Discrete Caffarelli-Kohn-Nirenberg Inequalities

• Discrete Caffarelli–Kohn–Nirenberg inequalities are weighted interpolation inequalities on lattice graphs that replace continuous singular weights with non-singular lattice weights.
• They control a weighted ℓ^q norm using a discrete Sobolev energy and a lower-order ℓ^r norm, offering relaxed parameter conditions compared to the continuous setting.
• The approach integrates finite-difference operators, continuum-transfer techniques, and discrete rearrangements to analyze extremals, symmetry, and compactness properties.

Discrete Caffarelli–Kohn–Nirenberg inequalities are weighted interpolation inequalities in non-continuous geometry, most explicitly formulated at present on the lattice graph $\mathbb Z^N$. In that setting, a weighted $\ell^q$ norm is controlled by a weighted discrete Sobolev energy and a lower-order weighted $\ell^r$ norm, with weights expressed through the graph distance from a distinguished origin. The direct graph-theoretic theory is comparatively recent, whereas a wider surrounding literature on metric measure spaces, symmetry of optimizers, non-doubling weights, weak endpoints, and quantitative stability supplies analytic templates rather than literal discrete formulations (Han et al., 5 Aug 2025, Tokura et al., 2017).

1. Lattice formulation and basic objects

The lattice model is the standard nearest-neighbor graph on $\mathbb Z^N$, with edge set

$$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$

The combinatorial distance from the origin is denoted by

$d(x)=d(x,0),$

and the weight is regularized as

$\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$

This shift by $1+d(x)$ is structurally important, because it removes the singularity at the origin that is present in the continuous $|x|^s$ model (Han et al., 5 Aug 2025).

For $1\le p<\infty$, the weighted sequence space is

$\ell^q$0

with $\ell^q$1 the corresponding class of functions. When $\ell^q$2, this reduces to the usual $\ell^q$3. The discrete differential structure is encoded by

$\ell^q$4

and by the pointwise $\ell^q$5-gradient

$\ell^q$6

The weighted discrete Sobolev energy is then

$\ell^q$7

The completion of $\ell^q$8 with respect to this norm is denoted $\ell^q$9 (Han et al., 5 Aug 2025).

This formulation already exhibits the two decisive modifications of the continuous CKN paradigm. First, the derivative is replaced by a nearest-neighbor difference operator. Second, the singular radial weight $\ell^r$0 is replaced by the nonsingular lattice weight $\ell^r$1. The resulting inequality is therefore not merely a formal finite-difference transcription of the Euclidean statement.

2. The discrete Caffarelli–Kohn–Nirenberg inequality

The main lattice theorem has the form

$\ell^r$2

for all

$\ell^r$3

The assumptions are

$\ell^r$4

together with

$\ell^r$5

$\ell^r$6

and the balance relation

$\ell^r$7

The theorem is first established in the critical case $\ell^r$8, and then extended to all $\ell^r$9 by the lattice embedding $\mathbb Z^N$0 for $\mathbb Z^N$1 (Han et al., 5 Aug 2025).

In the important special case $\mathbb Z^N$2, the critical exponent is

$\mathbb Z^N$3

The discrete result is presented as holding in a broader range of parameters than the classical continuous version. The reason is explicit. Because the weight $\mathbb Z^N$4 has no singularity at the origin, some continuous restrictions such as

$\mathbb Z^N$5

can be relaxed/removed. In addition, the discrete setting includes the supercritical case $\mathbb Z^N$6 (Han et al., 5 Aug 2025).

The comparison with the continuous model is exact at the structural level. The classical Euclidean inequality

$\mathbb Z^N$7

is transported to the lattice by replacing $\mathbb Z^N$8 with $\mathbb Z^N$9, $$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$0 with $$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$1, and Lebesgue norms with weighted $$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$2 norms. What changes is not the interpolation logic but the admissible range and the singularity structure (Han et al., 5 Aug 2025).

3. Proof architecture and Sobolev-space identification

The proof strategy is a continuum-transfer argument. The lattice function $$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$3 is extended to a piecewise linear function $$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$4 on $$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$5. On each elementary cube $$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$6, a point $$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$7 is written as

$$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$8

and the extension is defined by

$$E=\bigl\{\{x,y\}: x,y\in \mathbb Z^N,\ \sum_{i=1}^N |x_i-y_i|=1\bigr\}.$$9

This converts the discrete problem into a continuous one on $d(x)=d(x,0),$0 while preserving quantitative control (Han et al., 5 Aug 2025).

Two norm equivalences are then established: $d(x)=d(x,0),$1 They are obtained by finite-dimensional norm equivalence on each cube and summation over all cubes. After this reduction, the argument invokes a continuous weighted CKN inequality for non-homogeneous weights $d(x)=d(x,0),$2 and transfers the estimate back to the lattice (Han et al., 5 Aug 2025).

The same paper also identifies the appropriate discrete Sobolev space in the one-derivative case. When $d(x)=d(x,0),$3 and $d(x)=d(x,0),$4, it defines

$d(x)=d(x,0),$5

with

$d(x)=d(x,0),$6

and proves

$d(x)=d(x,0),$7

under

$d(x)=d(x,0),$8

The reverse inclusion is obtained by cutting off $d(x)=d(x,0),$9 with functions $\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$0, using logarithmic transitions between radii $\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$1, and proving

$\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$2

or $\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$3 in the endpoint $\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$4. This is the lattice analogue of the standard cutoff-density argument in weighted Sobolev spaces (Han et al., 5 Aug 2025).

4. Extremals, rearrangement, and nonlinear discrete elliptic equations

A distinct feature of the lattice theory is the use of discrete Schwarz rearrangement to recover compactness in supercritical regimes. The rearrangement employed there is built from one-dimensional rearrangements on $\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$5 or $\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$6, one-step rearrangements along special directions in the lattice, and iteration to obtain the discrete Schwarz symmetrization $\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$7. A function is called Schwarz symmetric if $\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$8. The key tools are the discrete Hardy–Littlewood inequality

$\mu_s(x)=(1+d(x))^s,\qquad s\in \mathbb R.$9

the discrete Pólya–Szegő inequality

$1+d(x)$0

and the compact embedding statement that if $1+d(x)$1 with $1+d(x)$2, then for any $1+d(x)$3 a subsequence converges in $1+d(x)$4 (Han et al., 5 Aug 2025).

These ingredients are used to prove existence of minimizers for the best constants in two supercritical cases. In the weighted Hardy-type case $1+d(x)$5, $1+d(x)$6, the variational problem

$1+d(x)$7

admits a minimizer under

$1+d(x)$8

In the unweighted mixed case $1+d(x)$9, the problem

$|x|^s$0

admits a minimizer when

$|x|^s$1

(Han et al., 5 Aug 2025).

The extremal theory is tied directly to nonlinear discrete elliptic equations. A minimizer for $|x|^s$2 yields a positive ground state solution of

$|x|^s$3

where

$|x|^s$4

Similarly, the minimizer for $|x|^s$5 yields a positive Schwarz symmetric ground state for

$|x|^s$6

with $|x|^s$7 determined by the parameters and the minimizer. The passage from minimizers to positive solutions uses the Lagrange multiplier rule and the discrete maximum principle (Han et al., 5 Aug 2025).

5. Continuum models that inform the discrete theory

The direct lattice theory sits inside a much larger CKN landscape, and several continuous developments explicitly state their relevance to discrete analogues. On proper metric measure spaces, one studies CKN inequalities of the form

$|x|^s$8

and under a doubling condition with exponent $|x|^s$9, the existence of the inequality with Euclidean exponent forces exact $1\le p<\infty$0-dimensional volume growth. That paper explicitly interprets its method as highly suggestive for discrete settings, where gradients would be replaced by edge differences, balls by graph metric balls, and volume growth by counting-measure growth (Tokura et al., 2017). An earlier metric-measure-space result proves the same rigidity principle for $1\le p<\infty$1 and applies it to Finsler manifolds, where the sharp inequality implies vanishing flag curvature and, in the Berwald case, Minkowski rigidity (Kristály et al., 2012).

Optimizer theory supplies another structural model. In the weighted Euclidean problem, symmetry breaking occurs in the Felli–Schneider region

$1\le p<\infty$2

while outside that region every nonnegative optimizer is radially symmetric and has the explicit form

$1\le p<\infty$3

The paper itself does not discuss discrete CKN inequalities directly, but it states that the mechanism suggests a possible discrete threshold between a symmetric branch and a symmetry-breaking branch (Dolbeault et al., 2016).

The theory of non-doubling weights gives a different type of template. On the half-line, a unified CKN-type inequality is proved with weights $1\le p<\infty$4 and Hardy-type profile

$1\le p<\infty$5

under the non-degenerate condition

$1\le p<\infty$6

That framework is explicitly presented as attractive for discrete analogues because it uses only radial or one-dimensional weight data, does not require exact scaling invariance, and permits non-doubling weights (Horiuchi, 2022).

Endpoint theory in weak Lebesgue spaces extends this picture further. Weak-type CKN inequalities of the form

$1\le p<\infty$7

remain valid at critical parameter values where strong inequalities fail, and the endpoint Hardy estimate

$1\le p<\infty$8

is singled out there as a template for discrete critical inequalities (Wang, 4 Feb 2026).

Quantitative stability theory contributes yet another analytic layer. Exact remainder identities for $1\le p<\infty$9-CKN inequalities, based on the $\ell^q$00-convexity remainder

$\ell^q$01

and on the Bregman-type quantity

$\ell^q$02

lead to deficit estimates, optimizer manifolds, and weighted Poincaré inequalities. These works are continuous, but they explicitly identify the convexity remainder, split-domain strategy, and weighted Poincaré framework as features that look transferable to discrete settings (Do et al., 2023, Chen et al., 28 Oct 2025).

6. Scope, boundaries, and present direction

Several neighboring CKN literatures are explicitly not graph-theoretic discrete theories. Improved Hardy and CKN inequalities based on the nonlocal functionals

$\ell^q$03

and their weighted variants remain continuous theories on $\ell^q$04 and smooth bounded domains; they contain no $\ell^q$05-inequalities, graph weights, or finite-difference discretizations in the usual sense (Nguyen et al., 2018). CKN inequalities on Lie groups of polynomial growth are likewise continuous, formulated through Carnot–Carathéodory distance, Hörmander vector fields, and Lorentz spaces $\ell^q$06, although their interpolation philosophy and weighted distance structure are close to what one would seek on discrete groups or graphs (Yacoub, 2017).

Other recent extensions are continuous by construction. The unified Hölder–Lebesgue framework $\ell^q$07 and its weighted higher-order variants $\ell^q$08 produce generalized CKN inequalities on bounded punctured domains, with endpoint logarithmic loss at $\ell^q$09, but they are stated as punctured-domain interpolation theorems rather than lattice results (Dong, 1 Oct 2025). Hyperbolic CKN space theory reformulates the weighted inequality on a conformal model with effective dimension

$\ell^q$10

proving sharpness for $\ell^q$11 and improved inequalities for $\ell^q$12; this is again a continuous conformal theory, not a discrete one (Devyver et al., 25 Nov 2025).

A recurring misconception is therefore that any nonlocal, metric, or weighted variant of CKN is already a discrete CKN inequality. The current literature is more sharply divided. One strand gives genuinely discrete lattice inequalities, extremals, and ground states on $\ell^q$13. A second strand gives continuum blueprints: rigidity from sharp inequalities, symmetry thresholds, non-doubling admissibility, weak endpoints, and stability of deficits. This suggests that the direct discrete theory is still concentrated on the lattice model, while its conceptual infrastructure is being assembled from several continuous directions whose methods are plausibly portable but not yet fully graph-theoretic.