Gagliardo–Nirenberg Constants

• Gagliardo–Nirenberg constants are optimal values in interpolation inequalities that bound one norm of a function by products of its derivative and lower-order norms.
• They are characterized via explicit extremal profiles in Euclidean spaces and variational methods in anisotropic and Lie group settings, providing sharp formulations for PDE analysis.
• Stability estimates and geometric interpretations of these constants reveal practical insights into nonlinear diffusion, entropy formulations, and spectral properties in various domains.

Searching arXiv for recent and foundational papers on Gagliardo–Nirenberg constants. Gagliardo–Nirenberg constants are the optimal constants in interpolation inequalities that control one norm of a function by a product of derivative and lower-order norms. In the sharp Euclidean family considered for $n\ge 2$ and admissible $t$, the inequality takes the form

$$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$

with $1<t<\frac{n}{n-2}$ if $n\ge3$ and $1<t<\infty$ if $n=2$. The best constant $A_{n,t}$ is attained, and equality holds exactly for the family

$$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$

up to multiplication by a nonzero constant. In this sense, Gagliardo–Nirenberg constants encode both the sharp inequality and the structure of its extremals (Seuffert, 2016).

1. Classical sharp constants and extremal profiles

For the Euclidean sharp family, the best constant $A_{n,t}$ is the smallest constant for which the inequality holds for all $t$0. If one fixes the basic profile

$t$1

then the constant is expressed by

$t$2

and, after normalizing $t$3,

$t$4

This makes the constant completely determined by the optimizer. Equality holds if and only if $t$5 for some $t$6, $t$7, and $t$8.

The sharp family has two distinguished limits. As $t$9, one has $$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$0, so the inequality degenerates to a trivial estimate without gradient control. At $$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$1, one has $$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$2, and the inequality becomes the Sobolev inequality up to the usual rewriting of exponents. The same extremal profile is also identified, after suitable rescaling and parameter matching, with the Barenblatt-type profiles arising in nonlinear diffusion, which is why the constants first appeared in work of Carrillo–Toscani and then in complete form in Del Pino–Dolbeault’s classification of sharp inequalities (Seuffert, 2016).

2. Variational characterizations beyond the Euclidean case

Outside the classical Euclidean setting, Gagliardo–Nirenberg constants are usually characterized variationally. A recurring pattern is that the sharp constant is the infimum of a quotient of Sobolev-type and Lebesgue-type terms, and the infimum is attained by a ground state of an associated nonlinear elliptic or subelliptic equation. This suggests a general principle: the constant is often more naturally attached to an Euler–Lagrange problem than to the inequality alone.

Setting	Constant	Characterization
Euclidean sharp family	$$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$3	Explicit extremal $$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$4
Stratified Lie groups	$$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$5	Ground state of a fractional subelliptic Schrödinger equation
Graded groups	$$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$6	Ground state of a nonlinear subelliptic equation with Rockland operators
BO–ZK anisotropic setting	$$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$7	Ground state of $$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$8

On stratified Lie groups, the sharp fractional constant is defined by an infimum over $$\|u\|_{L^{2t}(\mathbb{R}^n)} \le A_{n,t}\,\|\nabla u\|_{L^2(\mathbb{R}^n)}^{\theta}\,\|u\|_{L^{t+1}(\mathbb{R}^n)}^{1-\theta}, \qquad \theta=\frac{n(t-1)}{t[\,2n-(t+1)(n-2)\,]},$$9, and if $1<t<\frac{n}{n-2}$0 is a least energy solution of

$1<t<\frac{n}{n-2}$1

then $1<t<\frac{n}{n-2}$2 is expressed directly in terms of $1<t<\frac{n}{n-2}$3 or, equivalently, the ground state energy level $1<t<\frac{n}{n-2}$4. The same paper also gives the corresponding best Sobolev constant and a structural relation between the two constants (Ghosh et al., 2023).

On graded Lie groups, the constants depend on the precise Sobolev norm, hence on the chosen Rockland operators and their homogeneous orders. The spaces themselves are independent of the specific operator, but the norms and therefore the best constants are not. The sharp constant for

$1<t<\frac{n}{n-2}$5

is expressed either in variational form or through a ground state of the associated nonlinear subelliptic equation, and the paper records explicit formulas in terms of the least energy $1<t<\frac{n}{n-2}$6 (Ruzhansky et al., 2017).

For the anisotropic Gagliardo–Nirenberg-type inequality associated with the Benjamin–Ono–Zakharov–Kuznetsov equation,

$1<t<\frac{n}{n-2}$7

the sharp constant $1<t<\frac{n}{n-2}$8 is characterized by the ground state of the nonlocal elliptic equation

$1<t<\frac{n}{n-2}$9

Here the anisotropic structure is decisive: the constant reflects fractional regularity in $n\ge3$0, first-order regularity in $n\ge3$1, and the scaling of the BO–ZK dynamics (Esfahani et al., 2017).

3. Deficit functionals and stability constants

Sharpness alone does not describe how near-optimizers behave. A stability theory asks whether small deficit forces closeness to the extremal manifold, and the associated stability constants measure that coercivity. For the Euclidean family, the Gagliardo–Nirenberg deficit is

$n\ge3$2

and $n\ge3$3 exactly on the extremal manifold. Under the normalization $n\ge3$4, Seuffert proved that if the deficit is sufficiently small, then

$n\ge3$5

This is a quantitative rigidity statement: near-optimizers are close, in the natural density variable $n\ge3$6, to the exact extremals (Seuffert, 2016).

A central structural point is that the proof does not work directly at the Gagliardo–Nirenberg level. The whole family is encoded through a continuous-dimension Sobolev inequality in dimension $n\ge3$7, with

$n\ge3$8

and the stability estimate is deduced from a continuous-dimension Bianchi–Egnell inequality. Because the Bianchi–Egnell stability constant $n\ge3$9 is not known explicitly, the resulting constants $1<t<\infty$0 and thresholds $1<t<\infty$1 are non-explicit.

A different but closely related stability mechanism appears in the entropy formulation of sharp Gagliardo–Nirenberg inequalities. For the fast diffusion and porous medium families, the sharp inequality is rewritten as

$1<t<\infty$2

where $1<t<\infty$3 is a relative entropy with respect to a Barenblatt profile and $1<t<\infty$4 is the corresponding generalized Fisher information. After optimizing the scaling and fixing the second moment, one obtains an improved inequality of the form

$1<t<\infty$5

with $1<t<\infty$6 explicit and strictly convex. This yields an explicit stability measurement in terms of entropy and, through Csiszár–Kullback-type estimates, distances to the manifold of optimizers (Dolbeault et al., 2014).

4. Geometric, weighted, and domain-dependent constants

On compact Riemannian manifolds, the sharp constant becomes a pair $1<t<\infty$7. For

$1<t<\infty$8

the first best constant $1<t<\infty$9 is the smallest $n=2$0 for which some finite $n=2$1 makes the inequality valid, and $n=2$2 is the smallest such lower-order constant at $n=2$3. One always has $n=2$4, the Euclidean sharp constant, and for $n=2$5 the optimal inequality admits an extremal function (Ceccon et al., 2013).

For convex domains $n=2$6, the domain constant

$n=2$7

is bounded below by an explicit geometry-dependent quantity involving

$n=2$8

For cubes $n=2$9, the paper proves

$A_{n,t}$0

so the whole-space constant and the cube constant are linked, but not identical in general (Benguria et al., 2018).

On planar metric graphs, the best constant is

$A_{n,t}$1

Bounded graphs always admit minimizers. On unbounded graphs the situation is subtler: half-lines and several line-like configurations satisfy $A_{n,t}$2 but do not attain the infimum, while periodic graphs recover compactness modulo translations and do admit extremals (Esteban, 2021).

Weighted open cones provide a different geometric phenomenon. For three homogeneous weights $A_{n,t}$3, the paper proves a three-weight Gagliardo–Nirenberg inequality under a joint concavity condition. In the one-weight case $A_{n,t}$4, the sharp constants are explicit in terms of Beta functions and

$A_{n,t}$5

For a certain range of parameters, sharpness in the three-weight inequality implies a rigidity statement: if an extremal exists, then the weights must be equal up to constant multiplicative factors (Balogh et al., 2022).

5. Entropy, conformal invariants, and structural interpretations

One interpretation of Gagliardo–Nirenberg constants is information-theoretic. For the nonlinear diffusion equation $A_{n,t}$6, Toscani considers the Rényi entropy power $A_{n,t}$7 and the Fisher information of order $A_{n,t}$8, $A_{n,t}$9, and proves the isoperimetric inequality

$$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$0

where $$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$1 is the Barenblatt profile. After the substitution $$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$2, this becomes a sharp Gagliardo–Nirenberg inequality, so the sharp constant is determined by the Barenblatt profile through the scale-invariant product $$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$3. In this formulation, optimizers are the nonlinear analogues of Gaussians (Toscani, 2018).

A different structural interpretation is conformal. Case introduces conformal invariants $$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$4 on smooth metric measure spaces that generalize the relationship between the Yamabe constant and the Sobolev constant to the Gagliardo–Nirenberg–Sobolev setting. On Euclidean space, the metric and measure criticality analysis singles out exactly the two Del Pino–Dolbeault families, characterized by

$$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$5

This gives a geometric explanation for why those families, and not arbitrary exponent pairs, admit the explicit sharp constants and optimizer profiles found by Del Pino–Dolbeault (Case, 2011).

6. Applications, computability, and open problems

Gagliardo–Nirenberg constants play an operational role in PDE. Carlen–Figalli used a sharp stability inequality in the two-dimensional case $$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$6 to analyze the critical-mass Keller–Segel equation, and the later extension to the full family opens the same strategy for porous medium, fast diffusion, and other drift–diffusion systems (Seuffert, 2016). In the anisotropic BO–ZK setting, the sharp constant controls the nonlinear term in the conserved energy and yields threshold conditions for uniform bounds in the energy space (Esfahani et al., 2017). In the stratified Lie group setting, the same constants determine sharp logarithmic Sobolev inequalities and ground-state bounds for nonlinear fractional subelliptic Schrödinger equations (Ghosh et al., 2023). On cubes and convex domains, explicit GNS constants enter quantitative Lieb–Thirring arguments (Benguria et al., 2018).

From a computational viewpoint, explicitness varies sharply across the subject. Some sharp constants are fully explicit through extremals, Gamma functions, Beta functions, or weighted surface integrals; others are explicit only through ground-state norms or least-energy levels. Stability constants are often less tractable: in the Euclidean stability theory, the obstruction is the non-explicit Bianchi–Egnell constant $$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$7 (Seuffert, 2016). By contrast, the supplementary material to a later stability paper keeps track of constructive constants in local Sobolev/Gagliardo–Nirenberg inequalities on balls and even computes a two-dimensional disk constant numerically, $$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$8, for a specific $$v_{\lambda,x_0}(x)=\lambda^{\frac{2}{t-1}}\bigl(1+\lambda^2|x-x_0|^2\bigr)^{-\frac{1}{t-1}}$$9-type inequality (Bonforte et al., 2020).

Several open issues remain explicit in the literature. On stratified Lie groups, uniqueness of the ground state $A_{n,t}$0 is not known (Ghosh et al., 2023). For cubes, the exact value of $A_{n,t}$1 and the existence of minimizers in higher dimension remain unresolved (Benguria et al., 2018). More broadly, the contrast between explicit sharp constants and non-explicit stability constants indicates that the spectral theory of the linearized inequality is still less understood than the extremal problem itself.