{1,p}(\Omega, Q)$ with a quadratic form $Q(x, \xi)$ possibly vanishing on some directions,

$$\|f\|_{W^{1,p}(\Omega, Q)} = \|f\|_{L^p(\Omega)} + \left\| \sqrt{ Q(x, \nabla f) } \right\|_{L^p(\Omega)},$$

a degenerate Poincaré inequality suffices for compactness in the embedding into $L^q$.

2. Extensions to Non-Euclidean and Fractal Geometries

a. Hörmander Vector Fields

In the subelliptic setting, such as on domains equipped with Hörmander vector fields $X = (X_1, \ldots, X_m)$, Sobolev spaces are defined via iterated derivatives along these fields: $$\mathcal{W}_{X,0}^{k,p}(\Omega) = \overline{C_0^\infty(\Omega)}^{\| \cdot \|_{X,k,p}}, \qquad \|u\|_{X,k,p} = \left( \sum_{|J|\leq k} \|X_J u\|_{L^p(\Omega)}^p \right)^{1/p}.$$ A sharp Rellich–Kondrachov embedding holds (Chen et al., 30 Apr 2024): if $k p < \widetilde\nu$ (with $\widetilde\nu$ the generalized Étivier or nonisotropic dimension), then

$$\mathcal{W}_{X,0}^{k,p}(\Omega) \hookrightarrow L^s(\Omega) \quad \text{compactly for all }\, 1 \leq s < \frac{\widetilde\nu p}{\widetilde\nu - k p}.$$

This generalizes the classical dimension condition $k p < n$.

Key to the proof is the Rothschild–Stein lifting, which realizes the vector fields as approximating their nilpotent Lie group counterparts, and the saturation method, which bridges to a local representation formula: $$|f(x)| \le C \int_{W} \frac{ | X f(y) | + |f(y)| }{ | B(x, d(x,y)) | }\, d(x, y)\, dy.$$ This, combined with weak-$L^p$ kernel bounds, yields the desired compactness.

b. Fractal and Non-Lipschitz Domains

On domains with boundaries of positive Hausdorff dimension or fractal geometry, classical results do not apply directly. For "Sobolev admissible" $n$-sets $\Omega$ (where a suitable extension property holds), one retains

$$W_p^k(\Omega) \Subset L_q(\Omega) \qquad \text{if}\; k p < n.$$

The trace operator becomes compact, often into a Besov space $B^{2,2}_{\alpha}(\partial\Omega)$ with $\alpha = 1 - (n-d)/2$ when the boundary is a $d$-set (Rozanova-Pierrat, 2020).

3. Symmetry and Compactness: Manifolds and Radial Subspaces

On noncompact domains (manifolds, symmetric spaces, infinite lattices), loss of compactness is caused by translation or escape to infinity. By restricting to functions invariant under a sufficiently large symmetry group, one suppresses this defect.

• Hyperbolic space (symmetric spaces): For the fractional Laplacian $\Delta^\sigma$ on $\mathbb{H}^n$ and Sobolev norm

$$\| \phi \|_{\lambda,\sigma} = \| ( \Delta^\sigma - \lambda^\sigma )^{1/2} \phi \|_{2},$$

the embedding

$$\mathcal{H}_{\lambda,\sigma}^{\mathrm{rad}} \hookrightarrow L^q(\mathbb{H}^n)$$

is compact for $2 < q < 2n/(n - 2\sigma)$ (Bruno et al., 15 Sep 2025). - Local Compactness: Follows from Rellich's theorem on balls. - Decay at Infinity: Exponential decay for radial functions via Lions' lemma (see also (Skrzypczak et al., 2020)).

• Analogous phenomena occur for radially symmetric Sobolev spaces on $\mathbb{R}^n$ (Strauss' lemma), block-radial symmetries, and subspaces determined by group actions.

4. Compactness for Nonlinear and Vector Field Problems

The Rellich–Kondrachov principle extends to situations involving nonlinear gauge constraints and vector-valued functions. For example (Frank et al., 2021):

• For sequences $(A_n)$ of vector fields with nonlinear divergence-free constraint $\operatorname{div}(|A_n|A_n)=0$ and bounded curl,
• If $A_n$ is bounded in an appropriate norm and $\nabla \wedge A_n$ is weakly convergent, then strong convergence holds in $L^q$ for $q<3$ up to symmetries (translations, dilations, gauge transforms).

Such nonlinear versions are crucial for variational problems in mathematical physics, where the loss of linear structure requires careful compactness substitution.

5. Embedding Theorems on Groups, Discrete Spaces, and Function Spaces

• Locally Compact Abelian Groups: Sobolev spaces $H^s(G)$ defined via Fourier analysis have Rellich–Kondrachov-type compactness when appropriate conditions on the group and weight function (symbol) are met (Górka et al., 2012, Ccajma et al., 2022).
• Discrete Setting (lattices): Discrete analogues of Rellich's theorem for lattice Laplacians yield unique continuation and absence of embedded eigenvalues, with compactness modeled by lack of "tail" energy (Isozaki et al., 2012).
• Holomorphic Function Spaces: Sharp criteria for compact inclusions Bergman $\to$ Hardy and Hardy $\to$ Bergman spaces on the unit ball correspond to strict inequalities relating dimension, exponents, and weights (Bao et al., 12 Feb 2025). The notion of "tight fitting" embeddings (proper, contractive, non-compact) precisely characterizes the threshold for compactness.

6. Compactness in Fractional Sobolev and Trace Spaces

For fractional operators defined via extension problems (e.g., the Caffarelli–Silvestre approach to the fractional Laplacian), compactness of the trace operator is ensured under mild hypotheses (Bueno et al., 14 Apr 2025). If $X\subset Y$ is a dense compact embedding for Hilbert spaces, such as $H_0^1(\Omega)\subset L^2(\Omega)$,

• Interpolation methods (Lions–Magenes theory) imply that the trace space in the extension setting also embeds compactly.
• This creates a foundation for existence theory in nonlocal PDEs and variational problems for fractional operators.

7. Applications in PDEs, Variational Problems, and Spectral Theory

Rellich–Kondrachov-like compact embedding theorems are essential for:

• Proving existence and multiplicity of solutions to nonlinear and degenerate PDEs via the Palais–Smale condition and minimax arguments (see (Chen et al., 2022)).
• Establishing the discreteness of spectra for elliptic operators (including the Laplacian on the torus (Omenyi, 2018)), especially under general geometric and analytic settings.
• Ensuring the well-posedness of boundary value and obstacle problems on domains with low-regularity boundaries through compactness of the trace operator (Rozanova-Pierrat, 2020).

Table: Key Compact Embedding Variants

Setting/Space	Compactness Condition	Reference
$W^{1,p}(\Omega) \to L^q(\Omega)$	$\Omega$ bounded, $p<n$, $q<p^*$	(Chua et al., 2011)
$\mathcal{W}_{X,0}^{k,p}(\Omega) \to L^s(\Omega)$	$k p < \tilde{\nu}$, $1 \leq s < \tilde{\nu} p$	(Chen et al., 30 Apr 2024)
$H^s(G) \to L^p(G)$ (LCA group)	Technical symbol/integrability conditions	(Górka et al., 2012)
$$\mathcal{H}_{\lambda,\sigma}^{\mathrm{rad}} \to L^q(\mathbb{H}^n)$$	$2 < q < 2n/(n-2\sigma)$	(Bruno et al., 15 Sep 2025)
Trace spaces (fractional extension)	$X \subset\subset Y$ (compact)	(Bueno et al., 14 Apr 2025)
Piecewise $H^1$ on shape-regular triangulations	Uniform shape regularity	(Zhang, 2013)
Hardy/Bergman Embeddings	Strict dimension/exponent/weight inequalities	(Bao et al., 12 Feb 2025)

Conclusion

Rellich–Kondrachov-like compact embedding theorems provide foundational precompactness criteria in a wide range of settings—from degenerate and non-Euclidean geometries to holomorphic and discrete function spaces—and underpin a multitude of modern analytic techniques in PDE theory, harmonic analysis, and spectral theory. These results depend crucially on Poincaré-type inequalities, covering arguments, geometry-driven symmetry, and, in advanced cases, lifting, saturation, and interpolation methods. Their flexibility enables extension of compactness-based arguments to problems with degeneracies, nonstandard geometries, strong symmetries, and even to analytic function spaces of several complex variables.