Caffarelli-Kohn-Nirenberg Inequalities

• Caffarelli-Kohn-Nirenberg inequalities are foundational weighted interpolation estimates that extend Hardy and Sobolev inequalities through precise balance conditions.
• They enable control of weighted function norms via derivative norms across Euclidean spaces, Lie groups, and manifolds, with analysis of sharp constants and extremals.
• Recent advancements include fractional, Lorentz, and anisotropic generalizations that integrate functional analysis with PDEs, geometry, and nonlocal operator theory.

The Caffarelli-Kohn-Nirenberg (CKN) type inequalities comprise a foundational family of weighted interpolation inequalities, generalizing the Hardy and Sobolev inequalities and playing a central role in analysis, geometry, and the theory of partial differential equations. These inequalities, introduced in the seminal work of Caffarelli, Kohn, and Nirenberg (1984), control norms of functions weighted by powers of the distance from the origin via norms of their derivatives, also weighted, with sharp balance conditions relating the exponents and weights. Contemporary research has extended the CKN framework to Lorentz and anisotropic settings, generalized metric and group structures, fractional and nonlocal operators, and explored curvature dependence and rigidity on manifolds and metric spaces.

1. Classical Caffarelli-Kohn-Nirenberg Inequality: Origin and Structure

CKN inequalities on $\mathbb{R}^n$ are formulated for $u \in C_c^\infty(\mathbb{R}^n \setminus \{0\})$ with real parameters $a,b$, involving Lebesgue exponents $p,q$ determined by the scale invariance: $$\frac{1}{q} + \frac{b}{n} = \frac{1}{p} + \frac{a-1}{n}$$ Explicit versions include: $$\|\,|x|^{-b}u\|_{L^q(\mathbb{R}^n)} \leq C \|\,|x|^{-a}\nabla u\|_{L^p(\mathbb{R}^n)}$$ The balance covers critical exponents for Hardy ($b-a=1$), Sobolev ($a=b=0$, $p=2^*=2n/(n-2)$), and interpolates between them. The sharpness and attainability of constants are rigorously addressed in works such as (Cazacu et al., 2022), with extremals exhaustively classified for certain parameter ranges.

2. Extensions to Lie Groups, Homogeneous Spaces, and Lorentz Spaces

The scope of CKN-type inequalities advances significantly by formulation on non-Euclidean spaces:

• Stratified and Homogeneous Groups: On a stratified Lie group $G$ of homogeneous dimension $Q$, equipped with a homogeneous quasi-norm $|x|$ and the radial derivative $R$, the sharp $L^p$–CKN inequality reads (Ozawa et al., 2016):

$$\|\,|x|^{\alpha}f\|_{L^p(G)} \leq \frac{p}{|Q-\gamma p|} \|\,|x|^{\beta} Rf\|_{L^p(G)}^{1/p} \|\,|x|^{\beta} f\|_{L^p(G)}^{(p-1)/p}$$

All classical cases (Hardy, CKN, uncertainty principle) follow as specializations.

• Polynomial Growth Lie Groups, Lorentz Spaces: On $G$, a Lie group of polynomial volume growth with Carnot-Carathéodory distance $\rho(x)$ associated with a Hörmander system, Lorentz spaces $L^{p,q}(G)$ yield refinements:

$$\|\rho^{-\gamma} f\|_{L^{r,q_3}} \lesssim \|\,|Xf|\,\|_{L^{p,q_1}}^a \|f\|_{L^{q_2,q_3}}^{1-a}$$

where $|Xf|$ is the gradient norm over the vector fields (Yacoub, 2017). All classical Sobolev and Hardy-Sobolev inequalities are recovered in Lorentz variants.

• Anisotropic and Block-Radial Settings: Generalizations to multi-radial symmetries and cylindrical coordinates allow for embeddings with multi-singular weights $\rho_\gamma(x)$, and sharper singularity control (Tintarev et al., 2016, Li et al., 2021, Kalaman et al., 2024).

3. Manifolds, Metric Spaces, and Curvature Dependence

CKN inequalities are now established in the context of Riemannian, Finsler, and metric measure spaces:

• Riemannian and Finsler Manifolds: For Cartan–Hadamard manifolds (complete, simply connected, $K_M \leq 0$), the CKN inequality persists with the same Euclidean optimal constant (Nguyen, 2017):

$$\int_M d_P(x)^{-\gamma r} |f|^r \leq \left( \frac{r}{n - \gamma r} \right)^r \int_M d_P(x)^{-a p} |\nabla f|^p \left( ... \right)^{r-1}$$

Rigidity results establish that only globally flat spaces achieve equality; negative curvature induces strict stability via remainder terms. Analogous inequalities on Finsler manifolds track the influence of radial flag curvature (Wei et al., 2020).

• Metric Measure Spaces: For any $(X,d,\mathsf{m})$ with volume doubling, sharp CKN inequalities enforce exact $n$-dimensional volume growth; such spaces must essentially be Euclidean in both geometry and topology (Tokura et al., 2017).

4. Fractional, Nonlocal, and Lorentz Generalizations

Fractional order versions are constructed via Gagliardo semi-norms and have applications to nonlocal PDEs (Abdellaoui et al., 2016): $$\iint \frac{|u(x) - u(y)|^p}{|x-y|^{N+ps}|x|^\beta |y|^\beta}\ dx\,dy \geq S(N,p,s,\beta) \left( \int |u(x)|^{p_s^*} |x|^{2\beta p_s^*/p}\ dx \right)^{p/p_s^*}$$ The approach extends to Heisenberg groups and other non-commutative settings (Rawat et al., 2023).

CKN inequalities in Lorentz and Orlicz–Lorentz spaces refine interpolation estimates, allow for endpoint cases with logarithmic loss, and address sharpness/non-attainability of constants (Yacoub, 2017, Horiuchi, 2022, Horiuchi, 25 Dec 2025).

5. Anisotropic, Critical, and Non-Doubling Weight Regimes

The anisotropic and non-doubling-weight CKN inequalities address further complexities:

• Anisotropic CKN: Additional constraints emerge from weights like $|x'|^\alpha$, requiring refined endpoint and balance conditions. Nonlinear Poincaré inequalities permit extension to $q>0$ in the interpolation norms (Li et al., 2021, Kalaman et al., 2024).
• Critical CKN and Logarithmic Corrections: In the critical regime, iterated (super-)logarithmic weights are inserted in place of powers, yet the best constants remain unchanged (Ando et al., 2023). The equivalence of critical and subcritical best constants is explained via a unified weight class approach (Horiuchi, 2022).

6. Symmetry, Extremals, and Rigidity Phenomena

The existence and form of extremals (optimizing functions) is a central aspect:

• Symmetry Breaking and Classification: Rigidity and symmetry breaking are characterized in terms of parameters, especially for critical exponents; Liouville-type theorems and Riemannian reformulations yield explicit classification of positive critical points and radial profiles (Ciraolo et al., 2021, Dupaigne et al., 2021).
• Stability and Remainders: Exact remainder identities and stability estimates underpin sharp quantitative differences between Hardy and CKN inequalities, showing that nontrivial extremals exist only for certain parameter choices (Cazacu et al., 2022).
• Rigidity on Curved Spaces: The equality case in CKN inequalities identifies Euclidean (flat) geometry, with any curvature or measure distortion strictly precluding extremizers with compact support (Nguyen, 2017, Tokura et al., 2017).

7. Unified Interpolation Frameworks and Future Directions

Recent advances introduce unified function space scales (Hölder–Lebesgue), continuous interpolation theorems across regularity/integrability, and systematic extensions of CKN inequalities beyond the Lebesgue regime (Dong, 1 Oct 2025). Key innovations include control of weight singularities at the origin, explicit sharp constants in supercritical regimes, and non-doubling weight frameworks that unify critical and subcritical cases.

Further research explores:

• Existence and uniqueness of extremals, and spectral properties in non-Euclidean and variable-exponent regimes.
• Applications to sharp uncertainty principles, stability of functional inequalities, and optimal control in nonlocal and anisotropic PDEs.
• Extension of CKN-type inequalities to nonlinear evolution problems, geometric flows, and spaces with singular measure or curvature.

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Relevant Papers and Authors:

• (Yacoub, 2017) (CKN inequalities on Lie groups of polynomial growth – Yacoub)
• (Ozawa et al., 2016) (Sharp $L^p$–CKN on homogeneous groups – Ozawa, Ruzhansky, Suragan)
• (Li et al., 2021) (Anisotropic CKN inequalities – Li, Yan)
• (Ciraolo et al., 2021) (Symmetry for positive critical points – Ciraolo, Corso)
• (Ruzhansky et al., 2017, Ruzhansky et al., 2016, Kalaman et al., 2024) (Stratified groups, quasi-norms, cylindrical extensions – Ruzhansky et al.)
• (Nguyen, 2017, Wei et al., 2020, Tokura et al., 2017) (Riemannian/Finsler/metric space generalizations – Nguyen, Wei, Wu, Tokura et al.)
• (Horiuchi, 2022, Horiuchi, 25 Dec 2025, Ando et al., 2023) (Unified non-doubling weights, super-logarithmic critical CKN – Horiuchi et al.)
• (Deng et al., 2022, Ciraolo et al., 2021, Cazacu et al., 2022) (Optimal constants, stability, spectral estimates – Figalli, Zhang, Lam, Lu et al.)
• (Abdellaoui et al., 2016, Rawat et al., 2023) (Fractional and Heisenberg group extensions – Abdellaoui, Bentifour, Rawat, Roy)

This body of work demonstrates that the Caffarelli-Kohn-Nirenberg inequalities not only serve as fundamental interpolation estimates but also as a unifying theme linking functional analysis, PDE, metric geometry, and group representation theory.