A Lichnerowicz estimate for the spectral gap of a sub-Laplacian

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arXiv:1807.04453v1 [math.FA] 12 Jul 2018

Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below

Andrea Mondino

^∗

and Daniele Semola

^†

July 13, 2018

Abstract

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below byK > 0 and dimension bounded above byN ∈(1,∞) in a synthetic sense, the so called CD(K, N) spaces. We first establish a Polya-Szego type inequality stating that the W^1,p-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the p-Laplace operator with Dirichlet boundary conditions (on open subsets), for everyp∈(1,∞). This extends to the non-smooth setting a classical result of Bérard-Meyer [BM92] and Matei [Ma00]; remark- able examples of spaces fitting out framework and for which the results seem new include:

measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci≥K >0, finite dimensional Alexandrov spaces with curvature≥K >0, Finsler manifolds with Ricci≥K >0.

In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of RCD(K, N) spaces, which seem original even for smooth Riemannian manifolds with Ricci≥K >0.

1 Introduction

In 1884 Lord Rayleigh, in his book about the theory of sound [Ray], conjectured that, among all membranes of a given area, the disk has the lowest fundamental frequency of vibration. This was proven in 1920ies by Faber [Fa23] and Krahn [Kr25] for domains in the Euclidean plane and extended by Krahn [Kr26] to higher dimensions. The celebrated Rayleigh-Faber-Krahn inequality reads as follows.

Theorem 1.1 (Rayleigh-Faber-Krahn inequality [Fa23,Kr25,Kr26]). Let Ω⊂Rⁿ be a relatively compact open domain with smooth boundary. Then the first Dirichlet eigenvalue of Ω is bounded below by the first Dirichlet eigenvalue of a Euclidean ball having the same volume of Ω, moreover the inequality is rigid in the sense that equality is attained if and only ifΩis a ball.

The proof of the Rayleigh-Faber-Krahn inequality is based on two key facts: a variational characterisation for the first Dirichlet eigenvalue and the properties of symmetric decreasing rearrangements of functions. The variational characterisation of the first eigenvalue is given by

λ(Ω) := inf

u∈C_c¹(Ω)

R

Ω|∇u|²dx R

Ωu²dx . (1.1)

Let us now briefly recall few basics about decreasing rearrangements. Given an open subset Ω⊂Rⁿ, the symmetrized domain Ω^∗⊂Rⁿ is a ball with the same measure as Ω centred at the origin. Ifu is a real-valued Borel function defined on Ω, its spherical decreasing rearrangementu^∗is a function defined on the ball Ω^∗with the following properties: u^∗depends only on the distance from the origin, is decreasing along the radial direction and is equi-measurable withu(i.e. the super-level sets have the same volume: |{u > t}|=|{u^∗> t}|, for everyt∈R). Since the function and its spherical decreasing rearrangement are equi-measurable, theirL²-norms are the same. The key property that Faber and Krahn proved is that theL²-norm of the gradient of a function decreases under rearrangements. This last property was formalised, extended to everyL^p, 1< p <∞, and applied to several problems in mathematical physics by Polya and Szego in their book [PS51]; probably this is why it is now well known as the Polya-Szego inequality. The Polya-Szego inequality, combined with the variational characterization (1.1), immediately gives the Rayleigh-Faber-Krahn inequality.

Such a stream of ideas was extended in 1992 by Bérard-Meyer [BM92] to Riemannian manifolds (Mⁿ, g) with Ricg≥Kg,K >0. They proved the following result:

Theorem 1.2 (Bérard-Meyer [BM92]). Let (Mⁿ, g)be a Riemannian manifold with Ricg ≥Kg, K >0, and letΩ⊂M be an open subset with smooth boundary. LetSⁿ

K be the roundn-dimensional sphere of radiusp

(n−1)/K and letΩ^∗ ⊂Sⁿ

Kbe a metric ball having the same renormalized volume of Ω, i.e _|M^|Ω|_|= ^|Ω_|_Sn^∗^|

K|. Then λ(Ω)≥λ(Ω^∗)and equality is achieved if and only if M is isometric to Sⁿ

K andΩis a metric ball inSⁿ

K.

The two key ideas in [BM92] are the following. First, in the same spirit as above, for a function u∈ C_c¹(M) define a spherical decreasing rearrangementu^∗ on S; second, replace the Euclidean isoperimetric inequality by the Lévy-Gromov isoperimetric inequality [Gr07, Appendix C] in the proof of the corresponding Polya-Szego type inequality. Let us finally mention that, arguing along the same lines, the comparisonTheorem 1.2was generalized to the first Dirichlet eigenvalue of the p-Laplacian for anyp∈(1,∞) by Matei [Ma00].

The goal of the present paper is two-fold: first, we generalise the Polya-Szego and the Bérard- Meyer inequalities to non-smooth spaces with Ricci curvature bounded below in a synthetic sense;

second, we obtain a rigidity result for Polya-Szego inequality and an almost rigidity result for the Dirichletp-spectral gap which seem to be new even for smooth Riemannian manifolds.

1.1 Polya-Szego and p-spectral gap in CD(K, N ) spaces

In order to discuss the main results of the paper let us introduce some preliminaries about non- smooth spaces with Ricci curvature bounded below in a synthetic sense.

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A metric measure space (m.m.s. for short) is a triple (X,d,m) where (X,d) is a compact metric space endowed with a Borel probability measuremwith supp(m) =X, playing the role of reference volume measure. Using optimal-transport techniques, Lott-Villani [LV09] and Sturm [St06a,St06b]

introduced the so called curvature-dimension condition CD(K, N): the rough geometric picture is that a m.m.s. satisfying CD(K, N) should be thought of as a possibly non-smooth metric measure space with Ricci curvature bounded below byK∈Rand dimension bounded above byN ∈(1,∞) in a synthetic sense. The basic idea of this synthetic point of view is to consider weighted convex- ity properties of suitable entropy functionals along geodesics in the space of probability measures endowed with the quadratic transportation distance.

A first technical assumption throughout the paper is the so calledessentially non-branching property [RS14], which roughly amounts to require that the L²-optimal transport between two absolutely continuous (with respect to the reference measure m) probability measures moves along a family of geodesics with no intersections, i.e. a non-branching set of geodesics (for the precise definitions see Section2.1).

The class of essentially non-branching CD(K, N) spaces is very natural for extending the Polya- Szego/Bérard-Meyer results. Indeed a key ingredient for both is the isoperimetric inequality (via a coarea formula argument) and it was proved by Cavalletti with the first author [CM17a] that the Lévy-Gromov isoperimetric inequality extends to essentially non-branching CD(K, N) spaces (see Section2.2for the details).

Examples of essentially non-branching CD(K, N) spaces are Riemannian manifolds with Ricci curvature bounded below, finite dimensional Alexandrov spaces with curvature bounded below, Ricci limits and more generally RCD(K, N)-spaces, Finsler manifolds endowed with a strongly convex norm and with Ricci bounded below; let us stress that our results are new in all these celebrated classes of spaces (apart from smooth manifolds). A standard example of a space failing to satisfy the essential non-branching property isR² endowed with theL^∞norm.

In order to state the main theorems, let us introduce some notation about the model one- dimensional space and the corresponding monotone rearrangement.

For anyK >0 and 1< N <+∞we define the one dimensional model space (IK,N,d_eu,m_K,N) for the curvature dimension condition of parametersK andN by

IK,N := 0,

rN−1

K π

!

, m_K,N := 1 cK,N

sin

r K N−1t

!^N−1

L¹ IK,N, (1.2) whered_euis the restriction toIK,N of the canonical Euclidean distance over the real line,L¹is the standard Lebesgue measure, andcK,N :=R

I_K,Nsin p

Kt/(N−1)N−1

dL¹(t) is the normalizing constant.

We now introduce the corresponding monotone rearrangement. To this aim, given an open domain Ω ⊂ X and a non-negative Borel function u : Ω → [0,+∞) we define its distribution functionµ: [0,+∞)→[0,m(Ω)] by

µ(t) :=m({u > t}). (1.3)

It is not difficult to check that the distribution functionµis non increasing and left-continuous.

We will letu^#be the generalized inverse ofµ, defined in the following way:

u^#(s) :=

(ess supu ifs= 0, inf{t:µ(t)< s} ifs >0.

Definition 1.3(Rearrangement on one dimensional model spaces). Let (X,d,m) be a CD(K, N) space, for someK >0, 1< N <+∞, and let Ω⊂X be an open subset. Let (IK,N,d_eu,m_K,N) be the one-dimensional model space defined in (1.2) and consider [0, r]⊂IK,Nsuch thatm_K,N([0, r]) = m(Ω). For any Borel functionu: Ω→[0,+∞), the monotone rearrangement u^∗ : [0, r]→R⁺ is defined by

u^∗(x) :=u^#(m_K,N([0, x])), ∀x∈[0, r]. (1.4)

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For an arbitrary Borel functionu: Ω→(−∞+∞), letu^∗ be the monotone rearrangement of|u|.

Finally, we denote by W₀^1,p(Ω) the closure (with respect to the W^1,p-topology) of the set of Lipschitz functions compactly supported in Ω (see Section2 for more details).

We can now state the first main result of the paper.

Theorem 1.4(Polya-Szego inequality). Let (X,d,m)be an essentially non branching CD(K, N) space for someK >0,N ∈(1,+∞). LetΩ⊂X be an open subset and consider[0, r]⊂IK,N such that m_K,N([0, r]) =m(Ω).

Then the monotone rearrangement maps W₀^1,p(Ω) into W^1,p(([0, r],d_eu,m_K,N)) for any 1 < p <

+∞. Moreover for any u∈W₀^1,p(Ω) it holdsu^∗(r) = 0 and Z

Ω

|∇u|^p_wdm≥ Z r

0

|∇u^∗|^pdm_K,N. (1.5)

Theorem 1.4will be proved in Section3. The two main ingredients in the proof are the coarea formula and the Lévy-Gromov isoperimetric inequality, though the full argument requires some work and several intermediate results.

The second main result is a spectral gap for thep-Laplacian with Dirichlet boundary conditions, in the spirit of Berard-Meyer-MateiTheorem 1.2. In order to state it we need to introduce some more notation.

For everyv∈(0,1), let r(v)∈IK,N be such thatv=m_K,N([0, r(v)]). For any fixed 1< p <+∞, for anyv∈(0,1) and for any choice ofK >0 and 1< N <+∞, define

λ^p_K,N,v:= inf ( Rr(v)

0 |u^′|^pdm_K,N Rr(v)

0 u^pdm_K,N

: u∈LIP([0, r(v)]; [0,+∞)), u(r(v)) = 0 andu6≡0 )

.

For any metric measure space (X,d,m) with m(X) = 1, for any open subset Ω⊂X and for any 1< p <+∞, define

λ^p_X(Ω) := inf R

Ω|∇u|^pdm R

Ωu^pdm : u∈LIPc(Ω; [0,+∞)) andu6≡0

.

Observe that for any 2 ≤ N ∈ N and K > 0, λ^p_K,N,v = λ^p_SN K

(Bv), where S^N_K is the round N- dimensional sphere of radiusq

N−1

K andBv⊂S^N_K is a metric ball of volumev.

We can now state our second main result.

Theorem 1.5(p-Spectral gap with Dirichlet boundary conditions). Let(X,d,m)be an essentially non branchingCD(K, N)space for someK >0,1< N <+∞, and letΩ⊂X be an open domain withm(Ω) =v∈(0,1). Then it holds

λ^p_X(Ω)≥λ^p_K,N,v for any1< p <+∞.

The spectral gap in CD(K, N) spaces forNeumann boundary conditions, called Lichnerowicz inequality, was established by Lott-Villani [LV07] in casep= 2 (see also [EKS15] and [JZ16] for related results in RCD(K, N) spaces) and by Cavalletti with the first author [CM17b] for general p ∈ (1,∞). Let us stress that the techniques used in the aforementioned papers to establish Neumannspectral gaps seem not suitable for provingDirichletspectral gaps. This was indeed one of the motivations to write the present paper.

1.2 Rigidity and almost rigidity in RCD(K, N ) spaces

In order to discuss the rigidity statements associated toTheorem 1.4andTheorem 1.5let us recall the “Riemannian” refinement of the CD condition, called RCD. Introduced by Ambrosio-Gigli- Savaré [AGS14b] in case N = ∞ (see also [AGMR15]), the RCD condition is a strengthening

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of the CD condition by the requirement that the Sobolev space W^1,2((X,d,m)) is Hilbert (or, equivalently, the heat flow, or equivalently the laplacian, is linear). The main motivation is that the CD condition allows Finsler structures while the RCD condition isolates the “Riemannian” spaces.

A key property of the RCD condition is that, as well as CD, is stable under measured Gromov- Haudorff convergence [AGS14b, GMS15]. The finite dimensional refinement was subsequently proposed in [G15a] and extensively investigated in [EKS15,AMS15]. We refer to these papers and references therein for a general account on the synthetic formulation of the latter Riemannian-type Ricci curvature lower bounds; for a survey of results, see the Bourbaki seminar [V18] and the recent ICM-Proceeding [Am18].

We can now state the rigidity result associated to the Polya-Szego inequalityTheorem 1.4. In order to simplify the notation we will considerK=N−1, the case of a generalK >0 follows by a scaling argument (recall that (X,d,m) is an RCD(K, N) space for someK >0 and 1< N <+∞

if and only if the rescaled space (X,d^′,m), whered^′ :=q

N−1

K d, is an RCD(N−1, N) space).

Theorem 1.6(Rigidity in the Polya-Szego inequality). Let (X,d,m)be anRCD(N−1, N)space for someN ∈[2,+∞)with m(X) = 1 and letΩ⊂X be an open set such thatm(Ω) =v∈(0,1).

Assume that for some p∈(1,∞)there existsu∈W₀^1,p(Ω),u6≡0, achieving equality in the Polya- Szego inequality (1.5).

Then(X,d,m)is a spherical suspension, namely there exists anRCD(N−2, N−1)space(Y,d_Y,m_Y) withm_Y(Y) = 1such that (X,d,m)is isomorphic as a metric measure space to [0, π]×^N_sin⁻¹Y. If moreover the function uachieving equality in the Polya-Szego inequality (3.19)is Lipschitz and

|∇u|(x)6= 0 for m-a.e. x∈supp(u), then uis radial; i.e. u=f(d(·, x0)), where x0 is a tip of a spherical suspension structure ofX andf : [0, π]→Rsatisfies|f|=u^∗.

When specialized to the smooth setting, the last result reads as follows.

Corollary 1.7 (Rigidity in the Polya-Szego inequality-Smooth Setting). Let (M, g) be an N- dimensional Riemannian manifold, N ≥2, withRicg≥(N−1)g and denote bym the normalized Riemannian volume measure. LetΩ⊂X be an open subset withm(Ω)∈(0,1).

Assume that for some p∈(1,∞)there existsu∈W₀^1,p(Ω),u6≡0, achieving equality in the Polya- Szego inequality (1.5).

Then (M, g)is isometric to the round sphere S^N of constant sectional curvature one.

If moreover the function uachieving equality in the Polya-Szego inequality (3.19)is Lipschitz and

|∇u|(x)6= 0 for a.e. x∈ supp(u), then uis radial; i.e. u=f(d(·, x0)), for some x0 ∈S^N and f : [0, π]→Rsatisfying|f|=u^∗.

Let us mention that our proof of bothTheorem 1.6andCorollary 1.7builds on top of the almost rigidity in Lévy-Gromov inequality [CM17a] and seems new even in the smooth setting. The rough idea is that if the spaceX is not a spherical suspension then by the almost rigidity in Lévy-Gromov inequality, there is a gap in the isoperimetric profile of X and the model isoperimetric profile IN−1,N. Thus it is not possible to achieve almost equality in the Polya-Szego inequality for suitable approximationsun ∈LIPc(Ω) of uwith |∇un|(x)6= 0 m-a.e. x∈supp(un), hence contradicting that u ∈ W₀^1,p(Ω) achieves equality in Polya-Szego inequality. The rigidity statement in the function is more subtle and basically consists in proving that the structure of spherical suspension induced by the optimality in Lévy-Gromov by every super-level set{u > t}is independent oft.

Remark 1.8. A natural question aboutTheorem 1.6regards sharpness of the assumptions. Clearly, ifu≡0 also the decreasing rearrangementu^∗ vanishes; henceu, u^∗achieve equality in the Polya- Szego inequality but one cannot expect to infer anything on the space.

Let us also stress that the condition|∇u| 6= 0m-a.e. is necessary to infer thatu(·) =u^∗◦d(x0,·), even knowing a priori that the space is a spherical suspension with pole x0 and that u achieves equality in Polya-Szego inequality. Indeed letX=S^N be the round sphere, fix pointsx16=x2∈S^N and radii 0 < r1 < r2 < r3 such that Br1(x1) ⊂ Br2(x2) ⊂ Br3(x2). Consider a function u:S^N →[0,1] which is radially decreasing onBr1(x1) (with respect to the pole x1), constant on Br2(x2)\Br1(x1) and radially decreasing onBr3(x2)\Br2(x2) (w.r.t. the polex2). It is easy to check that such a function uachieves equality in the Polya-Szego inequality but is not globally radial.

Our second rigidity result concerns the Dirichletp-spectral gap.

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Theorem 1.9(Rigidity for the Dirichletp-spectral gap). Let(X,d,m)be anRCD(N−1, N)space.

LetΩ⊂Xbe an open subset withm(Ω) =vfor somev∈(0,1)and suppose thatλ^p_X(Ω) =λ^p_N_−1,N,v. Then

1. (X,d,m) is isomorphic to a spherical suspension: i.e. there exists an RCD(N −2, N−1) space(Y,d_Y,m_Y)such thatX ≃[0, π]×^N_sin⁻¹Y;

2. the topological closure Ω¯ of Ω ⊂ X coincides with the closed metric ball centred at one of the tips of the spherical suspension: i.e. either Ω = [0, R]¯ ×Y orΩ = [π¯ −R, π]×Y, where R∈(0, π)is such thatm_N_−1,N([0, R]) =v;

3. the eigenfunction u ∈ W₀^1,p(Ω) associated to λ^p_X(Ω) is unique up to a scalar factor and it coincides with the radial one: i.e. calledx0 the centre ofΩandw: [0, R]→[0,+∞)the first eigenfunction on ([0, R],d_eu,m_N−1,N) corresponding to λ^p_N−1,N,v (i.e. with the constraint w(R) = 0), it holds thatu(·) =w◦d(x0,·).

The proof ofTheorem 1.9builds on top of the rigidity in the Lévy-Gromov inequality proved in [CM17a]; indeed the rough idea to establish the first and second assertions is to prove that if λ^p_X(Ω) =λ^p_N−1,N,v, then the super-level sets of the firstp-eigenfunction are optimal in the Lévy- Gromov inequality. The proof of the third assertion requires more work. The rough idea is to show that the first Dirichletp-eigenfunction is unique thus, knowing already that Ω is almost a ball centred at a tip of the spherical suspension and hence there is already a naturalradial first Dirichletp-eigenfunction suggested by the model space, it follows that umust by radial. In the proof of the uniqueness of the first Dirichletp-eigenfunction we have been inspired by a paper of Kawhol-Lindqvist [KL06] dealing with smooth Riemannian manifolds and, in order to implement the arguments in non-smooth setting, we make use of the theory of tangent modules of m.m.s.

developed by Gigli [G18].

Let us also mention that the rigidity for theNeumannspectral gap, known as Obata Theorem, was established in case p= 2 by Ketterer [K15] and by Cavalletti with the first author [CM17b]

for generalp∈(1,∞).

We conclude the introduction with an almost-rigidity result which seems new even in the smooth framework, i.e. if (X,d,m) is anN-dimensional Riemannian manifold with Ricci curvature bounded below byN−1. A key point in the proof is that the class of RCD(N−1, N) spaces is compact with respect to mGH convergence, fact which clearly fails in the smooth setting as the limits usually present singularities.

We denote byd_mGH the measured Gromov Hausdorff distance between two normalized compact metric measure spaces.

Theorem 1.10 (Almost rigidity in thep-spectral gap). Fix 2 ≤N <+∞and v ∈(0,1). Then, for any ǫ > 0, there exists δ = δ(v, N) > 0 with the following property: let (X,d,m) be an RCD(N −1, N) m.m.s. with m(X) = 1 and Ω ⊂ X be an open domain with m(Ω) = v and λ^p_X(Ω)< λ^p_N−1,N,v+δ.

Then there exists a spherical suspension (Y,d_Y,m_Y) (i.e. there exists an RCD(N −2, N − 1) space (Z,d_Z,m_Z) with m_Z(Y) = 1 such that Y is isomorphic as a metric measure space to [0, π]×^N_sin⁻¹Z) such that

d_mGH((X,d,m),(Y,d_Y,m_Y))< ǫ.

Acknowledgement: Part of the work was developed while D.S. was visiting the Mathematics Institute at the University of Warwick. He would like to thank the institute for the excellent working conditions and stimulating environment.

A.M. is partly supported by the EPSRC First Grant EP/R004730/1 “Optimal transport and geometric analysis”.

The authors wish to thank L. Ambrosio for inspiring discussions around the topics of the paper.

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2 Preliminaries

Throughout the paper (X,d,m) will be a complete and separable metric measure space with supp(m) =X and m(X)<∞. We will denote byB(X) the family of Borel subsets ofX and by LIP(X) the space of real valued Lipschitz functions overX. For any open domain Ω⊂X, LIPc(Ω) and LIPloc(Ω) will stand for the space of Lipschitz functions with compact support in Ω and the space of locally Lipschitz functions in Ω. Givenu∈LIPloc(X), its slope|∇u|(x) is defined as

|∇u|(x) :=

(lim sup_y→x|u(x)−u(y)|

d(x,y) ifxis not isolated

0 otherwise,

moreover we introduce the notation Lip(u) for the global Lipschitz constant ofu∈LIP(X).

For any interval I ⊂ R we will denote by AC(I;X) the space of absolutely continuous curves γ:I→X. For anyγ∈AC(I;X), the metric derivative|γ^′|:I→[0,+∞] defined by

|γ^′|(t) := lim sup

s→t

d(γ(s), γ(t))

|t−s| , provides the following representation of the the length ofγ:

l(γ) = Z

I

|γ^′|(t) dt.

Next we introduce Sobolev functions and Sobolev spaces over (X,d,m). We refer for instance to [AGS14a,ACDM15] for a detailed discussion about this topic.

Definition 2.1 (Sobolev spaces and p-energy). Fix any 1 < p < +∞. The p-Cheeger energy Ch_p:L^p(X,m)→[0,+∞] is a convexL^p(X,m)-lower semicontinuous functional defined by

Ch_p(f) := inf

lim inf

n→∞

1 p

Z

|∇fn|^pdm:fn∈LIP(X)∩L^p(X,m),kfn−fk_Lp →0

. (2.1) Moreover, we defineW^1,p(X,d,m) :={Ch_p<+∞ }and we remark that, when endowed with the norm

kfk_W1,p := (kfk^p_Lp+pCh_p(f))

1 p, the Sobolev spaceW^1,p(X,d,m) is a Banach space.

By looking at the optimal approximating sequence in (2.1) one can find a minimal object called minimal weak upper gradient, providing the integral representation

Ch_p(f) =1 p

Z

|∇f|^p_wdm

for anyf ∈W^1,p(X,d,m). We remark that without further regularity assumptions on the metric measure space the minimal weak upper gradient depends also on the integrability exponent p;

nevertheless we will always omit this dependence in the notation.

Definition 2.2 (Local Sobolev spaces). Given an open set Ω⊂X, for any 1< p <+∞ we will denote byW₀^1,p(Ω) the closure of LIPc(Ω) inW^1,p(X,d,m), with respect to theW^1,pnorm.

2.1 Essentially non branching, CD(K, N ) and RCD(K, N ) metric measure spaces

Denote by

Geo(X) :={γ∈C([0,1], X) :d(γ(s), γ(t)) =|s−t|d(γ(0), γ(1)), for everys, t∈[0,1]}

the space of constant speed geodesics. The metric space (X,d) is ageodesic spaceif and only if for eachx, y∈X there exists γ∈Geo(X) so thatγ(0) =x, γ(1) =y.

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We denote withP(X) the space of all Borel probability measures overX and withP₂(X) the space of probability measures with finite second moment. The spaceP₂(X) can be endowed with theL²-Kantorovich-Wasserstein distanceW2 defined as follows: forµ0, µ1∈P₂(X), set

W₂²(µ0, µ1) := inf

π

Z

X×X

d²(x, y) dπ(x, y), (2.2)

where the infimum is taken over all π ∈P(X×X) with µ0 and µ1 as the first and the second marginal. The space (X,d) is geodesic if and only if the space (P₂(X), W2) is geodesic.

For anyt∈[0,1], let etbe the evaluation map:

et: Geo(X)→X, et(γ) :=γt.

Any geodesic (µt)t∈[0,1] in (P₂(X), W2) can be lifted to a measure ν ∈ P(Geo(X)), so that (et)♯ν =µtfor allt∈[0,1].

Givenµ0, µ1 ∈P₂(X), we denote by OptGeo(µ0, µ1) the space of all ν ∈P(Geo(X)) for which (e0,e1)♯ν realizes the minimum in (2.2). Such aν will be calleddynamical optimal plan. If (X,d)

is geodesic, then the set OptGeo(µ0, µ1) is non-empty for anyµ0, µ1∈P₂(X).

We will also consider the subspaceP₂(X,d,m)⊂P₂(X) formed by all those measures absolutely continuous with respect withm.

A setG⊂Geo(X) isa set of non-branching geodesicsif and only if for anyγ1, γ2∈G, it holds:

∃t¯∈(0,1) such that ∀t∈[0,¯t] γ1(t) =γ2(t) =⇒ γ1(s) =γ2(s), ∀s∈[0,1].

In the paper we will mostly consider essentially non-branching spaces, let us recall their definition (introduced by T. Rajala and Sturm [RS14]).

Definition 2.3. A metric measure space (X,d,m) is essentially non-branching (e.n.b. for short) if and only if for anyµ0, µ1 ∈ P₂(X), with µ0, µ1 absolutely continuous with respect tom, any element of OptGeo(µ0, µ1) is concentrated on a set of non-branching geodesics.

It is clear that if (X,d) is a smooth Riemannian manifold then any subset G ⊂ Geo(X) is a set of non branching geodesics, in particular any smooth Riemannian manifold is essentially non-branching.

In order to formulate curvature properties for (X,d,m) we recall the definition of the distortion coefficients: forK∈R, N∈[1,∞), θ∈(0,∞), t∈[0,1], set

τ_K,N^(t) (θ) :=t^1/Nσ_K,N^(t) ₋₁(θ)^(N^−1)/N, (2.3) where the σ-coefficients are defined as follows: given two numbersK, N ∈RwithN ≥0, we set for (t, θ)∈[0,1]×R₊,

σ^(t)_K,N(θ) :=











∞, ifKθ²≥N π², sin(tθp

K/N) sin(θp

K/N) if 0< Kθ²< N π²,

t ifKθ²<0 andN = 0, or ifKθ²= 0, sinh(tθp

−K/N) sinh(θp

−K/N) ifKθ²≤0 andN >0.

(2.4)

Let us also recall the definition of the Rényi Entropy functionalEN :P(X)→[0,∞]:

EN(µ) :=

Z

X

ρ^1−1/N(x) dm, (2.5)

whereµ=ρm+µ^swithµ^s⊥m.

The curvature-dimension condition was introduced independently by Lott-Villani [LV09] and Sturm [St06a,St06b], let us recall its definition.

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Definition 2.4 (CD condition). Let K ∈ Rand N ∈ [1,∞). A metric measure space (X,d,m) verifies CD(K, N) if for any two µ0, µ1 ∈ P₂(X,d,m) with bounded support there exist ν ∈ OptGeo(µ0, µ1) and π ∈ P(X ×X) W2-optimal plan such that µt := (et)♯ν ≪ m and for any N^′ ≥N, t∈[0,1]:

EN^′(µt)≥ Z

τ_K,N^(1−t)′(d(x, y))ρ^−1/N₀ ^′+τ_K,N^(t) ′(d(x, y))ρ^−1/N₁ ^′ dπ(x, y). (2.6) It is worth recalling that if (M, g) is a Riemannian manifold of dimension nand h∈ C²(M) withh > 0, then the m.m.s. (M,d_g, hVolg) (where d_g and Volg denote the Riemannian distance and volume induced byg) verifies CD(K, N) withN ≥nif and only if (see [St06b, Theorem 1.7])

Ricg,h,N ≥Kg, Ricg,h,N:= Ricg−(N−n)∇²_gh^N−n¹ h^N−n¹ .

In particular if N = n the generalized Ricci tensor Ricg,h,N = Ricg makes sense only if h is constant.

The lack of the local-to-global property of the CD(K, N) condition (for K/N6= 0) led in 2010 Bacher and Sturm to introduce in [BS10] the reduced curvature-dimension condition, denoted by CD^∗(K, N). The CD^∗(K, N) condition asks for the same inequality (2.6) of CD(K, N) to hold but the coefficientsτ_K,N^(s) (d(γ0, γ1)) are replaced by the slightly smallerσ^(s)_K,N(d(γ0, γ1)).

Since the CD condition allows Finsler geometries, in order to single out the “Riemannian”

structures Ambrosio-Gigli-Savaré [AGS14b] introduced the Riemannian curvature dimension condition RCD(K,∞) (see also [AGMR15] for the extension to σ-finite measures and for the present simplification in the axiomatization). The natural finite dimensional refinement RCD^∗(K, N) with N <∞has been subsequently proposed in [G15a] and extensively investigated in [EKS15,AMS15].

We refer to these papers and references therein for a general account on the synthetic formulation of the latter Riemannian-type Ricci curvature lower bounds; for a survey of results, see the Bour- baki seminar [V18] and the recent ICM-Proceeding [Am18]. Here we only briefly recall that it is a stable [AGS14b,GMS15] strengthening of the reduced curvature-dimension condition: a m.m.s.

verifies RCD^∗(K, N) if and only if it satisfies CD^∗(K, N) and is infinitesimally Hilbertian, meaning that the Sobolev spaceW^1,2(X,m) is a Hilbert space (with the Hilbert structure induced by the Cheeger energy).

To conclude we recall also that recently Cavalletti and E. Milman [CaMi16] proved the equivalence of CD(K, N) and CD^∗(K, N), together with the local-to-global property for CD(K, N), in the framework of essentially non-branching m.m.s. havingm(X)<∞. As we will always assume the aforementioned properties to be satisfied by our ambient m.m.s. (X,d,m), we will use both for- mulations with no distinction. It is worth also mentioning that a m.m.s. verifying RCD^∗(K, N) is essentially non-branching (see [RS14, Corollary 1.2]) implying also the equivalence of RCD^∗(K, N) and RCD(K, N).

For all the main results we will assume that the m.m.s. (X,d,m) is essentially non-branching and satisfies CD(K, N) from some K > 0 with supp(m) =X (or, more strongly, that (X,d,m) is a RCD(N −1, N) space). It follows that (X,d) is a geodesic and compact metric space with m(X)∈(0,∞). Since (X,d,m) is a CD(K, N) (resp. RCD(K, N)) space if and only if (X,d, αm) is so, without loss of generality we will also assumem(X) = 1.

2.2 Finite perimeter sets and Lévy-Gromov isoperimetric inequality

We now recall the definition of a finite perimeter set in a metric measure space (see [Am02,Mi03]

and the more recent [ADM14]).

Definition 2.5 (Perimeter and sets of finite perimeter). Given a Borel setE ⊂X and an open setAthe perimeter Per(E, A) is defined in the following way:

Per(E, A) := inf

lim inf

n→∞

Z

A

|∇un|dm:un∈LIPloc(A), un→χA in L¹_loc(A)

.

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We say thatE has finite perimeter if Per(E, X)<+∞. In that case it can be proved that the set functionA7→Per(E, A) is the restriction to open sets of a finite Borel measure Per(E,·) defined by

Per(E, B) := inf{Per(E, A) :B⊂A, Aopen}.

Below we recall the definition of the family of one dimensional model spaces for the curvature dimension condition of parametersK >0 and 1< N <+∞(cf. [Gr07, Appendix C] and [M15]).

Definition 2.6 (One dimensional model spaces). For anyK >0 and for any 1< N < +∞we define the one dimensional model space (IK,N,d_eu,m_K,N) for the curvature dimension condition of parametersK andN by

IK,N := 0,

rN−1

K π

!

, m_K,N := 1 cK,N

sin

r K N−1t

!N−1

L¹ IK,N, (2.7) wheredis the restriction toIK,N of the canonical Euclidean distance over the real line andcK,N :=

R

IK,Nsin q

K N−1tN−1

dL¹(t) is the normalizing constant.

In order to shorten the notation, we sethK,N(t) :=_c_K,N¹ sinq

K N−1t^N⁻¹

for allt∈IK,N. Let us recall that, for any normalized metric measure space (X,d,m), the isoperimetric profile I(X,d,m): [0,1]→[0,+∞) is defined by

I(X,d,m)(v) := inf{Per(E) : E∈B(X), m(E) =v}.

We will denote byIK,N the isoperimetric profile of the model space (IK,N,d_eu,m_K,N).

In [CM17a, CM18], exploiting the so-called localization technique (cf. [Kl17]), the following version of the Lévy-Gromov isoperimetric inequality [Gr07, Appendix C] for metric measure spaces was proven.

Theorem 2.7(Lévy-Gromov inequality). Let(X,d,m)be an essentially non branchingCD(K, N) metric measure space for someK >0and1< N <+∞. Then, for any Borel setE⊂X, it holds

Per(E)≥ IK,N(m(E)).

We next recall the notion of warped product between metric measure spaces, generalizing the well know Riemannian construction.

Given two geodesic metric measure spaces (B,d_B,m_B) and (F,d_F,m_F) and a Lipschitz function f : B → [0,+∞) one can define a length structure on the product B ×F as follows: for any absolutely continuous curveγ: [0,1]→B×F with components (α, β), define

L(γ) :=

Z 1 0

|α^′|²(t) + (f◦α(t))²|β^′|²(t)

1 2dt and consider the associated pseudo-distance

d((p, x),(q, y)) := inf{L(γ) :γ(0) = (p, x), γ(1) = (q, y)}. Thef-warped product ofB withF is the metric space defined by

B×fF := (B×F/∼,d),

where (p, x)∼(q, y) if and only if d((p, x),(q, y)) = 0. One can also associate a natural measure and obtain

B×^N_f F := (B×fF,m_C), m_C:=f^Nm_B⊗m_F,

that we will call warped product metric measure space of (B,d_B,m_B) and (F,d_F,m_F).

In [CM17a,CM18] also the rigidity problem for the Lévy-Gromov inequality was addressed in the framework of metric measure spaces. Before stating the result from [CM17a, CM18], observe that if (X,d,m) is an RCD(K, N) metric measure space for someK >0 and 1< N <+∞then the rescaled space (X,d^′,m), whered^′:=q

N−1

K d, is an RCD(N−1, N) space.

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Theorem 2.8 (Rigidity in Lévy-Gromov inequality). Let (X,d,m)be an RCD(N−1, N)metric measure space for some N ∈ [2,+∞) with m(X) = 1. Assume that there exists v¯ ∈ (0,1) such that I(X,d,m)(¯v) =IN−1,N(¯v). Then(X,d,m)is a spherical suspension: there exists anRCD(N− 2, N−1)m.m.s. (Y,d_Y,m_y)withm_Y(Y) = 1such that X is isomorphic as a metric measure space to[0, π]×^N_sin⁻¹Y. Moreover in this case the following hold:

(i) for anyv∈[0,1]we haveI(X,d,m)(v) =IN−1,N(v);

(ii) for anyv∈[0,1]there exists a Borel setA⊂X with m(A) =v and such that Per(A) =I(X,d,m)(v) =IN−1,N(v);

(iii) for any Borel setA⊂X such that m(A) =v, we havePer(A) =IN−1,N(v)if and only if m(A\ {(t, y)∈[0, π]×Y :t∈[0, r(v)]}) = 0

or

m(A\ {(t, y)∈[0, π]×Y :t∈[π−r(v), π]}) = 0, wherer(v)∈(0, π)is such that cNRr(v)

0 sin^N⁻¹(t) dt=v (with cN normalization constant).

2.3 BV functions and coarea formula in m.m.s.

As for the classical Euclidean case, in metric measure spaces one can introduce not only a notion of finite perimeter set but also a notion of function of bounded variation. We refer again to [Mi03]

and [ADM14] for more details about the topic.

Definition 2.9. A function f ∈ L¹(X,m) is said to belong to the space BV∗(X,d,m) if there exists a sequence of locally Lipschitz functionsfi converging tof inL¹(X,m) such that

lim sup

i→∞

Z

X

|∇fi|dm<+∞.

By localizing this construction one can define

|Df|_∗(A) := inf

lim inf

i→∞

Z

A

|∇fi|dm:fi∈LIPloc(A), fi →f inL¹(A)

for any openA⊂X. In [ADM14] it is proven that this set function is the restriction to open sets of a finite Borel measure that we calltotal variation of f.

For any Lipschitz functionf :X →Rit is easy to check that f ∈BV∗(X,d,m) and |Df|_∗ ≤

|∇f|m. In the following we will denote by |∇f|₁ the density of |Df|_∗ with respect to m. With a slight abuse of notation motivated by simplicity, we are going to use the same symbol|∇f|₁ to denote the equivalence class (underm-a.e. equality) and a Borel representative.

The following result is a simplified version of [APS15, Proposition 4.2].

Proposition 2.10. Let f ∈LIP(X). Then |∇f|₁(x) = 0for m-a.e. x∈ {f = 0}.

The identification result stated below is a consequence of the seminal work [Ch99] concerning Lipschitz functions on metric measure spaces satisfying doubling and Poincaré inequalities and of the identification result for p-minimal relaxed upper gradients obtained in [GH14] for proper RCD(K,∞) spaces.

Proposition 2.11. Let (X,d,m)be anRCD(K, N)space, for someK∈R, N∈(1,∞). Then for anyf ∈LIP(X)one has that|Df|_∗=|∇f|m.

The following coarea formula for functions of bounded variation on metric measure spaces is taken from [Mi03, Remark 4.3]. It will play a key role in the rest of the paper.

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Theorem 2.12(Coarea formula). LetΩ⊂X be an open domain in a m.m.s. (X,d,m).

Letv: Ω→[0,+∞)belong toBV∗(Ω,d,m). Then for any Borel functionf : Ω→[0,+∞)it holds Z

{s≤v<t}

fd|Dv|_∗= Z t

s

Z

fd Per({v > r})

dr, ∀s∈[0, t]. (2.8) CombiningProposition 2.11andTheorem 2.12we obtain the following.

Corollary 2.13. Let (X,d,m)be an RCD(K, N)space, for some K∈R, N∈(1,∞).

Let Ω⊂X be an open domain and v : Ω →[0,+∞) be Lipschitz. Then, for any Borel function f : Ω→[0,+∞)it holds that

Z

{s≤v<t}

f|∇v|dm= Z t

s

Z

fd Per({v > r})

dr, ∀s∈[0, t]. (2.9) The following result will be useful when dealing with the almost rigidity case in the spectral gap inequality.

Proposition 2.14. FixK >0andN ∈(1,∞). Let((Xn,d_n,m_n))_n∈Nbe a sequence of normalized RCD(K, N) spaces converging to(X,d,m)in the measured Gromov-Hausdorff sense.

Denote by In (resp. I) the isoperimetric profile of(Xn,d_n,m_n) (resp. of(X,d,m)).

Then, for anyt∈[0,1]and for any sequence(tn)n∈N withtn→t, it holds that I(t)≤lim inf

n→∞ In(tn). (2.10)

Proof. We refer to [GMS15,AH16] for the basic definitions and statements about convergence of functions defined over mGH-converging sequences of metric measure spaces.

First of all note that in order to prove (2.10), without loss of generality we can assume that sup_n∈NIn(tn)<+∞.

For anyn∈NletEn ⊂Xn be a Borel set such that Pern(En) =In(tn), whose existence follows as in the Euclidean case from standard lower semicontinuity and compactness arguments.

The sequence of the corresponding characteristic functions (χEn)n∈N satisfies the assumption of [AH16, Proposition 7.5], i.e.

sup

n∈N

nkχE_nk_L¹₍m_n)+|DχE_n|(Xn)o

= sup

n∈N

{tn+In(tn)}<+∞.

It follows from [AH16, Proposition 7.5] that, up to extracting a subsequence which we do not relabel, (χE_n)_n∈N strongly L¹-converges to a functionf ∈ L¹(X,m) (see [AH16, Section 3]). In particular we can say that

kfk_L¹₍m)= lim

n→∞kχEnk_L¹₍m_n)= lim

n→∞tn=t. (2.11)

We now claim thatf is the indicator function of a Borel setE⊂X, withm(E) =t. To this aim callgn :=χE_n(1−χE_n) and observe that (gn)n∈Nstrongly L¹-converges tog:=f(1−f) thanks to [AH16, Proposition 3.3]. Thusg= 0, sincegn = 0 for anyn∈Nand thereforegis the indicator function of a Borel set, as claimed.

We can now apply [AH16, Theorem 8.1] to get the Mosco convergence of the BV energies and conclude that

Per(E)≤lim inf

n→∞ Pern(En) = lim inf

n→∞ In(tn).

The lower semicontinuity for the isoperimetric profiles (2.10) easily follows, sinceEis an admissible competitor in the definition ofI(t).

3 Polya-Szego inequality

The working assumption of this section, unless otherwise stated, is that (X,d,m) is an essentially non branching CD(K, N) space for someK >0,N∈(1,+∞), withm(X) = 1 and supp(m) =X.

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Definition 3.1 (Distribution function). Given an open domain Ω⊂X and a non-negative Borel functionu: Ω→[0,+∞) we define its distribution functionµ: [0,+∞)→[0,m(Ω)] by

µ(t) :=m({u > t}). (3.1)

Remark 3.2. Suppose thatuis such thatm({u=t}) = 0 for any 0< t <+∞. Then it makes no difference to consider closed superlevel sets or open superlevel sets in (3.1).

It is not difficult to check that the distribution functionµis non increasing and left-continuous.

If moreoveruis continuous, thenµis strictly decreasing. We let u^# be the generalized inverse of µ, defined in the following way:

u^#(s) :=

(ess supu ifs= 0, inf{t:µ(t)< s} ifs >0.

Definition 3.3(Rearrangement on one dimensional model spaces). Fix anyK >0, 1< N <+∞, and let (IK,N,d,m_K,N) be the one-dimensional model space defined in (2.7). Let Ω⊂ X be an open subset and consider [0, r]⊂IK,N such thatm_K,N([0, r]) =m(Ω).

For any Borel functionu: Ω→[0,+∞), themonotone rearrangementu^∗_K,N : [0, r]→R⁺is defined by

u^∗_K,N(x) :=u^#(m_K,N([0, x])), ∀x∈[0, r].

For simplicity of notation we will often writeu^∗ in place ofu^∗_K,N.

Remark3.4. For simplicity of notation, throughout the paper we will consider monotone rearrangements ofnon-negative functions. Nevertheless, for an arbitrary Borel functionu: Ω→(−∞+∞) the analogous statements hold by settingu^∗ the monotone rearrangement of|u|.

In the next proposition we collect some useful properties of the monotone rearrangement, whose proof in the Euclidean setting can be found for instance in [K06, Chapter 1] and can be adapted with minor modifications to our framework.

Proposition 3.5. Let (X,d,m) with m(X) = 1 be an essentially non branching CD(K, N) space for someK >0,N ∈(1,+∞). Let Ω⊂X be an open subset and consider [0, r]⊂IK,N such that m_K,N([0, r]) =m(Ω). Let u: Ω→[0,+∞) be Borel and letu^∗: [0, r]→[0,+∞)be its monotone rearrangement.

Thenuandu^∗ have the same distribution function (we will often say that they are equimeasurable).

Moreover,

kuk_Lp(Ω,m)=ku^∗k_Lp([0,r],m_K,N), ∀1≤p <+∞, (3.2) and the monotone rearrangement operator L^p(Ω,m)∋u7→u^∗∈L^p([0, r],m_K,N)is continuous.

Motivated by the working assumptions ofLemma 3.10, we state and prove the following general result about approximation via functions with non vanishing minimal relaxed upper gradient.

Lemma 3.6 (Approximation with non vanishing gradients). Let (X,d,m) be a locally compact geodesic metric measure space and letΩ⊂X be an open subset withm(Ω)<+∞.

Then for any non-negativeu∈LIPc(Ω)withR

|∇u|^pdm<∞, there exists a sequence(un)n∈Nwith un ∈LIPc(Ω) non-negative,|∇un| 6= 0 m-a.e. on{un>0} for any n∈Nand such thatun→u inL^p(X,m),R

|∇un|^pdm→R

|∇u|^pdm asn→ ∞.

Proof. It is straightforward to check that there exists a sequence (ǫn)_n∈Nmonotonically converging to 0 from above such thatm({|∇u|=ǫn}) = 0 for anyn∈N.

Choose an open set Ω^′ containing the support of uand compactly contained in Ω. Let v : Ω→ [0,+∞) be the distance function from the complementary of Ω^′ inX, namely

v(x) := dist(x, X\Ω^′) for anyx∈Ω.

Observe thatv∈LIPc(Ω), moreover

|∇v|(x) = 1 for anyx∈Ω^′. (3.3)

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Indeed it suffices to observe that the restriction ofv to any geodesic connectingxwithy∈X\Ω^′ such thatv(x) =d(x, y) has slope equal to 1 atx.

Next we introduce the approximating sequence un := u+ǫnv and we claim that it has the desired properties. Indeed, ifu∈LIPc(Ω) is non-negative, then alsoun∈LIPc(Ω) is so. From the inequality

|∇(u+ǫnv)| ≥ ||∇u| −ǫn|∇v||

and from (3.3) it follows that{|∇un|= 0} ∩ {un>0} ⊂ {|∇u|=ǫn}. Since theǫn are chosen in such a way thatm({|∇u|=ǫn}) = 0, we infer thatm({|∇un|= 0} ∩ {un>0}) = 0 .

Clearlyun converge uniformly to uas n→ ∞, granting in particular thatun→uinL^p(Ω,m).

At the same time, the inequality

||∇un| − |∇u|| ≤ǫn|∇v|

grants that

Z

Ω

|∇un|^pdm→ Z

Ω

|∇u|^pdm, yielding the desired conclusion.

Corollary 3.7. Let (X,d,m)be a geodesic metric measure space verifying locally doubling and a weak 1-1 Poincaré inequality and letΩ⊂X be an open subset withm(Ω)<+∞.

Fix any1< p <+∞ and letu, un∈LIPc(Ω) be as in the statement and the proof ofLemma 3.6.

Then, for anyn∈N, it holds that |∇un|₁(x)6= 0for m-a.e. x∈ {un>0}.

Proof. One of the properties of the approximating sequence in Lemma 3.6 is that |∇un|(x)6= 0 form-a.e. x∈ {un>0}. The desired conclusion follows from [APS15] where it is proved that, under the locally doubling and Poincaré assumption, there existsc >0 such that

|∇f|(x)≤c|∇f|₁(x) form-a.e. x∈X, for any functionf ∈LIPloc(X).

Remark 3.8. Since any essentially non branching CD(K, N) metric measure space is (locally) doubling and verifies a weak 1-1 Poincaré inequality (see [VR08]),Corollary 3.7applies to the case of our interest.

InProposition 3.9below we extend to the non smooth setting [K06, Theorem 2.3.2]; the key idea is to replace the Euclidean isoperimetric inequality with the Lévy-Gromov isoperimetric inequality Theorem 2.7.

Proposition 3.9(Lipschitz to Lipschitz property of the rearrangement).Let(X,d,m)withm(X) = 1 be an essentially non branching CD(K, N)space for some K >0,N ∈(1,+∞). Let Ω⊂X be an open subset and consider [0, r]⊂IK,N such thatm_K,N([0, r]) =m(Ω).

Let u∈LIP(Ω) be non-negative with Lipschitz constantL≥0 and assume that |∇u|₁(x)6= 0 for m-a.e. x∈ {u >0}.

Then u^∗: [0, r]→[0,∞)isL-Lipschitz as well.

Proof. Let µ be the distribution function associated to u and denote by M := supu < +∞.

Observe that our assumptions grant continuity and strict monotonicity ofµ. Therefore for any s, k≥0 such thats+k≤m(Ω) we can find 0≤t−h≤t≤M in such a way thatµ(t−h) =s+k andµ(t) =s. Taking into account the assumption thatuisL-Lipschitz we can say that

Z

{t−h≤u≤t}

|∇u|₁dm≤L(µ(t−h)−µ(t)). (3.4) On the other hand, an application of the coarea formula (2.8) yields

Z

{t−h≤u≤t}

|∇u|₁dm= Z t

t−h

Per({u≥r}) dr. (3.5)

A Lichnerowicz estimate for the spectral gap of a sub-Laplacian

arXiv:1807.04453v1 [math.FA] 12 Jul 2018

Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below

Andrea Mondino

and Daniele Semola

July 13, 2018

Contents

1 Introduction

1.1 Polya-Szego and p-spectral gap in CD(K, N ) spaces

1.2 Rigidity and almost rigidity in RCD(K, N ) spaces

2 Preliminaries

2.1 Essentially non branching, CD(K, N ) and RCD(K, N ) metric measure spaces

2.2 Finite perimeter sets and Lévy-Gromov isoperimetric inequality

2.3 BV functions and coarea formula in m.m.s.

3 Polya-Szego inequality