Intrinsic dimensional functional inequalities on model spaces

We initiate a systematic study of intrinsic dimensional versions of classical functional inequalities which capture refined properties of the underlying objects. We focus on model spaces: Euclidean space, Hamming cube, and manifolds of constant curvature. In the latter settings, our intrinsic dimensional functional inequalities improve on a series of known results and lead to new Hamilton-type matrix inequalities. Our proofs rely on scaling, tensorization, and stochastic methods.


Introduction
This work focuses on the development of intrinsic dimensional versions of classical functional inequalities.In order to explain the meaning of "intrinsic" in this context it is best to start with an important example.The logarithmic Sobolev inequality in Gauss space [61,36] asserts that for every nice-enough absolutely continuous probability measure µ on R n , where γ n is the standard Gaussian measure on R n .Here, This material is based upon work supported by the NSF grant DMS-1929284 while A. E. was in residence at ICERM for the Harmonic Analysis and Convexity program.This material is based upon work supported by the National Science Foundation under Award Number 2002022.
is the relative entropy of µ with respect to ν and is the relative Fisher information of µ with respect to ν, provided that ν < < µ.Gross' motivation for (1) was to find a substitute for the Euclidean Sobolev inequalities holds in infinite-dimensional spaces (which was needed in constructive quantum field theories).Sobolev inequalities have the feature that the dimension n of the ambient space R n appears explicitly in the constants of the inequalities, which leads to triviality upon taking the limit n → ∞.In contrast, the constant 1/2 appearing in (1) is dimension-free, leading to (1) being well-defined in infinite dimensions.On the other hand, as was already observed by Stam [61], (1) can in fact be improved if the dimension n is taken into account.To see this improvement we first apply a standard change of measure (see [66]) which shows that (1) is equivalent to where λ n is the Lebesgue measure on R n .The dimensional log-Sobolev inequality of [20], improves upon (4) as can be seen from the inequality log s ≤ s − 1 for s ∈ (0, ∞).It is clear that when the Fisher information is large, (5) provides an exponential refinement over (4).Despite this quantitative improvement, (5) suffers from a lack of sensitivity to the intrinsic dimension of µ.To see this, suppose that µ is of the form dµ(x 1 , . . ., x n ) = d μ(x 1 , . . ., x k ) dγ n−k (x k+1 , . . ., x n ), where k < n and μ is an absolutely continuous probability measure on R k .Then (5) rephrased in terms of μ asserts that which deteriorates to (4) as the ambient dimension n increases, despite the fact that the intrinsic dimension k of µ is fixed.In other words, (5) is insensitive to the structure of µ.In [27, p. 12], Dembo showed that (5) can be further improved to an inequality which captures the intrinsic dimension of µ: where is the relative Fisher information matrix of µ with respect to ν. Observe that I(µ ν) = tr I(µ ν), (9) and thus (7) improves on (5) by the elementary inequality log det C ≤ n log tr C n which holds for every n × n positive semidefinite matrix C. In particular, both sides of (7) behave additively with respect to product measures: Plugging in dµ = d μ dγ n−k into (7) yields which captures correctly the intrinsic dimension of µ.More generally, by considering the eigenvalues of the Fisher information matrix, (7) can quantify the extent to which µ degenerates along each eigenvector direction.The goal of this work is to initiate a systematic study of intrinsic dimensional versions of classical functional inequalities.We focus on some important model spaces: Euclidean space, Hamming cube, and space forms (manifolds of constant sectional curvature).These model spaces have historically played a crucial role in the development of functional inequalities and their study has been the impetus leading to fruitful generalizations and abstractions; see the monograph [9].In view of the richness of the subject, our intrinsic dimensional functional inequalities on these spaces improve on multiple classical inequalities from the literature.The tools required to establish intrinsic dimensional functional inequalities in each of the models spaces will depend on the unique characteristics of the space itself: scaling (Euclidean space), tensorization (Hamming cube), and stochastic methods (space forms).In the rest of the introduction we will review each of these methods and present examples of the intrinsic dimensional functional inequalities which follow.We defer the statements of many of our results to the main body of the paper; see the following brief summary: Part 1. Euclidean and product spaces: scaling and tensorization • Logarithmic Sobolev inequalities for homogeneous measures (Section 2.2).
• Local logarithmic Sobolev inequalities and Hamilton's matrix inequalities on nonpositively curved space forms (Section 10).
1.1.Euclidean spaces: scaling.Most classical functional inequalities on R n are coordinatefree results phrased in a coordinate-dependent way.As such, they can often be substantially refined when expressed in a suitable basis.Concretely, the correct basis is found by performing a change of variables of the form x → Ax and then optimizing over a prescribed class of symmetries A ∈ G ⊆ GL n .Let us remark that explicit improvements of this form can be obtained only when it is possible to solve these optimization problems, which is not always the case.These improvements are moreover motivated by the study of equality cases.When a functional inequality has a non-constant function h : R n → R as an equality case, then the refined inequality obtained in the manner described above would be saturated by all functions of the form h A (x) = h(Ax), where A ∈ G.This principle has already been applied by Dembo [27] in the case of the Gaussian logarithmic Sobolev inequality (see also [30,14] and Section 2 below).In the first part of the paper we shall present more applications of this idea to other important functional inequalities in Euclidean space and further consequences.
1.1.1.Beckner inequalities.In [13], Beckner proved that any smooth function u ∈ C ∞ 0 (R n ) satisfies the estimates This family of inequalities interpolates between the Gaussian Poincaré inequality (corresponding to p = 1) and Gross' logarithmic Sobolev inequality [36] which arises as a limit when p → 2 − .We refer to the influential work of Latała and Oleszkiewicz [42] as well as [9, Section 7.6] for examples of Beckner-type inequalities satisfied by non-Gaussian measures.In [28,Corollary 4], Dolbeault and Toscani proposed a dimensional refinement of Beckner's inequality (11) for functions satisfying a second moment normalization condition.More specifically, they showed that if a function u ∈ C ∞ 0 (R n ) satisfies the normalization condition where the function ϕ p,n is given by Observe that (13) improves upon (11) up to the value of the implicit constant as The improvement (13) becomes particularly substantial when In the spirit of the matricial refinement (7) over the dimensional logarithmic Sobolev inequality (5), we present the following refinement of (13) for functions whose second moment matrix is appropriately normalized.
where δ ij is the Kronecker delta.Then, we have Applying the inequality det C ≤ trC n n and rearranging, we see that ( 17) strengthens (13).
and C p,q,r,r > 0 be such that (19) is satisfied for all functions u ∈ C ∞ 0 (R n ) with r = s under the constraint (18).Then, for every u ∈ C ∞ 0 (R n ), we have The inequality (21) improves on (19) by the arithmetic mean-geometric mean inequality so Theorem 2 asserts that Euclidean Gagliardo-Nirenberg-Sobolev inequalities, that is, inequalities of the form (19) with the choice of parameter r = 2, self-improve via scaling.In particular, (21) captures the fact (absent from (19)) that i u ≡ 0 on R n implies that u ≡ 0 under any L s -integrability assumption for u 1.2.Product spaces: tensorization.If (Ω, π) is a probability space, then for a measurable function f : Ω → R + we shall denote its entropy with respect to π by The usefulness of logarithmic Sobolev inequalities in probability and geometry stems largely from the fact that entropy satisfies a simple yet powerful tensorization principle, rendering them dimension-free estimates [43].In the interesting work [55], Polyanskiy and Samorodnitsky introduced a family of more general inequalities for Markov semigroups called nonlinear logarithmic Sobolev inequalities (see also [36,66,23,24,20,50,57] for previous occurrences of such estimates in the literature and applications).Let {P t } t≥0 be a Markov semigroup acting on measurable functions f : Ω → R with stationary measure π.Following [55], we say that {P t } t≥0 satisfies the (p, Φ)-LSI, where p ≥ 1 and Φ : R + → R + is a concave, continuous function with Φ(0) = 0, if for every measurable function f : where In [55, Theorem 1], the authors proved a dimensional tensorization property for nonlinear log-Sobolev inequalities asserting that if {P t } t≥0 satisfies the (p, Φ)-LSI, then for any n ≥ 1, the product semigroup {P ⊗n t } t≥0 with stationary measure π n satisfies the p, nΦ( By considering functions f of the form f (x 1 , . . ., x n ) = f (x 1 , . . ., x k ), for k < n, we see that (24) suffers from the problem of incorporating the ambient dimension n into the constant, thus ignoring the structure of f .In the Euclidean setting, we overcame this issue by finding the correct basis via an optimization procedure over the cone of positive semidefinite matrices.In contrast, such an approach is not suitable on the Hamming cube due to its discrete nature.Our solution to this problem is to refine tensorization instead of scaling.Indeed, as a consequence of a more general tensorization principle (see Theorem 18 below), we shall prove the following stronger nonlinear logarithmic Sobolev inequality for product spaces.
Theorem 3. Let (Ω, π, {P t } t≥0 ) be a stationary Markov semigroup satisfying the (p, Φ)-LSI for some p ≥ 1 and some concave, continuous function Φ : R + → R + with Φ(0) = 0.Then, for any n ≥ 1, every measurable function f : where E i (•, •) is the Dirichlet form associated with the i-th component of the semigroup {P ⊗n t } t≥0 .It follows readily from Jensen's inequality that where E(•, •) is the Dirichlet form associated to {P ⊗n t } t≥0 and thus (25) indeed strengthens (24).Moreover, in [55, Theorems 4 and 6], the authors found the optimal functions Φ p such that the (p, Φ p )-LSI is satisfied on the one-dimensional Hamming cube {0, 1} equipped with the uniform measure.Tensorizing their result via Theorem 3, one deduces an improved nonlinear logarithmic Sobolev inequality on the Hamming cube {0, 1} n .1.3.Space forms: stochastic methods.In order to explain our intrinsic dimensional functional inequalities on space forms we first recall the notion of local logarithmic Sobolev inequalities.Starting with the Euclidean setting, fix T ≥ 0, x ∈ R n , and let dµ dλ n = f P T δ x P T f (x) where δ x is the Dirac mass at x, f : R n → R is a nonnegative function, and {P t } t≥0 is the Euclidean heat semigroup given by P t h(x) := h(x + √ tz)dγ n (z).Plugging µ into (5) yields (after integration by parts and using the explicit form of P T δ x ), The inequality (27) is the local dimensional logarithmic Sobolev inequality on R n [10].While (27) provides an upper bound on the (local) entropy, the reverse local dimensional logarithmic Sobolev inequality [10] provides a lower bound, Analogously, we can use (7), instead of (5), to get the local intrinsic dimensional logarithmic Sobolev inequality on R n , which improves on (27).As for a reverse local intrinsic dimensional logarithmic Sobolev inequality in R n , we will establish below (Theorem 32) that which improves on (28).
Turning to the manifold setting, local dimensional logarithmic Sobolev inequalities exist on manifolds in forms which account for both the dimension of the manifold as well as the Ricci curvature [6].In light of the existence of the local intrinsic dimensional logarithmic Sobolev inequalities on Euclidean spaces (29) and (30), we wish to understand whether such inequalities can also exist on manifolds.Upon closer inspection, however, it is clear that inequalities such as (29) and (30) cannot hold if the only curvature information given pertains to the Ricci tensor.On a conceptual level, the difference between the dimensional and intrinsic dimensional inequalities is that the former provide information about the trace of the Fisher information matrix, while the latter provide information about the full spectrum.Hence, while information on the trace of the Riemann tensor, i.e., Ricci curvature, suffices to yield a dimensional inequality, information on the full Riemann tensor, i.e., sectional curvature, should be required to give an intrinsic dimensional inequality.
A concrete manifestation of this intuition is exhibited by the inequalities of Li-Yau and Hamilton [47,37].As was realized in [10], the reverse local dimensional logarithmic Sobolev inequality (28) since the argument in the log term of (28) must be nonnegative.Analogously, the reverse local intrinsic dimensional logarithmic Sobolev inequality (30) implies Hamilton's inequality, where is the order of positive semidefinite matrices.In the manifold setting, the Li-Yau inequality, which is a statement about the trace of the Hessian of log P T f , holds under a nonnegativity assumption on the trace of the Riemann tensor, namely the Ricci tensor [47,68].Indeed, Bakry and Ledoux [10] (see also the follow-up work [6]) established (reverse) local dimensional logarithmic Sobolev inequalities on manifolds with nonnegative Ricci curvature which imply the Li-Yau inequality.In contrast, Hamilton's inequality, which is a statement about the Hessian of log P T f , requires the manifold to have nonnegative sectional curvature (and also to be Einstein), which is an assumption on the full spectrum of the Riemann tensor [37].It follows that if local intrinsic dimensional logarithmic Sobolev inequalities were to hold, then information about the sectional curvature should be provided.
In this work we establish local intrinsic dimensional logarithmic Sobolev inequalities as well as Hamilton-type matrix inequalities for space forms: Euclidean spaces, spheres, and hyperbolic spaces.In addition to serving as the model spaces for functional inequalities on manifolds, these spaces are the simplest non-trivial examples of manifolds where we could hope for local intrinsic dimensional logarithmic Sobolev inequalities to hold.The methods of scaling and tensorization which worked, respectively, for Euclidean spaces and product spaces no longer apply on curved spaces as they lack product and homogeneity structures.Hence, we take a different route and build on the stochastic approach of Lehec [44,45] and Eldan, Lehec, and Shenfeld [30] towards logarithmic Sobolev inequalities.We start by stating our local intrinsic dimensional logarithmic Sobolev on space forms while deferring precise definitions to Part 2.
Then, we have the local intrinsic dimensional logarithmic Sobolev inequality and the reverse local intrinsic dimensional logarithmic Sobolev inequality As will become clear from the proof of Theorem 4, the theorem is not optimal and follows from a more powerful "master " matrix differential inequality (section 10.3).There are other inequalities which can be deduced from the master matrix differential inequality, specifically in space forms with nonpositive sectional curvature.In particular, we prove Hamilton-type matrix inequalities for the heat equation: Theorem 5. Let (M, g) be an n-dimensional Riemannian manifold with constant nonpositive sectional curvature κ ≤ 0. Let {P t } t≥0 be the associated heat semigroup and let f : M → R be a positive function.Then, for every x ∈ M and every T ≥ 0, if, either κ = 0, or κ < 0 and In flat space, where κ = 0, Theorem 5 reduces to (32), namely, Hamilton's matrix inequality [37,Corollary 4.4].In hyperbolic spaces, Theorem 5 is completely new.The constraint 4 n 2 κ ∆P T f (x) P T f (x) > 1 is natural.Indeed, Theorem 5 is a matrix version of the improved Li-Yau inequality of Bakry, Bolley, and Gentil-see Remark 34.
Going beyond matrix inequalities, we can use our master matrix differential inequality to obtain another form of local intrinsic dimensional logarithmic Sobolev inequalities.Theorem 6.Let (M, g) be the n-dimensional hyperbolic space with sectional curvature κ < 0 with the associated heat semigroup , and let µ be the probability measure with we have the local intrinsic dimensional logarithmic Sobolev inequality where {σ i } n i=1 are the eigenvalues of E µ [−∇ 2 log f ], and the reverse local intrinsic dimensional logarithmic Sobolev inequality where Acknowledgements.We are grateful to Dario Cordero-Erausquin, Max Fathi, Nathael Gozlan, and Yury Polyanskiy for useful pointers to the literature and to Georgios Moschidis for many helpful discussions.
Part 1. Euclidean and product spaces: scaling and tensorization

Logarithmic Sobolev inequalities in Euclidean spaces and Cramér-Rao bounds
In this section we discuss strengthenings of logarithmic Sobolev inequalities for measures on Euclidean spaces by means of scaling.In addition, we derive an application of these inequalities to Bayesian Cramér-Rao bounds.
2.1.Warm-up: Gross' inequality.The Euclidean reformulation (4) of the logarithmic Sobolev inequality in Gauss space [36] asserts that if f : R n → R + is a probability density, then Fix such a density f and consider the reparametrized density f A : R n → R + which is given by , where A ∈ GL n is a positive definite matrix.Applying (41) for f A we get which after rearranging becomes For the optimal choice of matrix A = I(µ λ n ) −1/2 , (43) readily becomes Dembo's inequality (7).
Observe that in this argument we made critical use of the change of variables formula for the Lebesgue measure, i.e., that Lebesgue is the only measure on Euclidean space satisfying such an invariance property under all linear transformations, in the next section we shall observe that a weaker self-improvement can be deduced for measures which behave well under diagonal linear maps.

2.2.
Logarithmic Sobolev inequalities for homogeneous measures.Let p 1 , . . ., p n ≥ 0. An absolutely continuous measure ρ on R n with density w : R n → R + is called (p 1 , . . ., p n )-homogeneous if for every t 1 , . . ., t n > 0, Theorem 7. Fix c 1 , c 2 > 0, n ∈ N, p 1 , . . ., p n ≥ 0 and let ρ be a (p 1 , . . ., p n )-homogeneous measure such that for any Borel probability measure µ on R n , Then, for any Borel probability measure µ on R n with positive differentiable density f , we have The existence of homogeneous measures ρ satisfying inequalities of the form (45), as well as more general entropy-energy inequalities follows, for instance, from [9,Proposition 7.3.1].
Proof of Theorem 7. Let f = dµ dρ be an arbitrary positive function with ρ-integral equal to 1 and fix t 1 , . . ., t n > 0. The measure µ t with density where we made the change of variables (y 1 , . . ., y n ) = (t 1 x 1 , . . ., t n x n ).We have, Similarly, assuming in addition that f is differentiable, for every k ∈ {1, . . ., n} we have Therefore, applying (45) for µ t and reorganizing the terms, we deduce that .
It is now elementary to check that the above infimum is attained when and plugging this choice of parameters completes the proof.

2.3.
A Bayesian Cramér-Rao bound.In [1], Aras, Lee, Pananjady and Courtade observed that logarithmic Sobolev inequalities formally imply Bayesian Cramér-Rao bounds, thus extending some results of Efroimovich [29] for Gaussian measures.In this section, we investigate similar applications of intrinsic dimensional log-Sobolev inequalities in the spirit of ( 46) and (7).
Following [1], we work in the setting of parametric statistics.Let {µ θ } θ∈R n be a family of probability measures on a measurable space (Ω, F).Assume moreover that there exists a dominating σ -finite measure λ on Ω such that µ θ has a positive density with respect to λ, We shall assume throughout that each function θ → f (x; θ) is smooth and that for almost every θ ∈ R n .The Fisher information of the parametric family {µ θ } θ∈R n is Finally, if π is a probability measure on R n , we denote the mutual information of π with the family {µ θ } θ∈R n by The main result of [1, Theorem 1] specified to the standard Gaussian measure γ n asserts that for every absolutely continuous probability measure π on R n , Inequality (55) implies the Gaussian logarithmic Sobolev inequality (1) since choosing µ θ = λ independently of θ, the terms I(π; {µ θ }) and J(θ) both vanish.We present inequalities in the spirit of (55) for homogeneous measures satisfying a log-Sobolev inequality of the form (45).
Then, for every parametric family {µ θ } θ∈R n and every absolutely continuous measure π on R n whose density with respect to ρ is h : R n → R + , we have Observe that the terms inside the logarithm on the right-hand side are the k-th component of the Fisher informations I(π|ρ) and J(θ) respectively, in analogy with Theorem 7.
Proof of Theorem 8. Consider the function f : Ω → R + given by and observe that Moreover, for x ∈ Ω, consider the function h x : R n → R + given by and notice that the measure ν x on R n with dν x (θ) = h x (θ) dρ(θ) is a probability measure since By Theorem 7 and the assumption on ρ, for every x ∈ Ω we have Integrating this inequality with respect to the probability measure f (x) dλ(x), we get where the last line follows from Jensen's inequality.Moreover, by definition we have Similarly, computing the integral on the right-hand side of ( 63), gives Combining everything, we deduce the desired inequality.
Remark 9.In the case of the Gaussian measure ρ = γ n , we have at our disposal the intrinsic dimensional logarithmic Sobolev inequality (7).Repeating the same proof mutatis mutandis while replacing (46) with (7), we conclude that for any probability measure π on R n whose density with respect to γ n is h : R n → R + , and for every parametric family {µ θ } θ∈R n , we have where M 2,π = θ ⊗2 dπ(θ).This recovers a result of Efroimovich [29, Theorem 5].Combining the inequalities log det C ≤ n log trC n and log y ≤ y − 1, which hold for all y > 0 and all n × n positive definite matrices C, we see that Efroimovich's inequality is a strengthening of (55).

Gagliardo-Nirenberg-Sobolev inequalities
In this section we shall prove Theorem 2: Then, for every u ∈ C ∞ 0 (R n ), we have Then, for s ≥ 1 we have and Therefore, applying (65) to u t and rearranging, we deduce that for every t 1 , . . ., t n > 0. Choosing gives the desired inequality (67).

Beckner inequalities
In this section we shall prove Theorem 1: where δ ij is the Kronecker delta.Then, we have For the proof of Theorem 11 we shall use the intrinsic dimensional logarithmic Sobolev inequality (7) which takes the following simple form for appropriately normalized functions in Gauss space.
Then, we have which is the density of a probability measure µ on R n .Then, we have On the other hand, for k ∈ {1, . . ., n}, we compute and thus for i, j ∈ {1, . . ., n}, we get For i j, integration by parts gives 2 whereas for i = j, again by integration by parts, 2 Plugging the above in (79) and using (75) again for the last term, we deduce that and the conclusion of the lemma follows from (7).
Equipped with Lemma 12, we proceed to the proof of Theorem 1.
Proof of Theorem 11.Assume, without loss of generality, that u L 2 (γ n ) = 1.Combining a lemma of Dolbeault and Toscani [28, Lemma 5] (see also [42]) with Lemma 12, we get that Therefore, which is the desired estimate under the normalization u L 2 (γ n ) = 1.

q-logarithmic Sobolev inequalities
Following Bobkov and Zegarlinski [18] (see also [11]) we say that a probability measure µ on the real line satisfies the q-logarithmic Sobolev inequality with constant C > 0 if for any Standard tensorization principles show that if (84) holds, then for any where In particular, it has been established in [18,Corollary 5.6] (see also [16,Section 5]) that the measure µ p with density 1 Z p e −|x| p , where p > 2, satisfies the q-logarithmic Sobolev inequality for q = p p−1 with some constant C q > 1.In order to investigate scale-invariant refinements of (85) for this family of measures in the spirit of (50), we first need to formulate them as Euclidean inequalities.
Theorem 13.For any q ∈ (1, 2), there exists a constant Cq > 0 such that for any n ∈ N and any probability measure µ on R n with positive differentiable density g, Proof.For p = q q−1 > 2 consider the probability measure dµ p (x) = e −|x| p Z p on R, where the normalizing constant is Z p = 2Γ (1 + 1/p) > 2. Let µ be a probability measure on R n with differentiable density g : R n → R + and consider the function f : R n → R + given by p g(x) 1/q e x p p /q , (87) which satisfies R n f (x) q dµ n p (x) = 1.Therefore, the q-logarithmic Sobolev inequality for µ n p applied to the function f implies that Observe that 1 and for i ∈ {1, . . ., n}, Therefore, rearranging (88) we deduce that for some different constant Cq > 0 and the proof is complete.
This Euclidean weakening of the q-logarithmic Sobolev inequality (85) for µ n p makes it amenable to refinements via scaling.Theorem 14.For any q ∈ (1, 2) and p = q q−1 , there exists a constant Cq > 0 such that for any n ∈ N and any probability measure µ on R n , Proof.Fix t 1 , . . ., t n > 0 and consider the probability measure µ t whose density is given by x → Then, we have and for every i ∈ {1, . . ., n}, Therefore, applying (86) to µ t and rearranging, we deduce that and taking an infimum over t 1 , . . ., t n > 0 completes the proof.

Beyond linear rescalings
The simple idea of the previous sections can be summarized as follows.Let be a functional inequality valid for regular enough functions f on R n and fix a subgroup of symmetries G ⊆ GL n .For a fixed f : R n → R for which inequality (96) is valid and A ∈ G, consider the function f A : R n → R given by f A (x) = f (Ax).If (96) applied to f A can be rearranged to an upper bound for K(f ) of the form then taking an infimum over A ∈ G yields a stronger inequality as (96) just amounts to the choice A = Id n .Observe that enhancing inequalities in this way, always produces a larger family of extremals.For instance, (4) becomes an equality only when µ is a translate of γ n , ( 5) becomes an equality when µ is a Gaussian measure with covariance matrix of the form σ Id n , where σ > 0, and ( 7) becomes an equality for any Gaussian measure on R n .
In this section, we will discuss the possibility of refining functional inequalities by using changes of variables via nonlinear maps and we shall illustrate this in the case of the logarithmic Sobolev inequality (4).Let T : R n → R n be a smooth diffeomorphism and for a measure µ on R n with a differentiable density f : R n → R + consider the measure µ T whose density is given by f T (x) = (f • T )(x)| det DT (x)|, where x ∈ R n and DT ∈ M n (R) is the differential of T .We need the following computations for the relative entropy and Fisher information of µ T .
Lemma 15.In the setting above, and The proof is a straightforward computation using a change of variables and is thus omitted.These formulas along with the fact that any absolutely continuous measure can be transported to γ n give rise to the following variational formula for relative entropy on R n .Theorem 16.Let µ be an absolutely continuous measure on R n .Then, with equality if T is a transport map from µ to γ n , where Proof.Applying the logarithmic Sobolev inequality (4) to µ T and using Lemma 15, we get with equality only if µ T = γ n .The existence of a map T transporting µ to γ n is a classical fact in optimal transport going back to at least [56,41] (see also [64]).
We are not aware of a proof of (100) which does not rely on the logarithmic Sobolev inequality (4).It remains very interesting to understand whether (100) can lead to stability estimates for (4), or even (7), in the spirit of [30,Theorem 3].Formula (100) becomes more tractable when specified to specific kinds of diffeomorphisms.For instance, when T is a product map of the form T (x) = (τ 1 (x 1 ), . . ., τ n (x n )), we get (102) A similar simplified formula can be derived if T is a rotationally invariant map of the form T (x) = σ (|x|)x.The equality cases of Theorem 16 show that if µ is a product measure or a rotationally invariant measure, then the inequalities obtained by optimizing over the corresponding class of nonlinear transformations become equalities.For the case of a general probability measure µ, we pose the following question.
A similar question can be asked for the optimal rotationally invariant change of variables.We have not investigated whether nonlinear changes of variables may give rise to variational formulas à la (100) when applied to other estimates like the Gagliardo-Nirenberg-Sobolev inequality (65) or Beckner's inequality (11).

Tensorization of nonlinear logarithmic Sobolev inequalities in product spaces
Let I be a countable set, {(X i , µ i )} i∈I a family of probability spaces where X i is countable and denote their product space by (X, µ) = ( i∈I X i , ⊗ i∈I µ i ).For a point x = (x i ) i∈I ∈ X and i ∈ I, we shall denote by x ∼i the point (x j ) j i ∈ j i X j and by µ ∼i def = ⊗ j i µ j .Moreover, for a point z ∈ j i X j and a function f : X → R, we shall denote by f z : X i → R the restriction of f given by For each i ∈ I, let B i be a functional acting on measurable functions g : j∈J X j → R for any J ⊆ I.We shall say that the family of functionals {B i } i∈I disintegrates if it satisfies the identities Our main tensorization principle for nonlinear entropy inequalities is the following.
Theorem 18. Fix a countable set I and two collections of functionals {Q i } i∈I , {M i } i∈I which disintegrate in the above sense.Let Φ : R → R be a concave function and suppose that, for any i ∈ I, every function f i : X i → R + satisfies the inequality Then, every function f : Proof.Combining the subadditivity of entropy and the assumptions of the theorem (including the disintegration of {Q i } i∈I ) we get that, for every f : , and E µ [f ] dµ ∼i (x ∼i ) defines a probability measure on j i X j .Hence, by Jensen's inequality and disintegration, we get This completes the proof of the theorem.
Remark 19.While Theorem 18 is stated in a general form which contains the disintegrating additive errors {Q i } i∈I , in its main application (Theorem 3) which refines the result of [55], these are assumed to be vanishing.We chose to include the deficits in the general formulation above as such terms often appear in modified logarithmic Sobolev-type inequalities, especially in discrete settings (see, for instance, [15,67,17,39]).
Proof of Theorem 3. The conclusion (25) directly follows from Theorem 18 with Q i (f ) = 0 and

Preliminaries
In this section we will introduce the necessary prerequisites from stochastic calculus on manifolds required to prove Theorem 4. We will be following the standard notation of [38,54].8.1.The frame bundle.Let (M, g) be a complete n-dimensional Riemannian manifold.The orthonormal frame bundle O(M) of M is the set of all pairs of the form (x, u), where x ∈ M and u : R n → T x M is a Euclidean isometry.We shall denote by π : O(M) → M the natural projection given by π(x, u) = x.Any scalar-valued function f : M → R admits a natural lift f : Abusing notation, we shall often identify the pair (x, u) ∈ O(M) with the isomorphism u.The frame bundle O(M) is equipped with Bochner's horizontal Laplacian and can be verified (see [38,Proposition 3.1.2])that the lift f of any function f : where ∆ is the Laplace-Beltrami operator of (M, g).
A curve {u t } t∈ [0,1] in O(M) is called horizontal if for every a ∈ R n , the vector field {u t a} t∈[0,1] is parallel along the curve {πu t } t∈[0,1] in M. A tangent vector X ∈ T u O(M) is called horizontal if it is the tangent vector of a horizontal curve passing from u.For any vector X ∈ T πu M there exists a unique horizontal vector X ∈ T u O(M) such that π * X = X; we say that X is the horizontal lift of X at u. Let {e 1 , . . ., e n } be the standard basis of R n .The i-th fundamental horizontal vector field H i evaluated at a point u ∈ O(M) is the horizontal lift of the vector ue i ∈ T πu M. Thus, for any i ∈ {1, . . ., n}, the lift f of a function f : A vector field on O(M) is called horizontal if it lies in the span of {H 1 , . . ., H n }.We denote by •, • hor the natural inner product on the space of horizontal vector fields on O(M) given by Moreover, we shall denote by ∇ hor f = (H 1 f, . . .H n f) ∈ R n the horizontal gradient of a given function f : O(M) → R. In this terminology, the horizontal Laplacian takes the form ∆ O(M) f = i H 2 i f.We record for future reference the following very useful expression for the action of the commutator of ∆ O(M) with H i on lifted functions.Lemma 21.If f : (M, g) → R is a smooth function, then for any i ∈ {1, . . ., n}, its lift f satisfies where Ric( Therefore, we have where in the last identity we used that V ab f = 0. Again, by [38,Lemma 5.5.1], if we denote by A k ab the number 1 2 for (a, b) = (k, ) and − 1 2 for (a, b) = ( , k), and zero otherwise, we obtain where the antisymmetry of Ω on the top indices follows from its definition in [38, p. 153] as it is an o(d)-valued tensor.Combining (116), ( 117), (118) and summing over k, we deduce that Now, observe that by the definition of Ω in terms of the Riemann tensor R of M in [38, p. 149], and the conclusion follows from the definition of Ricci curvature.

8.2.
Brownian motion on manifolds.Let W t = (W 1 t , . . ., W n t ) be a standard Brownian motion on R n and (M, g) be a complete n-dimensional Riemannian manifold.We consider the following stochastic differential equation on the frame bundle O(M), where the shorthand notation • refers to the Stratonovitch integral.In Itô terms, the above SDE asserts that for every smooth g : For any initial condition Φ 0 = u ∈ O(M), this equation has a strong solution which does not blow up in finite time if the Ricci curvature of M is bounded from below by any constant κ ∈ R (see [38,Theorem 4.2.4] and [63] for a sufficient and almost necessary condition for stochastic completeness).We denote by B t = πΦ t , where t ≥ 0, the Brownian motion on M whose starting point is x = πu ∈ M. Applying (122), we deduce that for any smooth function f : M → R, the Brownian motion 8.3.The Föllmer process and Lehec's formula.In this section we introduce an analogue of the classical Föllmer process [58,33,46] on Riemannian manifolds (see also [38,Section 5.4]).
We then present a result of Lehec [45] who used this process to give a stochastic proof of the dimensional logarithmic Sobolev inequality for manifolds with Ricci curvature bounded below (see [10,6] for more general statements proven via semigroup arguments).
Let W t = (W 1 t , . . ., W n t ) be a standard Brownian motion on R n and (M, g) be a complete ndimensional Riemannian manifold whose Ricci curvature is bounded from below.We shall denote by dx the volume measure on M and by {P t } t≥0 the heat semigroup on M. Recall that for a smooth function h : M → R, the action of the heat flow {P t } t≥0 on g is characterized by the ordinary differential equation with initial condition P 0 h = h on M. We recall that the heat semigroup and the Laplacian commute: ∆P t h = P t ∆h, and we write P t ∇ 2 f (x) for the 2-tensor on T x M identified with the symmetric matrix (P t ∇ 2 f (Φ 0 e i , Φ 0 e j )(x)) n i,j=1 .Note that P t and ∇ 2 do not commute (cf.Theorem 24).For a positive function f : M → R + and T > 0, we consider the following system of stochastic differential equations with respect to where the notation • again refers to the Stratonovitch integral.It is known (see [45,Theorem 7]) that if f is a smooth-enough positive function, then for any initial condition Ψ 0 = u ∈ O(M), the system (125) has a strong solution on [0, T ].In [45,Theorem 7], Lehec proved the manifold version of an important representation formula for relative entropy in terms of the Föllmer process X t , first proven in their earlier work [44].
Theorem 22 (Lehec).Let (M, g) be a complete n-dimensional Riemannian manifold whose Ricci curvature is bounded from below and fix a smooth enough positive density function f : M → R + and T > 0. If {X t } t∈[0,T ] is a solution of (125) with initial condition Ψ 0 = u and πu = x, then the relative entropy of the measure µ with density where It is worth pointing out that, in view of the decay and regularity of the heat kernel on space forms (see, e.g., [21,Chapter 6] and [48,25,35]), it suffices to assume that the functions for which we wish to prove the logarithmic Sobolev inequalities of Theorem 4 are Lipschitz and bounded away from 0. Therefore, the regularity conditions required for the function f in Lehec's theorem will always be tacitly assumed to hold.
We record for future reference the following computations (see also [38,Equations (5.5.2) -(5.5.4)]) on the SDE satisfied by partial derivatives of the logarithm of the heat kernel.
and by F t the lift of F t onto O(M).If {X t } t∈[0,T ] is a solution of (125) and {W t } t≥0 is a standard Brownian motion on R n , then for every i ∈ {1, . . ., n} we have Proof.Using Itô's formula and (125), we get (omitting the dependence on Ψ t on the right-hand side of (129)) Observe that the function F t satisfies the equation which, after applying H i on both sides, gives Moreover, we have where in the last identity we use that [H i , H k ]h = 0 for any lifted function h on O(M) [38, Lemma . Substituting ( 131) and ( 132) in (129), we finally obtain and the desired identity follows immediately from Lemma 21.

8.4.
The heat flow on space forms.The classical Bochner formula (see, e.g., [65]) implies that if (M, g) is a Riemannian manifold with constant Ricci curvature Ric = κ ∈ R, then for every smooth function f : M → R. In [65], Wang investigated commutation relations of this form for second order derivatives instead of the gradient ∇.We shall use the following result.
Theorem 24 (Wang).A Riemannian manifold (M, g) of dimension n has constant sectional curvature κ ∈ R if and only if the Hessian tensor of every smooth function f :

Intrinsic dimensional logarithmic Sobolev inequality in space forms
Having explained the necessary background we can now present Theorem 4. We first recall that when dµ dP T δ x = f , we have Theorem 25.Let (M, g) be an n-dimensional Riemannian manifold with constant sectional curvature κ ∈ R \ {0} with the associated heat semigroup {P t } t≥0 .Fix T > 0, x ∈ M, a smooth positive function f : M → R with M f dP T δ x = 1, and let µ be the probability measure with and let . Then, we have the local intrinsic dimensional logarithmic Sobolev inequality and the reverse local intrinsic dimensional logarithmic Sobolev inequality The proof of Theorem 25 (see also the stronger Theorem 30) is modeled after the stochastic proof by Eldan, Lehec, and Shenfeld [30] of the intrinsic dimensional logarithmic Sobolev inequality in flat space (7) (and a weaker reverse inequality [30,Theorem 3]).A basic ingredient of this approach is deriving a stochastic differential equation for the tensor whose trace is the term 2 in (126).This is the content of the next lemma for which we establish the following notation.Let {B t } t≥0 be a Brownian motion on M with B 0 = x.As before, we denote by F t the function log P T −t f and by F t its horizontal lift on O(M).Moreover, we shall denote by G t the function exp F t = P T −t f and by G t = exp F t its lift.Consider the random matrices Q(t), P(t) ∈ M n (R) (the space of n × n square matrices over R) given by We can now derive the aforementioned stochastic differential equation.
Lemma 26.Let (M, g) be a Riemannian manifold.In the terminology above, for every i, j ∈ {1, . . ., n}, there exists a martingale {M ij (t)} t∈[0,T ] such that for t ∈ [0, T ], we have Proof.Observe that by the chain rule, we have (omitting the dependence on Ψ t on the righthand side below) and by the definition and symmetry of the matrix Q(t), Combining Itô's product rule with Lemma 23, we get that for i, j ∈ {1, . . ., n}, where in the right-hand side we again omitted the dependence on Ψ t and X t .Denoting the term in the last line by dM ij (t), it is clear that {M ij (t)} t∈[0,T ] is a martingale and (142) becomes where we also used (141).This is the desired identity.
The stochastic differential equation of Lemma 26 will allow us to derive a differential equation for note that with this notation, (126) reads H(µ P T δ x ) = 1 2 T 0 tr v(t) dt.We will then turn the differential equation into a differential inequality from which Theorem 30 and Theorem 25 shall follow.To derive the differential equation for v(t) we start by defining Assuming that the underlying manifold M is Einstein and taking expectations, we deduce the following differential equation for v(t).
Lemma 27.Let (M, g) be an Einstein manifold with constant Ricci curvature Ric = ρ for some ρ ∈ R.
For every i, j ∈ {1, . . ., n} and t ∈ (0, T ), we have Proof.Since M has constant Ricci curvature ρ, we have Plugging this in the rightmost term of Lemma 26, we get that The result follows after taking expectation (since EM ij (t) = M ij (0) = 0) and differentiating.
In order to turn (145) into a differential inequality we will use Jensen's inequality n(t) m(t) 2 where we used P = Q 2 .To use the latter inequality we need to better understand the term m(t).On manifolds of constant curvature, m(t) takes the following simple form.
Lemma 28.Let (M, g) be an n-dimensional Riemannian manifold with constant sectional curvature κ ∈ R. For every i, j ∈ {1, . . ., n} and t ∈ (0, T ), we have Proof.Taking expectations in (140), we obtain It follows from ( 125) and (121) that Ψ t has law f (B T ) with respect to Φ t for every t ∈ [0, T ] (see also the proof of [45,Theorem 7] for an argument based on Girsanov's theorem).Therefore, by the tower property of conditional expectation, we have Recall that for any function h : M → R with horizontal lift h, we have By the definition [65, Equation (1. 2)] of the action of {P s } s≥0 on tensors, we have where the last identity follows from the definition of stochastic parallel transport given by {Φ s • Φ −1 0 } s≥0 (see [38,Section 2.3]).Similarly, we have and combining everything we deduce that Plugging ( 155) and ( 150) in (149) completes the proof.
We are now ready to derive the differential inequality for v(t).For simplicity, we shall denote by c T def = P T ∆f (x) and by J T the symmetric matrix with which satisfies trJ T = 0. Combining all of the above, we get the following matrix inequality: Proposition 29.Let (M, g) be an n-dimensional Riemannian manifold with constant sectional curvature κ ∈ R. For every t ∈ (0, T ), we have so in particular, where is the inequalities in the positive semidefinite ordering.
Proof.Combining the matrix Jensen inequality with (145), (148) and expanding, we get (157) The inequality (158) follows since the squared matrix is positive semidefinite.
Proposition 29 allows us to deduce the following local intrinsic dimensional logarithmic Sobolev inequalities which are, however, non-explicit.
Theorem 30.Let (M, g) be an n-dimensional Riemannian manifold with constant sectional curvature κ ∈ R. Fix T > 0, x ∈ M, a smooth positive function f : M → R with M f dP T δ x = 1, and let µ be the probability measure with dµ dP T δ x = f .Suppose there is a family of matrices U (t) ∈ M n (R) for t ∈ [0, T ] which solves the equation with either initial condition U (0) := v(0) or U (T ) := v(T ).Then, we have the local intrinsic dimensional logarithmic Sobolev inequality and the reverse local intrinsic dimensional logarithmic Sobolev inequality Proof.Lehec's formula (126) implies For the reverse local intrinsic dimensional logarithmic Sobolev inequality, we note that U (0) = v(0) so the result follows by (157) and standard comparison principles for matrix Ricatti equations, see [40].For the local intrinsic dimensional logarithmic Sobolev inequality, we have U (T ) = v(T ) and the conclusion follows by reversing time.
Theorem 30 provides sharp results which are, however, not explicit since the solutions of (161) are complicated.They are expressed in terms of special functions, except in the flat space case where they simplify considerably-see Section 9.1.To avoid the complication of Theorem 30 we will use (158), rather than the stronger inequality (157), which will lead to explicit bounds, namely Theorem 25.To this end, we shall need the following technical lemma on matrix Bernoulli differential inequalities.
for which every V (t) is a positive semi-definite matrix satisfies the ordinary differential inequality with boundary condition V (ε) = V ε , then it also satisfies the matrix inequalities and Moreover, the right-hand side of (167) is positive definite for every t ∈ (ε, T ).As V ε is positive definite, the same holds for V (t) for t near ε so let t max ∈ [ε, T ] be the supremum over t ∈ [ε, T ] where V (t) is positive definite.For t ∈ (ε, t max ), multiplying (165) by V (t) −1 on both sides, we deduce that where dt .Therefore, we have where in the last inequality we used that C(t) is symmetric.Integrating from ε to t, we get which can be rearranged to give, for every t ∈ [ε, t max ), Since the right-hand side of ( 172) is finite for every t ∈ [ε, T ], we can take the limit t ↑ t max to conclude that V (t max ) is positive definite, and hence t max = T .Since the function A → A −1 is operator decreasing on positive definite matrices, this proves (167) after some simple algebraic manipulations.Moreover, as a consequence of (172), the right-hand side of (167) is indeed positive definite.Similarly, integrating (170) from t to T and rearranging gives However, since V (t) −1 is positive definite for every t ∈ [ε, T ] this is equivalent to which concludes the proof of (166).
Proof of Theorem 25.Fix T > 0, ε > 0, and x ∈ M. Let f : M → R be a smooth positive function with M f dP T δ x = 1 and let µ be the probability measure on M with dµ dP T δ x = f .Without loss of generality, we can perturb f and assume that is a positive definite matrix.Following the terminology above, Lehec's formula (126) implies Since v(ε) = v ε is a positive definite matrix, Proposition 29 and Lemma 31 give where C(t) = e γt γ A + tB, for the matrices By the perturbation above, we have thus established the validity of (177) for an arbitrary smooth positive density f and for any ε > 0. Since v T = E µ ∇ log f ⊗2 , the logarithmic Sobolev inequality of Theorem 25 follows by combining (176) and (177) with ε → 0 + .The reverse logarithmic Sobolev inequality follows by using (167) since v 0 = ∇ log P T f (x) ⊗2 .9.1.Intrinsic dimensional local logarithmic Sobolev inequalities in flat spaces.Our next goal is to prove the intrinsic dimensional local logarithmic Sobolev inequalities in flat spaces, i.e., equations ( 29) and (30).In contrast to the proof of Theorem 25, which uses the weaker inequality (158), here we will use the stronger inequality (157) which in flat space has an explicit clear solution.
Theorem 32.Fix T > 0 and x ∈ R n .Let f : R n → R be a smooth positive function with R n f dP T δ x = 1 and let µ be the probability measure on R n with dµ dP T δ x = f .Then, we have the local intrinsic dimensional logarithmic Sobolev inequality and the reverse local intrinsic dimensional logarithmic Sobolev inequality Proof.The inequality (179) follows by setting dµ dλ n def = f P T δ x P T f (x) in (7).To prove (180), we may assume without loss of generality assume that −∇ 2 log P T f (x) is invertible.Set U (0) def = v(0) = (∇ log P T f (x)) ⊗2 and use the normalization assumption R n f dP T δ x = P T f (x) = 1 to conclude that is invertible.In flat space, using κ = 0 and The solution of (182) can be verified to be where we used Hamilton's matrix inequality (32) (see also Theorem 33 below) to justify the invertibility of [U (0) − P T ∇ 2 f (x)] −1 − t.Applying (163) of Theorem 30 yields where again we used normalization assumption P T f (x) = 1.To rewrite the right-hand side of (184) let {λ i } n i=1 stand for the eigenvalues of ∇ 2 log P T f (x) so (185) 10.1.Matrix inequalities.The main result of this section is the following Hamilton-type matrix inequality, namely, Theorem 5.
Theorem 33.Let (M, g) be an n-dimensional Riemannian manifold with constant nonpositive sectional curvature κ ≤ 0.Then, for every T ≥ 0, if, either κ = 0, or κ < 0 and Remark 34.To put Theorem 5 in a larger context, and also to shed light on the conditions regarding , let us recall the improved Li-Yau inequality of Bakry, Bolley, and Gentil [6, Corollaries 2.3,2.4]:Let (M, g) be an n-dimensional Riemannian manifold with lower bound (n−1)κ on its Ricci curvature.Let {P t } t≥0 be the associated heat semigroup and let f : M → R be a positive function.Then, for every x ∈ M and every T ≥ 0, and (n−1) 2 κ 2 T 2 we are able to obtain in hyperbolic spaces a matrix version of the improved Li-Yau inequality.
Proof of Theorem 33.We start by showing that satisfies the following differential inequality. where so, by (157), where the last inequality uses that κ ≤ 0 and that v(t) 0 (since it is a nonnegative sum of rank-one matrices).
The following technical lemma on matrix differential inequalities will allow us to further control the matrix m(t).
Lemma 36.Fix T > 0 and let W (t) be a family of matrices for t ∈ [0, T ] satisfying the differential inequality where with and with and Applying standard comparison theorems [49] we get that We are now ready for the proof of Theorem 33.Recall that Lemma 36 showed that the matrix m(t) satisfies (204) with α = nκ and β = κ∆P T f (x).In the following we let φ(t) def = m(t)θ, θ for θ ∈ S n−1 .Let us distinguish between the flat and negatively curved cases.
10.3.Discussion.We conclude this section by discussing the roles of matrix differential inequalities in our proofs.
Matrix differential inequalities.The master matrix differential inequality (157), which is at the core of all of our proofs, can be expressed either in terms of v(t), The inequalities (236) and ( 237) are equivalent and contain the same information.In particular, in flat space forms, where κ = 0, both inequalities are of the form dW (t) dt W (t) 2 .In curved spaces, there are two different ways to proceed from (236) and ( 237): (1) Omit the term u(t) 2  F(W (t)) unless θ i is an eigenvector of W (t).However, for the purpose of proving an inequality for the trace, there is no loss since the trace is invariant under rotations so for each t we can introduce a rotation R(t) which takes {θ i } to the eigenvectors of W (t) or U (t).

Theorem 4 .
Let (M, g) be an n-dimensional Riemannian manifold with constant sectional curvature κ ∈ R \ {0} with the associated heat semigroup {P t } t≥0 .Fix T > 0, x ∈ M, a smooth positive function f : M → R with M f dP T δ x = 1, and let µ be the probability measure with dµ dP T δ x = f .Define the 2-tensor C(t) = e nκt nκ A + tB for t ∈ R where A, B are the 2-tensors given by

Lemma 23 .
Let (M, g) be a complete n-dimensional Riemannian manifold and fix a smooth enough positive density function f : M → R + and T > 0. Denote by F t : M → R the function given by

Proof.
Since A and B commute, we have d dt e C(t) = e γt A + B e C(t) = e C(t) e γt A + B .(168)
The point of omitting u 2 (t) is that equation (239) can be solved explicitly, in contrast to the equation resulting if we keep the u 2 (t) term.1(2)Omit the term −κv(t) from (237), which can be done only in negatively curved space forms to get Again, the point of omitting −κv(t) is so that (241) can be solved explicitly.Note that in flat spaces, there is no loss in omitting −κv(t).Matrix vs. trace differential inequalities.The proofs of Theorem 40 and Theorem 25 proceed along similar but different lines.Both proofs start by establishing an inequality of the form The goal is to bound tr[W (t)] which can be achieve by two means.Letting {U (t)} be the solution to This is the method used to prove Theorem 25 and Theorem 32 (with different functionals F).(2)When F has scalar (rather than matrix) coefficients, it holds thatF(W (t))θ, θ ≥ F( W (t)θ, θ )(245)with strict inequality unless θ is an eigenvector of W (t). We can then {θ i } to be any basis and let φ i,W (t) := W (t)θ i , θ i , φ i,U (t) := U (t)θ i , θ i so 2 − u(t)U (t) − U (t)u(t) + (n − 1)κU (t).