Energy Bounds for Kantorovich Transport Distances

• The paper establishes sharp energy bounds for Kantorovich transport distances by relating optimal transport costs with dual Sobolev-type norms using general convex cost functions.
• The approach leverages Kantorovich duality and variational optimization techniques to extend classical Wasserstein estimates to a broader class of super-linear cost functions satisfying a Δ₂ condition.
• Results unify multiple functional inequalities in analysis, probability, and geometry, enabling applications in nonlinear PDEs, empirical measure convergence, and concentration phenomena.

Energy bounds for Kantorovich transport distances quantify the relationship between optimal transport costs for general convex cost functions and certain dual Sobolev-type norms of signed measures. These energy bounds unify and extend fundamental inequalities for Wasserstein distances, most notably estimates originating with Ledoux for power costs, to the full class of convex, super-linear cost functions satisfying a generalized Δ₂ condition. The structure of these results illuminates deep connections among optimal transport theory, functional inequalities, nonlinear PDEs, and regularity theory, and provides explicit constants dependent on the Young function geometry associated to the cost.

1. Mathematical Setup: Convex Costs and Young Functions

Let $\mathcal X=\mathbb R^n$, and define a Borel measurable cost function $c:\mathcal X\times\mathcal X\to[0,\infty)$ via

$c(x,y) = L(x-y),$

where $L:\mathbb R^n\to[0,\infty)$ is convex, $L(0)=0$, and strictly positive away from the origin. The function $L$ must satisfy super-linear growth,

$$\frac{L(x)}{\|x\|} \to +\infty \quad \text{as} \ \|x\| \to \infty,$$

as well as a two-sided $\Delta_2$–condition: there exists $C>0$ such that

$L(2x) \leq C\,L(x) \quad \forall x\in\mathbb R^n.$

Associated to $L$ is the “Young function”

$$\Phi_L(r) = \sup_{x\neq0} \frac{L(rx)}{L(x)} \quad (r\geq0),$$

which is finite, convex, increasing, with $\Phi_L(1)=1$. The one-sided derivatives at $1$,

$p_- = \Phi_L'(1-), \quad p_+ = \Phi_L'(1+)$

satisfy $1\le p_-\le p_+$ and play a central role in determining the constants in energy bounds.

The Kantorovich transport distance for $L$ is

$$W_L(\mu,\nu) = \inf_{\pi\in\Pi(\mu,\nu)} \int_{\mathbb R^n\times\mathbb R^n} L(x-y)\,d\pi(x,y),$$

for probability measures $\mu,\nu$ with finite $L$-energy.

The dual Sobolev (“energy”) norm is defined by using the Legendre transform $L^*$,

$$L^*(u) = \sup_{v\in\mathbb R^n} \{ \langle u,v\rangle - L(v) \},$$

so that for a reference probability $\lambda$,

$$\|\nu-\mu\|_{H^{-1,L}(\lambda)} = \sup \Big\{ \int f\,d(\nu-\mu) : \int L^*(\nabla f)\,d\lambda \leq 1,\ f\in C_c^\infty \Big\}.$$

When $L(v)=\|v\|^p$, this recovers the negative Sobolev norm $\|\cdot\|_{H^{-1,p}}$.

2. Main Energy Bounds and Sharp Constants

The two central theorems generalize Ledoux’s estimate from power costs to general convex $L$. For $\mu,\nu\in\mathcal P(\mathbb R^n)$ with $\iint L(x-y)\,d\mu(x)\,d\nu(y)<\infty$:

• Theorem 1 (absolutely continuous case): If $\nu\ll\mu\ll dx$,

$$W_L(\mu,\nu) \leq A\,\Phi_L\left(\|\nu-\mu\|_{H^{-1,L}(\mu)}\right), \quad A=\Phi_L(p_+).$$

For power costs $L(v)=\|v\|^p$, $\Phi_L(r)=r^p$, $A=p^p$, and this yields $W_p(\mu,\nu)\le p \|\nu-\mu\|_{H^{-1,p}(\mu)}$.

• Theorem 2 (general case): If $p_->1$, for arbitrary $\mu,\nu$,

$$W_L(\mu,\nu)\leq A\,\Phi_L\left(\|\nu-\mu\|_{H^{-1,L}(\mu)}\right),$$

with improved constant

$$A = \Phi_L(p_+)\,\Phi_L\left(\frac{1}{\gamma}\right), \qquad \gamma = \sup\left\{ a+br : a+br\le\min(r^{p_+},r^{p_-})\;\;\forall r\ge0 \right\}\in(0,1].$$

The exact form of $A$ reflects the local geometry determined by $L$ near the identity.

3. Proof Techniques: Dual Formulations and Optimization

The establishment of these energy bounds relies on several key analytic structures:

• Kantorovich duality and Hopf–Lax formula: The transport cost $W_L(\mu,\nu)$ admits the dual representation,

$$W_L(\mu,\nu) = \sup_{f\in C_b(\mathbb R^n)} \left\{ \int Q_1 f\,d\nu - \int f\,d\mu \right\},$$

where $Q_t$ is the Hopf–Lax semigroup,

$$Q_t f(x) = \inf_{y\in\mathbb R^n} \left\{ f(y) + t L\left(\frac{x-y}{t}\right) \right\},$$

satisfying the Hamilton–Jacobi equation $\partial_t Q_t f = -L^*(\nabla Q_t f)$.

• Coupling with dual-norm: For $\varphi=d\nu/d\mu$, $g=\varphi-1$, and smooth $f$, consideration of the path $u_x(t) = (1+\theta(t)g(x)) Q_t f(x) - f(x)$ with suitable path $\theta$ and exploitation of the dual Sobolev inequality enable bounding the dual formulation in terms of $\|\nu-\mu\|_{H^{-1,L}(\mu)}$.
• Variational optimization: Optimizing over the path $\theta:[0,1]\to[0,1]$ subject to a specific ODE yields the sharp constant $A$ and ensures the energy bound is attained.

These steps generalize the “infimum-convolution + dual-Sobolev” approach pioneered by Bobkov and Götze to arbitrary convex cost functions.

4. Special Cases and Connections to Orlicz and Mixed Norms

The framework encompasses and unites several important cases:

• Power costs $L(v)=\|v\|^p$: Recovers Ledoux’s sharp energy bound with $A=p^p$.
• Orlicz costs $L(v) = V(\|v\|)$, where $V$ is a convex Young function: The resulting bounds are governed by the associated Young function $\Phi_L(r) = \sup_{s>0}V(rs)/V(s)$, extending energy bounds into the Orlicz–Wasserstein regime.
• Mixed costs such as $L(v) = \|v\|^p\log(1+\|v\|)$: Provided the two-sided $\Delta_2$-condition holds, energy bounds apply, capturing a broader class of transportation metrics beyond polynomial growth.

These generalizations reveal how the underlying geometry induced by the cost $L$ dictates both analytic and probabilistic properties of the associated optimal transport metric.

5. Applications in Analysis, Probability, and Geometry

Energy bounds for transport distances have deep implications across several domains:

• Quantitative rates for empirical measures: The inequalities control the expected optimal transport error, e.g., for $W_L(\mu_N, \lambda)$ where $\mu_N$ is an empirical measure, in high-dimensional statistical settings.
• Nonlinear PDEs: The Hamilton–Jacobi formalism allows derivation of gradient-flow structures and large deviation-type estimates for models of diffusion with non-quadratic mobility or transport cost.
• Geometric analysis: The transport-inequality framework extends isoperimetric and concentration results to metric-measure spaces with convex cost structure, pushing beyond classical $W_2$ settings.
• Functional inequalities: The approach simultaneously encompasses Poincaré, log–Sobolev, and Orlicz-type inequalities, integrating them in a single transport-energy perspective, unifying previously distinct analytic inequalities.

A plausible implication is a more unified treatment of concentration and stability phenomena in geometric analysis, driven by the explicit energy control afforded by these bounds (Bobkov et al., 20 Dec 2025).

6. Related Developments and Perspectives

The work on energy bounds for Kantorovich distances is situated within a wider context that includes:

• The Kantorovich–Rubinstein duality, which for $W_1$ connects integration error to $L^\infty$ bounds of gradients, and recent work interpolating between such dual formulations and endpoint spaces, notably via Lorentz and Orlicz norms (Steinerberger, 2020).
• The scaling limits and convergence of discrete optimal transport metrics, where analogous energy and Dirichlet-type functionals govern convergence of discrete-to-continuous transport structures in both finite and infinitesimal mesh settings (Gladbach et al., 2018).
• The extension of transport inequalities to new metric-measure spaces, pointing toward broader classes of geometric or functional inequalities derivable via transport-energy methods.

Open questions persist regarding the precise spectrum of intermediate spaces linking various $W_p$ and dual norms, and the potential to further generalize energy bounds to more singular or degenerate transport costs.

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This synthesis encapsulates the sharp and general structure of energy bounds for Kantorovich transport distances, highlighting the explicit dependence on cost geometry through Young functions, the duality-driven analytic machinery, and the substantial breadth of applications in mathematics and statistics (Bobkov et al., 20 Dec 2025).