Scale-Dependent Brascamp–Lieb Inequalities

• Scale-dependent Brascamp–Lieb inequalities are a family of statements where optimal constants, functional forms, and metrics vary with parameters like scale, curvature, and dimension.
• They extend the classical multinear inequality framework to include nonlinear, log-concave, and Fourier formulations, with applications in convex and algebraic geometry.
• The analysis employs techniques such as Hessian metrics, induction on scales, and operator scaling to manage anisotropy and multiscale degeneracies in functional inequalities.

Scale-dependent Brascamp–Lieb inequalities are not a single canonical class of inequalities, but rather a family of Brascamp–Lieb-type statements in which the optimal constant, the admissible functional form, or the underlying metric structure varies with a scale parameter, a localization radius, curvature, dimension, or a multiscale decomposition. In the classical multilinear setting, the Brascamp–Lieb inequality is scale-invariant once the balance condition is satisfied; in later developments, scale enters through Hessian metrics for log-concave measures, explicit dimension-dependent deficits, induction-on-scales recursions for nonlinear data, algebraic weights that compensate for local degeneracies, and multiscale singular-integral or Fourier-dual formulations (Böröczky, 2024).

1. Classical Brascamp–Lieb structure and the emergence of scale

The classical multilinear Brascamp–Lieb inequality is formulated for a datum $(B,p)$, where $B_i : \mathbb{R}^n \to H_i$ are surjective linear maps onto Euclidean spaces $H_i$ of dimensions $n_i$, and $p_i>0$. It states that for nonnegative integrable $f_i \in L^1(H_i)$,

$$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$

with optimal constant $\mathrm{BL}(B,p)$. A necessary balance condition is

$\sum_{i=1}^k p_i n_i = n,$

and finiteness is characterized by the additional subspace inequalities

$$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$

(Böröczky, 2024).

A decisive structural fact is Lieb’s Gaussian characterization: the optimal constant is obtained by restricting to centered Gaussian inputs, and may be written in matrix form as

$B_i : \mathbb{R}^n \to H_i$0

This already exhibits a scale mechanism: positive definite matrices encode ellipsoidal scales and anisotropies through their eigenvalues (Böröczky, 2024).

The classical inequality is scale-invariant if one rescales the functions appropriately, but when restricting to subsets, imposing geometric constraints, or replacing linear maps by nonlinear or weighted structures, constants can become scale-sensitive (Böröczky, 2024). That observation underlies most later uses of the phrase “scale-dependent Brascamp–Lieb inequalities.”

2. Curvature, Hessian metrics, and dimensional refinements

In the variance form associated with a log-concave probability measure

$B_i : \mathbb{R}^n \to H_i$1

the Brascamp–Lieb inequality reads

$B_i : \mathbb{R}^n \to H_i$2

Here the inverse Hessian furnishes a pointwise metric on gradients, so local curvature directly determines the scale at which fluctuations are penalized (Bolley et al., 2015).

A higher-resolution covariance theory is developed for strictly convex $B_i : \mathbb{R}^n \to H_i$3 and $B_i : \mathbb{R}^n \to H_i$4. For $B_i : \mathbb{R}^n \to H_i$5, $B_i : \mathbb{R}^n \to H_i$6, and $B_i : \mathbb{R}^n \to H_i$7 the least eigenvalue of $B_i : \mathbb{R}^n \to H_i$8, one has

$B_i : \mathbb{R}^n \to H_i$9

with sharp constant $H_i$0 (Carlen et al., 2011). The same work proves

$H_i$1

linking global divided differences to local gradient control (Carlen et al., 2011).

The paper does not explicitly use “scale-dependent” terminology, but it identifies $H_i$2, fractional inverse powers $H_i$3, and $H_i$4 as the weights governing the inequality. This suggests a scale-dependent interpretation in which low-curvature directions correspond to larger effective fluctuation scales, while strong curvature corresponds to short confinement scales (Carlen et al., 2011).

A different notion of scale dependence appears in dimensional refinements. For the same variance-form inequality, one has sharpened estimates such as

$H_i$5

and, for $H_i$6,

$H_i$7

These deficits have the right scale with respect to the dimension and behave correctly under tensorisation (Bolley et al., 2015).

Homogeneity provides another explicit scale parameter. For symmetric log-concave measures with $H_i$8-homogeneous potential $H_i$9, a strong Brascamp–Lieb inequality with constant $n_i$0 for even functions is equivalent to a local $n_i$1-Brunn–Minkowski inequality for the level sets of $n_i$2, with

$n_i$3

In the extremal case $n_i$4, one obtains $n_i$5, where $n_i$6 (Kolesnikov et al., 16 Jul 2025). This makes the homogeneity exponent itself a scale parameter linking Hessian spectral gaps to convex-geometric $n_i$7-Brunn–Minkowski behavior.

3. Nonlinear Brascamp–Lieb inequalities and induction on scales

A central nonlinear direction replaces linear maps $n_i$8 by $n_i$9 submersions $p_i>0$0 with $p_i>0$1. For simple linear data—meaning the exponent vector lies in the interior of the finiteness polytope—the nonlinear Brascamp–Lieb inequality asserts that for every $p_i>0$2 there exists a neighborhood $p_i>0$3 of $p_i>0$4 such that

$p_i>0$5

The proof is a tight induction on scales that uses Gaussian extremisers for the linearized datum in a fundamental way (Bennett et al., 2018).

The local scale parameter is encoded by quantities $p_i>0$6, the best constants on balls $p_i>0$7 for inputs that are $p_i>0$8-constant at scale $p_i>0$9. At sufficiently small scales one has a base-case linearization estimate

$f_i \in L^1(H_i)$0

while the recursive step takes the form

$f_i \in L^1(H_i)$1

for suitable $f_i \in L^1(H_i)$2 and positive $f_i \in L^1(H_i)$3 (Bennett et al., 2018). This is a genuinely scale-dependent Brascamp–Lieb inequality: the constant at one scale is controlled by constants at a smaller scale, with summable multiplicative losses.

A related global formulation is obtained through heat flow. For nonlinear data $f_i \in L^1(H_i)$4 on manifolds, Duncan introduces heat-flow operators $f_i \in L^1(H_i)$5 and scale-comparison constants $f_i \in L^1(H_i)$6, satisfying the submultiplicative relation $f_i \in L^1(H_i)$7, together with the one-step estimate

$f_i \in L^1(H_i)$8

for small $f_i \in L^1(H_i)$9 (Duncan, 2021). The corresponding nonlinear Ball inequality is

$$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$0

which yields a global near-monotonicity statement for the nonlinear Brascamp–Lieb functional under a variable-coefficient heat flow (Duncan, 2021). This formalizes scale dependence as an explicit recursion in the smoothing parameter.

4. Homogeneous, convex-geometric, and algebraic manifestations

Convex geometry supplies a geometric meaning for scale through John’s position, isotropic position, support functions, and ellipsoids. In Ball’s applications and related survey work, positive definite matrices represent ellipsoids whose eigenvalues encode directional scales, and geometric Brascamp–Lieb data with

$$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$1

give a scale-free inequality with optimal constant $$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$2 (Böröczky, 2024). Once localization, regularization, or non-geometric perturbations are introduced, the Brascamp–Lieb constant becomes sensitive to those geometric scales (Böröczky, 2024).

An algebraic global nonlinear Brascamp–Lieb inequality internalizes scale in a different way. For quasialgebraic maps $$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$3 on an algebraic variety $$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$4, one has

$$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$5

In the polynomial case $$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$6, this becomes

$$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$7

All fine-scale geometric degeneracies are absorbed into the affine-invariant weight $$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$8, while the global constant depends only on degrees and dimensions (Duncan, 2020). This yields a uniform constant across scales, with local scale information encoded entirely in the weight.

5. Beyond Hölder scaling: generalized size functionals and singular integrals

Some scale-dependent Brascamp–Lieb phenomena arise precisely when the standard Hölder scaling is absent. A five-linear singular integral of Brascamp–Lieb type is shown to satisfy the full range of estimates under the non-Hölder homogeneity condition

$$\int_{\mathbb{R}^n} \prod_{i=1}^k f_i(B_i x)^{p_i}\,dx \le \mathrm{BL}(B,p)\,\prod_{i=1}^k \left(\int_{H_i} f_i(y)\,dy\right)^{p_i},$$9

Its analysis is built from localized analysis on lower-dimensional subspaces, tensor-type stopping-time decompositions, hybrid maximal and square functions, and multiscale size/energy estimates (Muscalu et al., 2020). In this setting, scale dependence is not a perturbative refinement of a scale-free inequality; it is the central organizing principle of the proof.

A different generalization replaces fixed $\mathrm{BL}(B,p)$0 norms by a size functional $\mathrm{BL}(B,p)$1. For surjective maps $\mathrm{BL}(B,p)$2, a function $\mathrm{BL}(B,p)$3 is an HBL function if

$\mathrm{BL}(B,p)$4

The admissible $\mathrm{BL}(B,p)$5 are characterized by the HBL polytope $\mathrm{BL}(B,p)$6: for all $\mathrm{BL}(B,p)$7,

$\mathrm{BL}(B,p)$8

and

$\mathrm{BL}(B,p)$9

This makes the scaling behavior of $\sum_{i=1}^k p_i n_i = n,$0 polytope-controlled rather than monomial (O'Neill, 2017). In the Young-convolution case, near-extremizers are shown to concentrate at a single dyadic scale, and the dominant scales of the different factors must align (O'Neill, 2017).

6. Fourier, geodesic, and algorithmic structure of the constant

The Brascamp–Lieb constant itself admits scale-sensitive structural formulations. On the manifold of positive definite matrices, endowed with the Riemannian metric induced by the Hessian of $\sum_{i=1}^k p_i n_i = n,$1, the function

$\sum_{i=1}^k p_i n_i = n,$2

is geodesically concave, and

$\sum_{i=1}^k p_i n_i = n,$3

The geodesics

$\sum_{i=1}^k p_i n_i = n,$4

interpolate scales multiplicatively through eigenvalues, so geodesic convexity is a natural analytic formulation of scale balance for Brascamp–Lieb data (Vishnoi et al., 2018).

Algorithmically, operator scaling turns Brascamp–Lieb normalization into an alternating sequence of isotropy and projection normalizations. Equivalent data satisfy the determinant transformation law

$\sum_{i=1}^k p_i n_i = n,$5

for appropriate invertible changes of variables (Böröczky, 2024). In the explicit algorithmic reduction, one obtains polynomial-time procedures for feasibility, approximation of the optimal constant, and weak separation for the BL polytope, together with continuity bounds on the BL constant as a function of the input datum (Garg et al., 2016). The paper emphasizes that this continuity is important for developing non-linear BL inequalities (Garg et al., 2016).

Fourier duality supplies another structural scale symmetry. For finitely-generated discrete abelian groups,

$\sum_{i=1}^k p_i n_i = n,$6

identifying the Brascamp–Lieb constant on a subgroup $\sum_{i=1}^k p_i n_i = n,$7 with the constant on the annihilator $\sum_{i=1}^k p_i n_i = n,$8 in the Pontryagin dual (Bennett et al., 2020). In the broader setting of locally compact abelian groups, a structure theorem factors the BL constant over four components $\sum_{i=1}^k p_i n_i = n,$9, separating compact connected, vector, totally disconnected bounded, and discrete torsion-free behavior (Bennett et al., 2024). This amounts to a componentwise decomposition of scales: Euclidean dimensions, subgroup growth indices, and compact-open subgroup structure each contribute their own BL factor (Bennett et al., 2024).

7. Quantitative stability and recent directions

Recent work adds a stability layer to the scale-dependent picture. For the variance-form Brascamp–Lieb inequality

$$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$0

with $$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$1, equality holds precisely when

$$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$2

A quantitative stability theorem gives

$$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$3

where $$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$4 depends only on the dimension and is independent of the convex function $$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$5 (Machado et al., 27 Nov 2025). This uniformity is notable because the scale information carried by $$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$6 may vary widely, yet the stability modulus is dimension-only.

The same work derives uniform stability results for moment measures, again with dimension-only constants and a sharp exponent $$\dim V \le \sum_{i=1}^k p_i \dim(B_iV)\qquad \forall\,V\subset\mathbb{R}^n$$7 (Machado et al., 27 Nov 2025). This does not define a new scale-dependent BL inequality by itself, but it shows that the deficit-to-extremizer mechanism can be robust across families of convex potentials, including rescaled and anisotropic ones. A plausible implication is that future scale-dependent BL theories may combine explicit multiscale constants with equally robust stability statements.

Taken together, these developments show that scale dependence in Brascamp–Lieb theory appears in several distinct but connected senses: through curvature and Hessian eigenvalues, through dimension-dependent deficits, through induction-on-scales and heat-flow recursions, through algebraic weights compensating for local degeneracy, through non-Hölder singular integral analysis, and through matrix, Fourier, and operator-scaling formulations of the constant itself. The subject is therefore less a single inequality than a framework for tracking how Brascamp–Lieb structure survives localization, perturbation, anisotropy, and passage across spatial, geometric, and frequency scales.