Indeterminacy estimates and the size of nodal sets in singular spaces
aa r X i v : . [ m a t h . DG ] J a n Indeterminacy estimatesand the size of nodal sets in singular spaces
Fabio Cavalletti ∗ and Sara Farinelli ∗ January 12, 2021
Abstract
We obtain the sharp version of the uncertainty principle recently introduced in [47], and im-proved by [13], relating the size of the zero set of a continuous function having zero mean and theoptimal transport cost between the mass of the positive part and the negative one. The result isactually valid for the wide family of metric measure spaces verifying a synthetic lower bound onthe Ricci curvature, namely the
MCP ( K, N ) or CD ( K, N ) condition, thus also extending the scopebeyond the smooth setting of Riemannian manifolds.Applying the uncertainty principle to eigenfunctions of the Laplacian in possibly non-smoothspaces, we obtain new lower bounds on the size of their nodal sets in terms of the eigenvalues.Those cases where the Laplacian is possibly non-linear are also covered and applications to linearcombinations of eigenfunctions of the Laplacian are derived. To the best of our knowledge, noprevious results were known for non-smooth spaces.
Contents
RCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
MCP ( K, N ) densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
MCP and CD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 The infinitesimally Hilbertian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 ∗ Mathematics Area, SISSA, Trieste (Italy) [email protected], [email protected] Introduction
This paper is motivated by the recent emerging interest on uncertainty estimates and their applicationsto the behaviour of solutions of certain elliptic equations.To be more precise: given a continuous function f : Ω ⊂ R n → R with zero mean R Ω f = 0, withΩ compact, it is natural to interpret f + , the positive part of f , and f − , the negative part of f , astwo distributions of mass one can compare evaluating their Wasserstein distance W (even thoughthey do not have total mass 1). Then, if it is cheap to transport f + dx to f − dx (meaning theirWasserstein distance is small), necessarily most of the mass of f + must be close to most of the massof f − . Continuity of f implies then that necessarily the nodal set { x ∈ Ω : f ( x ) = 0 } has to be large.Uncertainty estimates will quantify this relation.This question was firstly investigated by Steinerberger in dimension 2 [53] and later in any dimensionby Sagiv and Steinerberger [47] proving that any continuous function f : [0 , n → R having zero meansatisfies the following inequality W ( f + dx, f − dx ) · H n − ( { x ∈ X : f ( x ) = 0 } ) ≥ C (cid:18) k f k L k f k L ∞ (cid:19) − n k f k L . (1.1)The constant C depends only on n and H n − denotes the Hausdorff measure of dimension n − − /n and extending the range ofapplicability to continuous functions defined on any smooth and compact Riemannian manifold.The first main result of this note is the following sharp (in the exponent) uncertainty principlevalid for real valued, continuous or Sobolev functions defined over a metric measure spaces (m.m.s. forshort) ( X, d , m ) verifying synthetic (meaning not requiring any smoothness assumption on X ) lowerbound on the Ricci curvature called Curvature-Dimension condition and denoted by CD ( K, N ), with K mimiking the lower bound on the Ricci curvature and N the upper bound on the dimension. Withthe terminology metric measure space we intend a complete and separable metric space ( X, d ) endowedwith a non-negative Radon measure m (all the other terminology and notations will be introduced inSection 2). Theorem 1.1 (Sharp indeterminacy estimate) . Let
K, N ∈ R with N > . Let ( X, d , m ) be anessentially non-branching metric measure space verifying CD ( K, N ) . Let f ∈ L ( X, m ) be a continuousfunction, or alternatively f ∈ W , ( X, d , m ) , such that R X f m = 0 and assume the existence of x ∈ X such that R X | f ( x ) | d ( x, x ) m ( dx ) < ∞ .Then the following indeterminacy estimate is valid: W ( f + m , f − m ) · Per ( { x ∈ X : f ( x ) > } ) ≥ k f k L ( X, m ) k f k L ∞ ( X, m ) k f k L ( X, m ) C K,D , (1.2) where D = diam ( X ) and C K,D := ( K ≥ ,e − KD / K < . The essentially non-branching assumption in Theorem 1.1 is to prevent branch-like behaviour ofgeodesics and it is trivially satisfied by Riemannian manifolds and verified by the more regular classof
RCD spaces. The notation
Per ( A ) is used to denote the Perimeter of the set A (see Section 2 for itsdefinition in this abstract setting). In the smooth setting, i.e. an n -dimensional Riemannian manifoldendowed with the volume measure, it coincides thanks to De Giorgi’s Theorem with the H n − -measureof the reduced boundary of A (same result has been recently extended to the setting of non-collapsed RCD ( K, N ) spaces, see [2]).We will now list few detailed comments on Theorem 1.1.
Setting and Sharpness: CD ( K, N ) condition, see Section 2.1 for theprecise definition and for a list of class of spaces falling within this theory. Here we mention thatgiven a complete Riemannian manifold (
M, g ) one can naturally consider the m.m.s. ( M, d g , Vol g )where where d g is the geodesic distance and Vol g the volume measure both induced by the metric g . Then ( M, d g , Vol g ) verifies CD ( K, N ) if and only if Ric g ≥ Kg and dim( M ) ≤ N . In par-ticular any compact smooth weighted (meaning with m = e − V Vol g with V smooth) Riemannianmanifold is included in our setting. Thanks to the well-known stability property of the CD ( K, N )conditions in the measured-Gromov-Hausdorff sense, Theorem 1.1 applies also to any possiblelimit space of sequences of manifolds having Ric bounded from below uniformly and dimensionbounded from above uniformly.- The estimate does not require the space X to be compact nor the reference measure m to be finite,at least when K ≥
0. If
K <
0, then to have a meaningful estimate necessarily the diameter of X has to be finite.- Inequality (1.2) does not depend on the dimension. In particular the same statement is validfor m.m.s. satisfying CD ( K, ∞ ) (i.e. no synthetic upper bound on the dimension) for which alocalization paradigm is at disposal. In particular if ( X, d , m ) satisfies CD ( K, ∞ ) and the weaker MCP ( K ′ , N ′ ) with some other K ′ , N ′ , then (1.2) is still valid. Referring to Theorem 4.1 for theprecise statement, here we underline that for any continuous function f : R n → R having zeromean and satisfying the growth assumptions with respect to e − V dx for some smooth convexfunction V , then (1.2) holds true.- As pointed out in [13] by Carroll, Massaneda and Ortega-Cerd`a, their version of (1.1) cannot beimproved by lowering the exponent 2 − /n below 1; exponent 1 was known to be reached onlyin dimension 2. Hence the exponent 1 in (1.2) is sharp .- Theorem 1.1 will be also valid for spaces verifying another synthetic curvature notion calledmeasure-contraction property MCP ( K, N ) (see Section 2.1). A long list of subRiemanninanspaces, including the Heinseberg group, verifies this latter condition while failing CD ( K, N ). Inthis framework the constant in the inequality (1.2) will depend on the dimension. See Theorem4.3 for the precise statement.Our approach to obtain to Theorem 1.1 will be via a dimensional-reduction argument. In particular,the L -optimal transport problem between the positive and the negative part of f gives, as a byproduct,a foliation of the ambient space X into a family of geodesics obtained by considering the integralcurves of the gradient of a Kantorovich potential u , i.e. a solution of the dual problem. This non-smooth foliation has few pleasant properties that are summarised in Theorem 2.4 (see Section 2 for allpreliminaries). Here we mention that the integral of the function f along almost every geodesics of thefoliation is still zero, where the integral is with respect to the corresponding marginal measure. Alsothe curvature properties of the space are inherited by the one-dimensional “weighted” geodesic in asuitable sense. These two properties permit to reduce the proof of Theorem 1.1 to a one-dimensionalanalysis (Section 3) and, in turn, to obtain the sharpness. Our main application of Theorem 1.1 will be a lower bound on the size of nodal sets for eigenfunctionsof the Laplacian (and linear combination of them) in possibly singular spaces verifying synthetic Riccicurvature bounds.The whole list of topics related to the geometry of Laplace eigenfunctions (for instance the Courantnodal domain theorem or the quasi-symmetry conjecture) goes beyond the scope of this short introduc-tion. However, to put the problem into perspective, we will now recall the long series of contributionsto Yau’s conjecture and the solution to it. 3au conjectured in [57] that for any n -dimensional C ∞ -smooth closed Riemannian manifold M , hencewithout boundary and compact, any Laplace eigenfunction − ∆ f λ = λf λ satisfies c √ λ ≤ H n − ( { f λ = 0 } ) ≤ C √ λ, with c, C depend solely on M and not on λ .First Br¨uning [12] proved the validity of the lower bound for n = 2. Then Donnelly and Feffermanin 1988 [26] established Yau’s conjecture in the case of real analytic metrics (for instance sphericalharmonics). In the case of smooth manifold Nadirashvili in 1988 [40] proved for n = 2 that H ( { f λ =0 } ) ≤ Cλ log λ and later improved [27, 25] to H ( { f λ = 0 } ) ≤ Cλ / . For general n >
2, Hardt andSimon [32] obtained the non-polynomial bound H n − ( { f λ = 0 } ) ≤ Cλ C √ λ .Few years later the lower bound has been improved to H n − ( { f λ = 0 } ) ≥ cλ − n , (1.3)in independent contributions by Colding and Minicozzi [23], Sogge and Zelditch [48, 49] and by Steiner-berger [51]. Finally, a breakthrough has been obtained by Logunov in 2018, proving, in the smoothcase and for any n ∈ N , a polynomial upper bound [34] and the lower bound [35] in Yau’s conjecture.For an overview on all these result we refer to [36].To the best of our knowledge there are no results on the size of nodal sets of eigenfunctions ofthe Laplacian whenever a singularity on the ambient manifold is allowed. Following Steinerberger[53], upper bounds on the W -distance between the positive and the negative parts of a Laplaceeigenfunctions will yield lower bounds on the size of their nodal sets. This indeed is the content of thefollowing results. The first one will be for spaces verifying CD ( K, N ); at this level of generality theLaplacian may not be even a linear operator (see Section 2.4).
Theorem 1.2 (Nodal sets on CD -spaces) . Let
K, N ∈ R with N > . Let ( X, d , m ) be an essentiallynon-branching m.m.s. verifying CD ( K, N ) and such that m ( X ) < ∞ . Let f λ be an eigenfunction ofthe Laplacian of eigenvalue λ > (see Definition 2.17) and assume moreover the existence of x ∈ X such that R X | f λ ( x ) | d ( x, x ) m ( dx ) < ∞ .Then the following estimate on the size of the its nodal set holds true: Per ( { x ∈ X : f λ ( x ) > } ) ≥ √ λ C K,D p m ( X ) · k f λ k L ( X, m ) k f λ k L ( X, m ) k f λ k L ∞ ( X, m ) , where D = diam ( X ) and C K,D is the same of Theorem 1.1.
As before, Theorem 1.2 is actually valid also in other frameworks: for spaces verifying CD ( K, ∞ )and MCP ( K ′ , N ′ ) (Theorem 5.3) or for spaces verifying MCP ( K, N ) (Theorem 5.4) with dimensiondependent constant.If in addition we assume the Laplacian to be linear (more precisely the Sobolev space W , ( X, d , m )to be an Hilbert space), then more techniques from the classical setting come into play (for instancecontraction estimates for the heat flow) permitting to obtain more refined results. Theorem 1.3 (Nodal sets on
RCD spaces I) . Let
K, N ∈ R with N > . Let ( X, d , m ) be a m.m.s.satisfying RCD ( K, N ) , and such that diam ( X ) = D < ∞ . Let f λ be an eigenfunction of the Laplacianof eigenvalue λ > . Then the following estimate is valid: Per ( { x ∈ X : f λ ( x ) > } ) ≥ C K,D,N s λ log λ · k f λ k L ( X, m ) k f λ k L ∞ ( X, m ) , (1.4) where ¯ C K,D,N grows linearly in D if K ≥ and exponentially if K < and grows with power / in N . W ( f + λ m , f − λ m ) ≤ C ( K, N, D ) r log λλ k f λ k L , (1.5)(see Proposition 5.6), already obtained in the smooth setting by Steinerberger [52], and recently im-proved in [13] removing the log λ term but with an approach that seems confined to the smooth setting.Finally, we notice that using already available L ∞ estimates for Laplace eigenfunctions one canobtain an explicit lower bound on the size of the nodal set of an eigenfunction. Theorem 1.4 (Nodal sets on
RCD spaces II) . Let
K, N ∈ R with N > . Let ( X, d , m ) be a m.m.s.verifying RCD ( K, N ) , and with diam ( X ) = D < ∞ ; finally pose m ( X ) = 1 . Let f λ be an eigenfunctionof the Laplacian of eigenvalue λ > max { , D − } . Then the following estimate is valid: Per ( { x ∈ X : f λ ( x ) > } ) ≥ C K,D,N √ log λ λ − N , (1.6) where ¯ C K,D,N grows linearly in D if K ≥ and exponentially if K < and grows with power / in N . To conclude we mention that if the Laplacian is linear, like for instance in the
RCD setting, Theorem1.2 and Theorem 1.3 extend also to linear combinations of eigenfunction giving a non-smooth analogueof Sturm-Hurwitz’ Theorem, see [9]; see Section 6 for all the results.
Consider a compact, smooth, N -dimensional Riemannian manifold M endowed with the geodesicdistance d g and the volume measure Vol g . For this class of spaces the improved version of (1.5)obtained in [13] holds true. Hence our Theorem 1.1 gives the following inequality H N − ( { x ∈ X : f λ ( x ) = 0 } ) ≥ C K,D,N √ λ · k f λ k L ( M ) k f λ k L ∞ ( M ) , One can then use the inequality k f λ k ∞ ≤ λ N − k f λ k by Sogge and Zelditch [49] (which is known tobe sharp on spherical harmonics) and obtain H N − ( { x ∈ X : f λ ( x ) = 0 } ) ≥ λ − N , reproving the estimate (1.3) by Colding and Minicozzi [23], Sogge and Zelditch [48, 49] and by Steiner-berger [51]. It is therefore plausible to expect (1.3) (or its counterpart with the Perimeter) to holds truealso for compact RCD -spaces, provided the the validity of the following two inequalities is established W ( f + λ m , f − λ m ) ≤ C ( K, N, D ) 1 √ λ k f λ k L , k f λ k L ∞ ≤ λ N − k f λ k L . that are left for a future investigation. Similar investigation will be also carried out for the quasi-symmetry property of eigenfunction in the non-smooth setting. In what follows, ( X, d , m ) will be a complete and separable metric measure space that is ( X, d ) is acomplete and separable metric space and m is a non-negative Radon measure on X . Also, throughoutthe note, the various curvature conditions we will assume will imply X to be proper (bounded andclosed sets are compact). In various situation this will simplify the presentation (see Section 2.4).5 .1 Synthetic Curvature conditions We briefly recall the main definitions of curvature bounds for metric measure spaces that we will usethroughout the paper referring for more details to the original papers [37, 55, 56].In the following P ( X ) is the space of Borel probability measures on X and, for p ≥ P p ( X ) is thespace of Borel probability measures with finite p -moment.The p-Wasserstein distance W p on P p ( X ) is defined for any µ , µ ∈ P p ( X ) as follows: W p ( µ , µ ) p := inf π ∈ Π( µ ,µ ) Z X × X d p ( x, y ) π ( dxdy ) , (2.1)where Π( µ , µ ) := n π ∈ P ( X × X ) : P (1) ♯ π = µ , P (2) ♯ π = µ o is the set of admissible transport plans between µ and µ and P ( i ) is the projection on the i -thcomponent, for i = 1 ,
2. We will only considering in this note W and W . Geo( X ) denotes the spaceof constant speed geodesics on X :Geo( X ) := { γ ∈ C ([0 , , X ) : d ( γ ( s ) , γ ( t )) = | s − t | d ( γ (0) , γ (1)) for any s, t ∈ [0 , } . For any t ∈ [0 , t is defined on Geo( X ) by e t ( γ ) := γ ( t ). For any pair of measures µ , µ in P ( X ), the set of dynamical optimal plans is defined byOptGeo( µ , µ ) := { ν ∈ P (Geo( X )) : (e , e ) ♯ ν realizes the minimum in (2.1) } . Definition 2.1 (Essentially non-branching) . A subset G ⊂ Geo( X, d ) of geodesics is called non-branching if for any γ , γ ∈ G the following holds: ∃ t ∈ (0 ,
1) : γ s = γ s ∀ s ∈ [0 , t ] = ⇒ γ = γ s ∀ s ∈ [0 , . ( X, d ) is called non-branching if Geo( X, d ) is non-branching. ( X, d , m ) is called essentially non-branching if for any µ , µ ≪ m with µ , µ ∈ P ( X ) any ν ∈ OptGeo( µ , µ ) is concentrated ona Borel non-branching subset G ⊂ Geo( X, d ).The above definition was introduced in [45] by Rajala and Sturm. The restriction to essentiallynon-branching spaces is natural and facilitates avoiding pathological cases. One example is the failureof the local-to-global property for a general CD ( K, N ) in [44], property that has been recently verifiedin [19] under the assumption of essentially non-branching (and finite m ).Given K ∈ R and N ∈ (0 , ∞ ], define: D K, N := ( π √ K/ N K > , N < ∞ + ∞ otherwise . (2.2)In addition, given t ∈ [0 ,
1] and 0 < θ < D K, N , define: σ ( t ) K, N ( θ ) := sin( tθ q K N )sin( θ q K N ) = sin( tθ √ K N )sin( θ √ K N ) K > , N < ∞ t K = 0 or N = ∞ sinh( tθ √ − K N )sinh( θ √ − K N ) K < , N < ∞ , and set σ ( t ) K, N (0) = t and σ ( t ) K, N ( θ ) = + ∞ for θ ≥ D K, N . Given K ∈ R and N ∈ (1 , ∞ ], the distortioncoefficients are defined as: τ ( t ) K,N ( θ ) := t N σ ( t ) K,N − ( θ ) − N . When N = 1, set τ ( t ) K, ( θ ) = t if K ≤ τ ( t ) K, ( θ ) = + ∞ if K >
R´enyi entropy functional E : P ( X ) → [0 , ∞ ] is defined as E ( µ ) := Z X ρ − N m , where µ = ρ m + µ s and µ s ⊥ m , and the Boltzman entropy
Ent : P ( X ) → [0 , ∞ ] defined byEnt( µ ) := Z X ρ log( ρ ) m , if µ = ρ m , and Ent( µ ) := ∞ otherwise Definition 2.2 ( CD conditions) . ( X, d , m ) verifies the CD ( K, N ) (resp. CD ( K, ∞ )) condition for some K ∈ R , N ∈ (1 , ∞ ) if for any pair of probability measures µ , µ ∈ P ( X ) with µ , µ ≪ m (andEnt( µ i ) < ∞ , i = 0 , ν ∈ OptGeo( µ , µ ) and an optimal plan π ∈ Π( µ , µ ) such that µ t := (e t ) ♯ ν ≪ m and E N ′ ( µ t ) ≥ Z n τ (1 − t ) K,N ′ ( d ( x, y )) ρ − N ′ + τ ( t ) K,N ′ ( d ( x, y )) ρ − N ′ o π ( dxdy )for any N ′ ≥ N , t ∈ [0 ,
1] (resp .Ent( µ t ) ≤ (1 − t )Ent( µ ) + t Ent( µ ) − K t (1 − t ) W ( µ , µ ) ).For our purposes we also need to introduce a weaker variant of CD called Measure-Contractionproperty, MCP ( K, N ) in short, introduced separately by Ohta [41] and Sturm [56] with two definitionsthat slightly differ in general metric spaces, but that coincide on essentially non-branching spaces.
Definition 2.3 ( MCP ( K, N )) . A m.m.s. ( X, d , m ) is said to satisfy MCP ( K, N ) if for any o ∈ supp( m )and µ ∈ P ( X, d , m ) of the form µ = m ( A ) m x A for some Borel set A ⊂ X with 0 < m ( A ) < ∞ (andwith A ⊂ B ( o, π p ( N − /K ) if K > ν ∈ OptGeo( µ , δ o ) such that:1 m ( A ) m ≥ (e t ) ♯ (cid:0) τ (1 − t ) K,N ( d ( γ , γ )) N ν ( dγ ) (cid:1) ∀ t ∈ [0 , . (2.3)If ( X, d , m ) is a m.m.s. verifying MCP ( K, N ), then (supp( m ) , d ) is Polish, proper and it is a geodesicspace. With no loss in generality for our purposes we will assume that X = supp( m ).To conclude this part we include a list of notable examples of spaces fitting in the assumptions ofour results. The class of essentially non branching CD ( K, N ) spaces includes many remarkable familyof spaces, among them: • Measured Gromov Hausdorff limits of Riemannian N -dimensional manifolds satisfying Ric g ≥ Kg and more generally the class of RCD ( K, N ) spaces . Indeed measured Gromov Hausdorfflimits of Riemannian N -manifolds satisfying Ric g ≥ Kg are examples of RCD ( K, N ) spaces (seefor instance [28] and for the definition of
RCD see Section 2.4) and, in particular, are essentiallynon-branching and CD ( K, N ) (see [45]). • Alexandrov spaces with curvature ≥ K . Petrunin [43] proved that the lower curvature boundin the sense of comparison triangles is compatible with the optimal transport type lower boundon the Ricci curvature given by Lott-Sturm-Villani. Moreover geodesics in Alexandrov spaceswith curvature bounded below do not branch. It follows that Alexandrov spaces with curvaturebounded from below by K are non-branching CD ( K ( N − , N ) spaces. • Finsler manifolds where the norm on the tangent spaces is strongly convex, and which satisfylower Ricci curvature bounds.
More precisely we consider a C ∞ -manifold M , endowed witha function F : T M → [0 , ∞ ] such that F | T M \{ } is C ∞ and for each p ∈ M it holds that F p := T p M → [0 , ∞ ] is a strongly-convex norm, i.e. g pij ( v ) := ∂ ( F p ) ∂v i ∂v j ( v ) is a positive definite matrix at every v ∈ T p M \ { } . M, F ) to be geodesicallycomplete and endowed with a C ∞ probability measure m in a such a way that the associatedm.m.s. ( X, F, m ) satisfies the CD ( K, N ) condition. This class of spaces has been investigatedby Ohta [42] who established the equivalence between the Curvature Dimension condition and aFinsler-version of Bakry-Emery N -Ricci tensor bounded from below.While CD ( K, N ) implies the weaker
MCP ( K, N ), the latter is capable to capture the behaviour ofmore general family of spaces. In particular, for a complete list of subRiemannian spaces verifying the
MCP ( K, N ) (and not CD ( K, N )), we refer to the recent [38].
One of the key tools of our approach to obtain a sharp indeterminacy estimate is the dimensionalreduction argument furnished by localization theorem. In its various forms, the following theorem goesback to [10] for the
MCP case (with a slightly different presentation), while to [17] for the CD ( K, N ) casewith m ( X ) < ∞ and to [21] for a general Radon measure. We refer to the aforementioned referencesfor all the missing details. Theorem 2.4.
Let ( X, d , m ) be an essentially non-branching metric measure space with supp( m ) = X .Let f : X → R be m -integrable such that R X f m = 0 and assume the existence of x ∈ X such that R X | f ( x ) | d ( x, x ) m ( dx ) < ∞ .Assume also ( X, d , m ) verifies CD ( K, N ) (resp. MCP ( K, N ) ) condition for some K ∈ R and N ∈ [1 , ∞ ) .Then the space X can be written as the disjoint union of two sets Z and T with T admitting apartition { X α } α ∈ Q and a corresponding disintegration of m x T such that: m x T = Z Q m α q ( dα ) , where q is a Borel probability measure over Q ⊂ X such that Q ♯ ( m x T ) ≪ q , with Q the quotient mapassociated to the partition and the map Q ∋ α m α ∈ M + ( X ) satisfying the following properties: • for any m -measurable set B , the map α m α ( B ) is q -measurable; • for q -a.e. α ∈ Q , m α is concentrated on Q − ( α ) = X α (strong consistency); • For q -almost every q ∈ Q , it holds R X q f m q = 0 and f = 0 m -a.e. in Z . • For q -almost every q ∈ Q , the set X q is a geodesic (even more a transport ray) and the onedimensional m.m.s. ( X α , d , m α ) verifies CD ( K, N ) (resp. MCP ( K, N ) ).Moreover, fixed any q as above such that Q ♯ ( m x T ) ≪ q , the disintegration is q -essentially unique. Remark 2.5.
Via the ray map g associated to the transport set of f + m into f − m (see for instance[10]), we have that for q -a.e. q ∈ Q m q = g ( q, · ) ♯ (cid:0) h q · L (cid:1) , for some function h q : Dom ( g ( q, · )) ⊂ R → [0 , ∞ ) where Dom ( g ( q, · )) is an interval I q ⊂ R and( I q , | · | , h q · L ) is isomorphic to ( X α , d , m α ); in particular it verifies CD ( K, N ) (resp.
MCP ( K, N )).We will therefore spend few lines on one-dimensional m.m.s. verifying curvature bounds.
Definition 2.6 ( CD ( K, N ) density) . Given
K, N ∈ R and N ∈ (1 , ∞ ), a non-negative function h defined on an interval I ⊂ R is called a CD ( K, N ) density on I , if for all x , x ∈ I and t ∈ [0 , h ( tx + (1 − t ) x ) N − ≥ σ ( t ) K,N − ( | x − x | ) h ( x ) N − + σ (1 − t ) K,N − ( | x − x | ) h ( x ) N − , σ from Section 2.1). The case N = ∞ request insteadlog h ( tx + (1 − t ) x ) ≥ t log h ( x ) + (1 − t ) log h ( x ) + K t (1 − t )( x − x ) , obtained from the previous one subtracting 1 from both sides, multiplying by N −
1, and taking thelimit as N → ∞ . For completeness, we will say that h is a CD ( K,
1) density on I iff K ≤ h isconstant on the interior of I . Definition 2.7 ( MCP ( K, N ) density) . Given
K, N ∈ R and N ∈ (1 , ∞ ), a non-negative function h defined on an interval I ⊂ R is called a MCP ( K, N ) density on I if for all x , x ∈ I and t ∈ [0 , h ( tx + (1 − t ) x ) ≥ σ (1 − t ) K,N − ( | x − x | ) N − h ( x ) . (2.4)The link between one dimensional m.m.s. with curvature bounds and densities is contained in thenext straightforward result. Theorem 2.8. If h is a CD ( K, N ) (resp. MCP ( K, N ) ) density on an interval I ⊂ R then the m.m.s. ( I, |·| , h ( t ) dt ) verifies CD ( K, N ) (resp. MCP ( K, N ) ).Conversely, if the m.m.s. ( R , |·| , µ ) verifies CD ( K, N ) (resp. MCP ( K, N ) ) and I = supp( µ ) is nota point, then µ ≪ L and there exists a version of the density h = dµ/d L which is a CD ( K, N ) (resp. MCP ( K, N ) ) density on I . The estimate (2.4) implies several known properties that we collect in what follows. To write themin a unified way we define for κ ∈ R the function s κ : [0 , + ∞ ) → R (on [0 , π/ √ κ ) if κ > s κ ( θ ) := (1 / √ κ ) sin( √ κθ ) if κ > ,θ if κ = 0 , (1 / √− κ ) sinh( √− κθ ) if κ < . (2.5)For the moment we confine ourselves to the case I = ( a, b ) with a, b ∈ R ; hence (2.4) implies (cid:18) s K/ ( N − ( b − x ) s K/ ( N − ( b − x ) (cid:19) N − ≤ h ( x ) h ( x ) ≤ (cid:18) s K/ ( N − ( x − a ) s K/ ( N − ( x − a ) (cid:19) N − , (2.6)for x ≤ x . Hence denoting with D = b − a the length of I , for any ε > (cid:26) h ( x ) h ( x ) : x , x ∈ [ a + ε, b − ε ] (cid:27) ≤ C ε , (2.7)where C ε only depends on K, N , provided 2 ε ≤ D ≤ ε . In particular, MCP ( K, N ) densities will belocally Lipschitz in the interior of their domain and continuous on its closure (see [21] for details).To conclude we present here a folklore result about localization paradigm in the setting of CD ( K, ∞ )spaces. So far Theorem 2.4 is not known for a general CD ( K, ∞ ) spaces, the missing ingredient beinggood behaviour of W -geodesics. Additionally assuming the space to satisfy MCP ( K ′ , N ′ ) for some K ′ , N ′ ∈ R (with possibly K ′ different from K ) excludes all the technical issues and the proof of thefollowing localization result just follows as the one of Theorem 2.4. Theorem 2.9.
Let ( X, d , m ) be an essentially non-branching metric measure space with supp( m ) = X .Let f : X → R be m -integrable such that R X f m = 0 and assume the existence of x ∈ X such that R X | f ( x ) | d ( x, x ) m ( dx ) < ∞ .Assume also ( X, d , m ) verifies CD ( K, ∞ ) and MCP ( K ′ , N ′ ) conditions for some K, K ′ ∈ R and N ′ ∈ [1 , ∞ ) .Then the space X can be written as the disjoint union of two sets Z and T with T admitting apartition { X α } α ∈ Q and a corresponding disintegration of m x T such that: m x T = Z Q m α q ( dα ) , here q is a Borel probability measure over Q ⊂ X such that Q ♯ ( m x T ) ≪ q and the map Q ∋ α m α ∈ M + ( X ) satisfies the following properties: • for any m -measurable set B , the map α m α ( B ) is q -measurable; • for q -a.e. α ∈ Q , m α is concentrated on Q − ( α ) = X α (strong consistency); • For q -almost every q ∈ Q , it holds R X q f m q = 0 and f = 0 m -a.e. in Z . • For q -almost every q ∈ Q , the set X q is a geodesic (even more a transport ray) and the onedimensional m.m.s. ( X α , d , m α ) verifies CD ( K, ∞ ) .Moreover, fixed any q as above such that Q ♯ ( m x T ) ≪ q , the disintegration is q -essentially unique. Given a metric measure space ( X, d , m ), one can introduce a notion of perimeter which extends theclassical one on R n . The following presentation follows [39] and the more recent [3]. We start byrecalling the notion of slope (or local Lipschitz constant) of a real-valued function. Definition 2.10 (Slope) . Let ( X, d ) be a metric space and u : X → R be a real valued function. Wedefine the slope of f at the point x ∈ X as |∇ u | ( x ) := ( lim sup y → x | u ( x ) − u ( y ) | d ( x,y ) if x is not isolated0 otherwise . To fix notations, the space of Lipschitz maps on ( X, d ) will be denoted by Lip( X ) = Lip( X, d ) whileLip c ( X ) = Lip c ( X, d ) will be the subspace of compactly supported Lipschitz maps. If the function islocally Lipschitz in an open set A, i.e. for every x ∈ A , the function is Lipschitz in a neighborhood ofx, then we use the notation Lip loc ( A ) Definition 2.11 (Perimeter) . Let E ∈ B ( X ), where B ( X ) denotes the class of Borel sets of ( X, d ),and let A ⊂ X be open. We define the perimeter of E relative to A as: Per ( E ; A ) := inf (cid:26) lim inf n →∞ Z A |∇ u n | m : u n ∈ Lip loc ( A ) , u n → χ E in L loc ( A, m ) (cid:27) , where |∇ u | ( x ) is the slope of u at the point x . If Per ( E ; X ) < ∞ , we say that E is a set of finiteperimeter . We denote Per ( E ; X ) with Per ( E ).When E is a fixed set of finite perimeter, the map A Per ( E ; A ) is the restriction to open sets ofa finite Borel measure on X , defined as Per ( E ; B ) := inf { Per ( E ; A ) : A open , A ⊃ B } . For some recent progress on the extension of De Giorgi’s rectifiability theorem (relating the perimeterand the Hausdorff measure of codimension 1) to the setting of non-collapsed
RCD ( K, N ) spaces, werefer to [2] and references therein.We observe the following fact.
Lemma 2.12.
Let ( X, d , m ) be a metric measure space, E ⊆ X be a Borel set. Assume that we aregiven a strongly consistent disintegration of m associated to a zero mean function as given in Theorem2.9: m x T = Z Q m α q ( dα ) , where q is a Borel probability measure over Q ⊂ X such that and m α ∈ M + ( X ) . Then it holds Per ( E ) ≥ Z Q Per α ( E α ) q ( dα ) , where E α = E ∩ X α and Per α is the perimeter functional in the space ( X α , d , m α ) . roof. Let { f n } n ∈ Lip loc ( X ) be a sequence of functions converging in L ( X, m ) to χ E . Then, bydisintegration0 = lim n → + ∞ Z X | f n ( x ) − χ E ( x ) | m ( dx ) = lim n → + ∞ Z Q Z X α | f n ( x ) − χ E ( x ) | m α ( dx ) q ( dα ) , so up to extracting a subsequence, that we call again { f n } , we have that for q -a.e. q ∈ Q lim n → + ∞ Z X α | f n ( x ) − χ E ( x ) | m α ( dx ) = 0 . Recalling that each m α is concentrated on X α and denoting E α := E ∩ X α , we have that f n x X α converges on L ( X α , m α ) to χ E α for q -a.e α ∈ Q . We observe in addition that if f n is Lipschitz then f n x X α is Lipschitz as well with a smaller local Lipschitz constant. Hence, taken { f n } n ∈ Lip loc ( X ) asequence of functions attaining in the limit Per ( E ), we have that Per ( E ) = lim inf n →∞ Z X |∇ f n | m ≥ lim inf n →∞ Z Q Z X α |∇ f n | m α q ( dα ) ≥ lim inf n →∞ Z Q Z X α |∇ f n x X α | m α q ( dα ) ≥ Z Q lim inf n →∞ Z X α |∇ f n x X α | m α q ( dα ) ≥ Z Q Per α ( E α ) m α q ( dα ) , and the claim follows.We also include the following easy fact about the perimeter in the weighted one dimensional case.For a proof we refer to [20, Proposition 3.1] where there is an analogous statement for CD ( K, N ) den-sities.
Lemma 2.13.
Let m = h L be a non-negative measure on R , with h a CD ( K, ∞ ) density on itssupport, which in particular is an interval. Let E be an open set in supp( m ) . Let C , . . . , C n be itsconnected components, with n possibly + ∞ . We consider the set ∪ nk =0 ¯ C k . We observe that ∪ nk =0 ¯ C k = ∪ mk =0 [ a k , b k ] , with [ a k , b k ] disjoint, with m possibly + ∞ , a k , b k ∈ R ∪ {±∞} . Then, setting B ( E ) := ∪ mk =0 { a k , b k } \ { inf(supp( m )) , sup(supp( m )) } , it holds Per h ( E ) = X x ∈ B ( E ) h ( x ) = m X k =1 h ( a k ) + h ( b k ) , where Per h is the Perimeter functional in the space (supp( m ) , | · | , h L ) . RCD
The main references for this part are [22, 4, 5, 29, 28, 7, 16] or [1] for a survey on the subject.We recall the definition of the Cheeger energy of an L function, which will be used to defineSobolev spaces on metric measure spaces. We will be only concerned with the case p = 2.Let f ∈ L p ( X, m ), the Cheeger energy of f is defined as Ch ( f ) := inf (cid:26) lim inf n →∞ Z |∇ f n | m : f n ∈ Lip( X ) ∩ L ( X, m ) , k f n − f k L → (cid:27) , (2.8)where |∇ f n | ( x ) is the slope of f n at the point x . Then W , ( X, d , m ) is defined as the space of functions f ∈ L ( X, m ) with finite Cheeger energy, endowed with the norm k f k W , ( X, d , m ) := n k f k L ( X, m ) + Ch ( f ) o W , ( X, d , m ) a Banach space. For any f ∈ W , ( X, d , m ), one can single out a distin-guished object |∇ f | w ∈ L ( X, m ), which plays the role of the modulus of the gradient and providesthe integral representation Ch ( f ) = 12 Z |∇ f | w m ;this function is called the minimal weak upper gradient (after its identification with the minimal relaxedgradient). For Lipschitz functions Lip( f ) is a weak upper gradient for f .Next we review the definition of Laplacian. Throughout the note, the various curvature conditionswe will assume will always imply X to be proper (bounded and closed sets are compact) thus simplifyingthe presentation.We recall that a Radon functional over X is a linear functional T : LIP c ( X ) → R such that forevery compact subset W ⊂ X there exists a constant C W ≥ | T ( f ) | ≤ C W max W | u | , for all u ∈ LIP c (Ω) with supp( u ) ⊂ X. Finally for any f, u locally in the Sobolev space (see [29]), define the functions D ± f ( ∇ u ) : X → R by D + f ( ∇ u ) := inf ε> | D ( u + εf ) | w − | Du | w ε , while D − f ( ∇ u ) is obtained replacing inf ε> with sup ε< . Definition 2.14 (General definition) . Let ( X, d , m ) be a m.m.s. and f : X → R be a Borel function.The function f ∈ W , ( X, d , m ) is in the domain of the Laplacian in X , f ∈ D ( ∆ ), provided thereexists a Radon functional T : Lip c ( X ) → R such that Z D − u ( ∇ f ) m ≤ − T ( u ) ≤ Z D + u ( ∇ f ) m , for each u ∈ Lip c ( X ). In this case we write T ∈ ∆ ( f ). Definition 2.15 (Eigenfunction) . Let ( X, d , m ) be a m.m.s. and f be in W , ( X, d , m ). The function f is an eigenfunction for − ∆ if there exists λ > − λf m ∈ ∆ f ;i.e. Z D − u ( ∇ f ) m ≤ λ Z f u m ≤ Z D + u ( ∇ f ) m , for each u ∈ Lip c ( X ). Remark 2.16.
It is straightforward to check that any eigenfunction has zero mean, provided m ( X ) < ∞ . Here we only sketch the argument when X is proper. Consider any sequence ( χ n ) of 1-Lipschitzfunctions with bounded support and values in [0 ,
1] such that χ n ≡ B n (¯ x ), for some fixed ¯ x ∈ X .Since we are assuming X to be proper, χ n ∈ Lip c ( X ) and therefore Z D − χ n ( ∇ f ) m ≤ λ Z χ n f m ≤ Z D + χ n ( ∇ f ) m ;for both quantities, (cid:12)(cid:12)(cid:12)(cid:12)Z D ± χ n ( ∇ f ) m (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z X \ B n (¯ x ) |∇ f | w m that are both converging to zero, provided m ( X ) < ∞ , giving R f m = 0 by dominated convergencetheorem. 12rom the lower semi-continuity and convexity of Ch : D ( Ch ) ⊂ L ( X, m ) → [0 , ∞ ), it is natural toconsider an alternative definition of Laplacian related to the sub-differential ∂ − Ch of convex analysis.We recall that the sub-differential ∂ − Ch is the multivalued operator in L ( X, m ) defined at all f ∈ D ( Ch ) by the family of inequalities g ∈ ∂ − Ch ( f ) ⇐⇒ Z X g ( h − f ) m ≤ Ch ( h ) − Ch ( f ) , ∀ h ∈ L ( X, m ) . Definition 2.17 ( L -Laplacian) . The Laplacian − ∆ f ∈ L ( X, m ) of a function f ∈ W , ( X, d , m )is the element of minimal L ( X, m )-norm in the sub-differential ∂ − Ch ( f ), provided the latter is non-empty. Accordingly, f ∈ W , ( X, d , m ) is an eigenfunction provided − ∆ f = λf , for some λ > f, g ∈ L ( X, m ) with Ch ( f ) < ∞ and − g ∈ ∂ − Ch ( f ), then g ∈ D ( ∆ )and g m ∈ ∆ f .Again from the lower semi-continuity and convexity of Ch , invoking the classical theory of gradientflows of convex functionals in Hilbert spaces, it follows that: for any f ∈ L ( X, m ) there exists a uniquecontinuous curve ( f t ) t ≥ in L ( X, m ) locally absolutely continuous in [0 , ∞ ) with f = f , such that ddt f t ∈ ∂ − Ch ( f t ) for a.e. t > f ∈ L ( X, m ) stems from the density of D ( Ch ) in L ( X, m ). Thisgives rise to a semigroup ( H t ) t ≥ on L ( X, m ) defined by H t f = f t , where f t is the unique L -gradientflow of Ch .It follows that f t ∈ D (∆) and d + dt f t = ∆ f t , ∀ t ∈ (0 , ∞ ) , according to Definition 2.17.On the other hand, one can study the metric gradient flow of the Boltzmann entropy Ent in P ( X, d ).If ( X, d , m ) satisfies CD ( K, ∞ ), it has been proven in [4] that for any µ ∈ D (Ent) there exists a uniquegradient flow of Ent starting from µ (for details we refer to [4]). This gives rise to a semigroup ( H t ) t ≥ on P ( X, d ) defined by H t µ = µ t where µ t is the unique gradient flow of Ent starting from µ .One of the main result of [4] is the identification of the two gradient flows: if ( X, d , m ) is a CD ( K, ∞ )space and f ∈ L ( X, m ) such that f m = µ ∈ P ( X, d ), then H t µ = ( H t f ) m , ∀ t ≥ . (2.9)In particular we will only use the notation H t for both semi-groups. Definition 2.18 ( RCD condition) . We say that ( X, d , m ) is infinitesimally Hilbertian if the Cheeger en-ergy Ch defined in (2.8) is a quadratic form on W , ( X, d , m ). Finally ( X, d , m ) satisfies the RCD ( K, N )condition if it satisfies the CD ( K, N ) condition and it is infinitesimally Hilbertian.Under the
RCD condition, powerful contraction estimates for the heat flow are at disposal.
Theorem 2.19 (Theorem 3 of [28]) . Let ( X, d , m ) be a metric measure space verifying RCD ( K, N ) ,then for any µ, ν ∈ P ( X ) and s, t > W ( H t µ, H s ν ) ≤ e − Kτ ( s,t ) W ( µ, ν ) + 2 N − e − Kτ ( s,t ) Kτ ( s, t ) ( √ t − √ s ) , (2.10) where τ ( s, t ) = 2( t + s + √ ts ) / . X is infinitesimally Hilbertian then the two notions of Laplacian coincide and the previousimplication can be reversed. Indeed if f, g ∈ L ( X, m ) with Ch ( f ) < ∞ and f ∈ D ( ∆ ) with g m ∈ ∆ f then − g ∈ ∂ − Ch ( f ).Finally if W , ( X, d , m ) is an Hilbert space (hence in the RCD case) and f ∈ W , ( X, d , m ) is aneigenfunction for the Laplacian in the sense of Definition 2.15, then H t f = e − λt f . It is indeed enoughto check that − λe − λt f ∈ ∂ − Ch ( e − λt f ) that is equivalent to − λf ∈ ∂ − Ch ( f ) and this follows from [29,Proposition 4.12]. In this section we will obtain the one-dimensional version of the uncertainty principle we will thenintegrate via Disintegration Theorem. A slightly different version of the following Proposition 3.1 wasalready present in the literature [54, Theorem 4].Throughout this section we will tacitly assume the Wasserstein distance to be defined on any coupleof non-negative Borel measures having the same finite mass (not necessarily coinciding with 1).We fix some notations that will be useful in this section and in the following one.Given a function f : I → R with zero mean, with I real closed interval, possibly of infinite length,satisfying the hypotheses of Theorem 2.4 we say that the trasport of f goes along a unique trasportray if applying Theorem 2.4 one has that the partition in { X α } α is made of only one element.We underline in addition that, recalling the notations of Lemma 2.13, in the case of f being a continuousfunction B ( { x | f ( x ) > } ) is a subset of the zero set of f . Proposition 3.1.
Let f : [0 , → R be a continuous function having zero mean w.r.t Lebesgue measure,i.e. Z (0 , f + ( x ) dx = Z (0 , f − ( x ) dx, and assume that the transport of f goes along a unique transport ray, (see notations above). Then itholds: W ( f + L , f − L ) H ( B ( { x | f ( x ) > } )) ≥ k f + k L (0 , {k f + k L ∞ (0 , , k f − k L ∞ (0 , } . (3.1) Proof.
Step 1.
We claim that given two non-negative functions f, g ∈ L ∞ (0 ,
1) such that R [0 , f ( x ) dx = R [0 , g ( x ) dx and which satisfy the following condition on the supports: there exists¯ x ∈ (0 ,
1) such that supp { f } ⊆ [0 , ¯ x ] , supp { g } ⊆ [¯ x, , (3.2)then one has W ( f, g ) ≥ k f k L min {k f k L ∞ , k g k L ∞ } . (3.3)Indeed we can consider the two following rearrangement of the masses r f L := k f k L ∞ χ (¯ x − τ f , ¯ x ) L , r g L := k g k L ∞ χ (¯ x, ¯ x + τ g ) L , with τ f and τ g chosen so that the total mass of r f L is the same total mass of f L , and the same for r g L and g L . We notice that by direct calculation it holds W ( r f L , r g L ) = 12 k f k L (0 , k f k L ∞ (0 , + k g k L (0 , k g k L ∞ (0 , ! , (3.4)and then we observe that W ( f L , g L ) ≥ W ( r f L , r g L ) . (3.5)14ndeed for any π optimal transport plan between f L and g L , one has W ( f L , g L ) = Z | x − y | π ( dxdy ) = Z (¯ x, ( y − ¯ x ) g ( y ) dy + Z (0 , ¯ x ) (¯ x − x ) f ( x ) dx = Z (¯ x, yg ( y ) dy + Z (0 , ¯ x ) − xf ( x ) dx ≥ Z (¯ x, yr g ( y ) dy + Z (0 , ¯ x ) − xr f ( x ) dx where the last inequality follow from the two following observations: • g − r g ≤ x, ¯ x + τ g ) and g − r g ≥ x + τ g ,
1) , • f − r f ≤ , ¯ x − τ f ) and f − r f ≥ x − τ f , ¯ x ),and the fact that for a function ψ : (0 , + ∞ ) → R with zero mean and such that ψ ≤ , a ) and ψ ≥ a, + ∞ ) it holds that R (0 , + ∞ ) xψ ( x ) dx ≥
0. So finally putting (3.4) and (3.5) togheter weobtain W ( f L , g L ) ≥ k f k L (0 , {k f k L ∞ (0 , , k g k L ∞ (0 , } . Step 2.
Let f : [0 , → R be continuous and such that R (0 , f ( x ) dx = 0.Let C , . . . , C n be the connected components of { x ∈ [0 , | f ( x ) > } , with n possibly + ∞ . As inLemma 2.13 we consider the set ∪ nk =0 ¯ C k . We observe that it is the union of disjoint closed intervals D k : ∪ nk =0 ¯ C k = ∪ mk =0 D k . If m = + ∞ then H ( B ( { x | f ( x ) > } )) = + ∞ and the statement is triviallytrue. Hence we can assume that m < + ∞ . Let T : [0 , → R be an optimal transport map for theproblem. We observe that T ( f + x D k L ) ≤ f − L , so in particular it is absolutely continuous withrespect to the Lebesgue measure and its density dT ♯ ( f + x Dk L ) d L satisfies k dT ♯ ( f + x Dk L ) d L k L ∞ ≤ k f − k L ∞ .In addition we observe that since the transport of f goes along a unique transport ray, we have thateither u ( x ) := − x or u ( x ) := x is a Kantorovich potential for the problem, so assuming without loss ofgenerality u ( x ) = − x . Using the definition of Kantorovich potential we have that each couple ( x, T ( x ))with x ∈ supp f + satisfies u ( x ) − u ( T ( x )) = | x − T ( x ) | so in particular T ( x ) = x + | x − T ( x ) | and T ( x ) ≥ x . This means that for each k the couple f + x D k and dT ♯ ( f + x Dk L ) d L satisfies the hypothesesof the previous step. This is because the second function is concentrated on T ( D k ). So in particularapplying the previous estimate to each of f + x D k L and its pushforward through the map T one hasthat W ( f + x D k L , T ♯ ( f + x D k L )) ≥ k f + x D k k L (0 , min {k f + k L ∞ (0 , , k f − k L ∞ (0 , } . Being in addition the sets C k disjoint, we have that W ( f + L , f − L ) = m X k =1 W ( f + x D k L , T ♯ ( f + x D k L ))) ≥ m X k =1 k f + x D k k L (0 , min {k f + k L ∞ (0 , , k f − k L ∞ (0 , } . Applying Cauchy-Schwartz inequality we get W ( f + L , f − L ) ≥
12 min {k f + k L ∞ (0 , , k f − k L ∞ (0 , } m m X k =1 k f + x D k k L (0 , ! = 12 m k f + k L (0 , min {k f + k L ∞ (0 , , k f − k L ∞ (0 , } . Remark 3.2.
We observe that in the preceeding proposition the fact that the interval in which weare working in is exactly [0 ,
1] plays no role, so it analogously holds for an interval [ a, b ] o in generalfor intervals of infinite length provided that the function f is in L .15 .1 One dimensional densities with curvature bounds We now obtain the one-dimensional estimate also for a reference measure other than the Lebesgue one.As before, we will first consider the case of functions defined on a compact interval [0 , D ] and thenwe will discuss the non-compact case in the following Remark 3.4.
Proposition 3.3.
Let h : [0 , D ] → [0 , + ∞ ) be a CD ( K, ∞ ) -density (recall Definition 2.6).Let f : [0 , D ] → R be a continuous function having zero mean w.r.t the measure h L : R (0 ,D ) f ( x ) h ( x ) dx = 0 .Assume also that the transport of f h goes along a unique transport ray. Then it holds W ( f + h L , f − h L ) X x ∈ B ( { f> } ) h ( x ) ≥ k f h k L (0 ,D ) C K,D k f k L ∞ (0 ,D ) , (3.6) (see Lemma 2.13 for the definition of B ( { f > } ) ) where C K,D := ( K ≥ ,e − KD / K < . (3.7) Proof.
Step 1.
We make the following preliminary observation. From CD ( K, ∞ ) assumption it followsthat the map [0 , D ] ∋ x log h ( x ) + K ( x − ¯ x ) , is concave. In particular, for each ¯ x ∈ (0 , D ) either is increasing in [0 , ¯ x ] or is decreasing in [¯ x, D ].Hence in the first case log h ( x ) + K ( x − ¯ x ) ≤ log h (¯ x ) , ∀ x ∈ [0 , ¯ x ];while in the second case: log h ( x ) + K ( x − ¯ x ) ≤ log h (¯ x ) , ∀ x ∈ [¯ x, D ];The combination of the two previous inequalities yieldsmin {k h k L ∞ [0 , ¯ x ] , k h k L ∞ [¯ x,D ] } ≤ h (¯ x ) C K,D . (3.8)where C K,D is the defined in (3.7).Similarly to Step 1 of the previous proof we make a base estimate that we will use in the next step:we take two non negative bounded functions f, g : [0 , D ] → R such that R [0 ,D ] f h dx = R [0 ,D ] gh dx ,satisfying supp { f } ⊆ [0 , ¯ x ] , supp { g } ⊆ [¯ x, D ] . (3.9)We can now apply (3.3) to f h, gh (recalling that h is positive) and W ( f h L , gh L ) ≥ k f h k L (0 ,D ) {k f h k L ∞ (0 ,D ) , k gh k L ∞ (0 ,D ) }≥ k f h k L (0 ,D ) C K,D max {k f k L ∞ (0 ,D ) , k g k L ∞ (0 ,D ) } h (¯ x ) , (3.10)where the second inequality follows from (3.8). Step 2.
Consider C , . . . , C n the connected components of { f > } with n possibly + ∞ . As inLemma 2.13 we consider the set ∪ nk =0 ¯ C k . We observe that it is the union of disjoint closed intervals:16 nk =0 ¯ C k = ∪ mk =0 [ a k , b k ], with m possibly + ∞ . We will proceed as in the proof of Proposition 3.1: weconsider an optimal trasport map T and we obtain W ( f + h L , f − h L ) = m X k =0 W ( f + h L x ( a k ,b k ) , T ♯ ( f + h L x ( a k ,b k ) )) . Then we can apply (3.10) (as in the previous proof using the fact that the transport of f + h into f − h goes along a unique transport ray) to obtain: W ( f + h L , f − h L ) = m X k =0 W ( f + h L x ( a k ,b k ) , T ♯ ( f + h L x ( a k ,b k ) )) ≥ m X k =0 k f + h k L ( a k ,b k ) C K,D k f k L ∞ ( a k ,b k ) ( h ( a k ) + h ( b k )) ≥ C K,D k f k L ∞ (0 ,D ) m X k =0 k f + h k L ( a k ,b k ) ( h ( a k ) + h ( b k )) ≥ k f h k L (0 ,D ) C K,D k f k L ∞ (0 ,D ) P mk =0 ( h ( a k ) + h ( b k )) , with the convention that if a k = 0 (resp. b k = D ) the term h ( a k ) (resp. h ( b k )) does not appear. Fromthis we get 8 C K,D W ( f + h L , f − h L ) m X k =0 ( h ( a k ) + h ( b k )) ! ≥ k f h k L (0 ,D ) k f k L ∞ (0 ,D ) , with the same convention on h (0), h ( D ) as above, from which the conclusion follows. Remark 3.4.
The case of non-compact intervals of definition holds without modifications. The onlyrelevant case is K ≥ D = ∞ indeed for K < D = ∞ , the claim becomes empty. Noticethat D plays a role only in (3.8) where, in the relevant cases, it becomes independent on D . MCP ( K, N ) densities We now address the case of an
MCP ( K, N )-density. As it is clear from the proof of Proposition 3.3, theonly place where the CD ( K, ∞ ) assumption has been used is to ensure h > , D ) and to derive(3.8). A similar estimate, with suitable variations, can be obtained also for MCP ( K, N )-densities.
Lemma 3.5.
Let h : [0 , D ] → [0 , ∞ ] be an MCP ( K, N ) -density for some real parameters K, N with N ≥ . Then for any ¯ x ∈ [0 , D ] the following estimates holds true: min {k h k L ∞ [0 , ¯ x ] , k h k L ∞ [¯ x,D ] } ≤ h (¯ x ) C K,N,D , (3.11) where C K,N,D := ( N − K ≥ N − e √ − K ( N − D K < . (3.12) Proof.
The claim will follow from simple manipulations of (2.6). For clarity we recall it: for all0 ≤ x ≤ x ≤ D (cid:18) s K/ ( N − ( D − x ) s K/ ( N − ( D − x ) (cid:19) N − ≤ h ( x ) h ( x ) ≤ (cid:18) s K/ ( N − ( x ) s K/ ( N − ( x ) (cid:19) N − ;indeed for x ∈ [0 , ¯ x ] h ( x ) ≤ (cid:18) s K/ ( N − ( D − x ) s K/ ( N − ( D − ¯ x ) (cid:19) N − h (¯ x ) ≤ h (¯ x ) s K/ ( N − ( D − ¯ x ) N − sup ≤ x ≤ ¯ x s K/ ( N − ( D − x ) N − x ∈ [¯ x, D ] h ( x ) ≤ (cid:18) s K/ ( N − ( x ) s K/ ( N − (¯ x ) (cid:19) N − h (¯ x ) ≤ h (¯ x ) s K/ ( N − (¯ x ) N − sup ¯ x ≤ x ≤ D s K/ ( N − ( x ) N − . Then if K ≥
0, in particular h will be MCP (0 , N ) givingsup ≤ x ≤ ¯ x h ( x ) ≤ h (¯ x ) (cid:18) DD − ¯ x (cid:19) N − , sup ¯ x ≤ x ≤ D h ( x ) ≤ h (¯ x ) (cid:18) D ¯ x (cid:19) N − , and thereforemin {k h k L ∞ [0 , ¯ x ] , k h k L ∞ [¯ x,D ] } ≤ h (¯ x ) D N − min { / ( D − ¯ x ) , / ¯ x } N − ≤ N − h (¯ x ) , proving the inequality if K ≥
0. If
K <
0, arguing analogously one getsmin {k h k L ∞ [0 , ¯ x ] , k h k L ∞ [¯ x,D ] } ≤ h (¯ x )2 N − e √ − K ( N − D , concluding the proof.Putting together the proof of Proposition 3.3 and Lemma 3.5 we straightforwardly obtain the next Proposition 3.6.
Let h : [0 , D ] → [0 , + ∞ ) be an MCP ( K, N ) -density. Let f : [0 , D ] → R be a contin-uous function having zero mean w.r.t the measure with density h : R (0 ,D ) f ( x ) h ( x ) dx = 0 . Assume alsothat the transport of f h goes along a unique transport ray: R (0 ,s ) f ( x ) h ( x ) dx ≥ for all s ∈ [0 , D ] .Then it holds W ( f + h L , f − h L ) X { x ∈ B ( { f> } ) } h ( x ) ≥ k f h k L (0 ,D ) C K,N,D k f k L ∞ (0 ,D ) , (3.13) where C K,N,D is given by (3.12) . Remark 3.7.
The case of non-compact intervals of definition holds again without modifications. Theonly relevant case here will be K = 0 and D = ∞ ; if K >
0, then
MCP implies that
D < D K, N (see(2.2)) while if K < D = ∞ , the claim becomes empty. Notice that D plays a role only in (3.8)that is the content of Lemma 3.5. We now use the one-dimensional estimates of the previous section to deduce the following sharp inde-terminacy estimates.
Theorem 4.1.
Let
K, K ′ , N ∈ R with N > . Let ( X, d , m ) be an essentially non-branching m.m.s.satisfying either CD ( K, N ) or MCP ( K ′ , N ) and CD ( K, ∞ ) . Let f ∈ L ( X, m ) a continuous functionor, alternatively, f ∈ W , ( X, d , m ) be such that R X f m = 0 . Assume also the existence of x ∈ X such that R X | f ( x ) | d ( x, x ) m ( dx ) < ∞ . Then the following indeterminacy estimate is valid: W ( f + m , f − m ) · Per ( { x ∈ X : f ( x ) > } ) ≥ k f k L ( X, m ) C K,D k f k L ∞ ( X, m ) , (4.1) where D = diam ( X ) and C K,D := ( K ≥ ,e − KD / K < . emark 4.2. Notice that curvature assumptions CD ( K, N ) and
MCP ( K, N ) imply
D < ∞ only inthe range K > N ∈ (1 , ∞ ). Hence under the second set of assumptions ( MCP ( K ′ , N ) and CD ( K, ∞ )), the result (4.1) for K ≥ D = ∞ . Proof.
Given f as in the assumptions, we can invoke localization paradigm (Theorem 2.4 and Theorem2.9) yielding a decomposition of the space X as X = Z ∪ T , where f is zero m -a.e. in Z and T can bepartitioned into { X α } α with α in a Borel set Q ⊂ X , and a disintegration of m , m x T = Z Q m α q ( dα ) , with q Borel probability measure with q ( Q ) = 1 and Q ∋ α m α ∈ M + ( X ) satisfying the proper-ties of Theorem 2.4; in particular, ( X α , d , m α ) is a CD ( K, N ) space (or CD ( K, ∞ ) see Theorem 2.9), R X α f m α = 0 and every X α is a transport ray associated to the L -optimal trasport of f + m into f − m . Step 1.
As proven in [21, Proposition 4.4] for the case of signed distance functions, q can beidentified with a test plan, see [4, Definition 5.1]; hence, if f ∈ W , ( X, d , m ), by the identificationbetween different definitions of Sobolev spaces [4, Theorem 6.2], for q -a.e. α the function f restrictedto the geodesic X α is Sobolev and therefore continuous.As said in Remark 2.5, we have an isomorphism between each space ( X α , d , m α ) and spaces( I α , | · | , h α · L ), with I α a real interval (of possible infinite length) satisfying the same CD ( K, N )(or CD ( K, ∞ )) condition, R I α f α ( x ) h α ( x ) dx = 0 being f α the corresponding of f x X α through theisomorphism and I α transport ray for f α . Whenever possible, for simplicity of notation, we will use f = f α .So now we can apply Proposition 3.3 and we have that q -a.e. α ∈ Q it holds W ( f + α h α L , f − α h α L ) X x ∈ B ( { f α > } ) h α ( x ) ≥ k f k L ( X α , m α ) C K,D k f k L ∞ ( X α , m α ) . (4.2)By Lemma 2.13 P x ∈ B ( { f α > } ) h α ( x ) = Per h α ( { x ∈ I α : f α ( x ) > } ), hence using the isomorphisms ofmetric measure spaces, we have W ( f + m α , f − m α ) Per α ( { x ∈ X α : f ( x ) > } ) ≥ k f k L ( X α , m α ) C K,D k f k L ∞ ( X α , m α ) , where Per α is the perimeter in ( X α , d , m α ) and Per h α in ( I α , | · | , h α · L ). In the previous factor wehave tacitly used that C K,D ≥ C K,D α , where D α is the length of X α . Integrating the square root ofthe inequality with respect to the measure q on Q and applying Holder inequality, we get (cid:18)Z Q W ( f + m α , f − m α ) q ( dα ) (cid:19) (cid:18)Z Q Per α ( { x ∈ X α : f ( x ) > } ) q ( dα ) (cid:19) ≥ Z Q (cid:0) W ( f + m α , f − m α ) · Per α ( { x ∈ X α : f ( x ) > } ) (cid:1) q ( dα ) ≥ Z Q k f k L ( X α , m α ) p C K,D k f k L ∞ ( X α , m α ) q ( dα ) ≥ p C K,D k f k L ∞ ( X, m ) Z Q Z X α | f ( x ) | m α ( dx ) q ( dα )= k f k L ( X, m ) p C K,D k f k L ∞ ( X, m ) . R Q W ( f + m α , f − m α ) q ( dα ) = W ( f + m , f − m ); therefore W ( f + m , f − m ) (cid:18)Z Q Per α ( { x ∈ X α : f ( x ) > } ) q ( dα ) (cid:19) ≥ k f k L ( X, m ) p C K,D k f k L ∞ ( X, m ) . The conclusion follows using Lemma 2.12.Repeating the same argument of the previous proof and using Proposition 3.6, we also obtainthe analogous estimate for spaces verifying the weaker
MCP ( K, N ); as expected, weaker curvatureassumptions yields a dependence on the dimension of the estimate.
Theorem 4.3.
Let
K, N ∈ R with N > . Let ( X, d , m ) be an essentially non-branching m.m.s.verifying MCP ( K, N ) .Let f ∈ L ( X, m ) a continuous function or, alternatively, f ∈ W , ( X, d , m ) be such that R X f m =0 . Assume also the existence of x ∈ X such that R X | f ( x ) | d ( x, x ) m ( dx ) < ∞ . Then the followingindeterminacy estimate is valid: W ( f + m , f − m ) · Per ( { x ∈ X : f ( x ) > } ) ≥ k f k L ( X, m ) C K,N,D k f k L ∞ ( X, m ) , (4.3) where diam ( X ) = D and C K,N,D := ( N − K ≥ , N − e √ − K ( N − D K < . The plan for this section is to obtain lower bounds on the nodal set of eigenfunctions under curvatureassumptions. Building on the previous Theorem 4.1 and Theorem 4.3, this will reduce to find an upperbound on the W distance between the positive and the negative part of the eigenfunction. MCP and CD Here, as throughout the paper, the W distance is understood to be tacitly extended between any finitenon-negative measure with the same total mass. Lemma 5.1.
Let ( X, d , m ) be a m.m.s. verifying MCP ( K, N ) and with finite total mass, m ( X ) < ∞ .Let f be an eigenfunction of the Laplacian with eigenvalue λ = 0 accordingly to Definition 2.17 andassume moreover the existence of x ∈ X such that R X | f ( x ) | d ( x, x ) m ( dx ) < ∞ .Then W ( f + m , f − m ) ≤ p m ( X ) √ λ k f k L ( X, m ) Proof.
First from Remark 2.16, R f m = 0 and, by definition, f ∈ W , ( X, d , m ). By assumptionKantorovich duality has a solution and therefore exists a 1-Lipschitz Kantorovich Potential u : X → R such that W ( f + m , f − m ) = Z X ( f + ( x ) − f − ( x )) u ( x ) m ( dx ) = Z X f ( x ) u ( x ) m ( dx ) . (5.1)Since f is a eigenfunction in the sense of Definition 2.17, then the following integration by-parts formula Z X D − g ( ∇ f ) m ≤ λ Z X gf m ≤ Z X D + g ( ∇ f ) m , is valid for any g ∈ W , ( X, d , m ) (see for instance the proof of [29, Proposition 4.9]).20rom m ( X ) < ∞ it follows that u ∈ W , ( X, d , m ), hence together with (5.1) gives W ( f + m , f − m ) ≤ λ Z X D + u ( ∇ f ) m ≤ λ Z X | Du | w | Df | w m ≤ Lip( u ) λ Z X | Df | w m , where we used the fact that | D ± u ( ∇ f ) | ≤ | Du | w | Df | w and that Lip( u ) is a weak upper gradient for u . Then by Holder inequality we have Z X | Df | w m ≤ m ( X ) (cid:18)Z X | Df | w m (cid:19) = m ( X ) (cid:18)Z X D − f ( ∇ f ) m (cid:19) ≤ m ( X ) √ λ (cid:18)Z X f m (cid:19) noticing that Df + ( ∇ f ) = | Df | w (see [29, (3.6)]) and f itself as test-function. Remark 5.2.
The same claim can be obtained assuming f to be an eigenfunction for the more generalnotion of Laplacian of Definition 2.14, provided one additionally knows f to be Lipschitz regular,yielding integration by-parts formula against any Sobolev functions (and in particular yielding f to bean eigenfunction for the Laplacian of Definition 2.17.)Putting together Lemma 5.1 and the previous results we obtain the next Theorem 5.3.
Let ( X, d , m ) be an essentially non-branching m.m.s. verifying either CD ( K, N ) or MCP ( K ′ , N ′ ) and CD ( K, ∞ ) and such that m ( X ) < ∞ .Let f be an eigenfunction of the Laplacian of eigenvalue λ > accordingly to to Definition 2.17 andassume moreover the existence of x ∈ X such that R X | f ( x ) | d ( x, x ) m ( dx ) < ∞ . Then the followingestimate on the size of the its nodal set holds true: Per ( { x ∈ X : f ( x ) > } ) ≥ √ λ C K,D p m ( X ) · k f k L ( X, m ) k f k L ( X, m ) k f k L ∞ ( X, m ) , where D = diam ( X ) and C K,D := ( K ≥ ,e − KD / K < . Proof.
Theorem 4.1 and Lemma 5.1 imply the claim.Using Theorem 4.3, we obtain the following analogous statement for spaces verifying the weaker
MCP ( K, N ) condition with dimension-dependent constant appearing. The proof, being completely thesame is omitted.
Theorem 5.4.
Let ( X, d , m ) be an essentially non-branching m.m.s. verifying MCP ( K, N ) and suchthat m ( X ) < ∞ .Let f be an eigenfunction of the Laplacian of eigenvalue λ > accordingly to to Definition 2.17and assume moreover the existence of x ∈ X such that R X | f ( x ) | d ( x, x ) m ( dx ) < ∞ .Then the following estimate on the size of the its nodal set holds true: Per ( { x ∈ X : f ( x ) > } ) ≥ √ λ C K,N,D p m ( X ) · k f k L ( X, m ) k f k L ( X, m ) k f k L ∞ ( X, m ) , where D = diam ( X ) and C K,N,D := ( N − K ≥ , N − e √ − K ( N − D K < . .2 The infinitesimally Hilbertian case Assuming the heat flow to be linear yields more sophisticated argument and sharper estimtes. We startwith the following folklore result whose proof is included as no proof as been found in the literature.
Lemma 5.5.
Let ( X, d , m ) be a m.m.s. with diam ( X ) < D . Let f, g : X → [0 , ∞ ) be functions with k f k L ( X, m ) = k g k L ( X, m ) . Then W ( f m , g m ) ≤ D k f − g k L ( X, m ) . Proof.
Construct an admissible plan ¯ π ∈ Π( f m , g m ), with ¯ π = π + π by defining π := ( Id, Id ) ♯ (cid:0) g m x { g ≤ f } (cid:1) + ( Id, Id ) ♯ (cid:0) f m x { g>f } (cid:1) and considering any π ∈ Π(( f − g ) + m , ( f − g ) − m ). Then it is straightforward to check that W ( f m , g m ) ≤ Z X × X d ( x, y ) π ( dxdy ) ≤ D π ( X × X ) = D Z X ( f − g ) + m ( dx ) , proving the claim. Proposition 5.6.
Let ( X, d , m ) be a m.m.s. verifying RCD ( K, N ) and such that diam ( X ) = D < ∞ .Let f be an eigenfunction of eigenvalue λ > . Then W ( f + m , f − m ) ≤ C ( K, N, D ) r log λλ k f k L ( X, m ) , with C ( K, N, D ) growing linearly in D and as square root in N .Proof. We define µ ± := f ± m , µ ± t := H t µ ± , where H t is the heat flow (see Section 2.4) and by triangular inequality W ( µ +0 , µ − ) ≤ W ( µ +0 , µ + t ) + W ( µ + t , µ − t ) + W ( µ − t , µ − ) , notice indeed that µ + t ( X ) = µ +0 ( X ) = µ − ( X ) = µ − t ( X ). Then by Theorem 2.19 we deduce that W ( µ ± t , µ ± ) = (cid:18)Z X f + m (cid:19) W ( µ ± t /µ ± t ( X ) , µ ± /µ ± ( X )) ≤ k f k L ( X, m ) W ( µ ± t /µ ± t ( X ) , µ ± /µ ± ( X )) ≤ √ t k f k L ( X, m ) C ( t, K, N ) , where C ( t, K, N ) := (cid:16) N − e − K t/ K t/ (cid:17) / , (with C ( t, K, N ) ≤ √ N if K ≥ W ( µ + t , µ − t ) we use Lemma 5.5. Call g t the evolution of a function g through the heatflow ( g t = H t g ), by the identification (2.9), it follows that (recall that f ∈ W , ( X, d , m ) by definition) µ ± t = ( H t f ± ) m = f ± t m . Notice that by infinitesimal Hilbertianity f + t − f − t = H t ( f + − f − ) = H t ( f ) = e − λt f, where the last identity is a consequence of f being an eigenfunction (see Section 2.4). Then we havethat W ( µ + t , µ − t ) ≤ D k f + t − f − t k L ( X, m ) = D k f t k L ( X, m ) = De − λt k f k L ( X, m ) .
22o finally W ( µ +0 , µ − ) ≤ (cid:16) √ tC ( t, K, N ) + De − λt (cid:17) k f k L ( X, m ) . Choosing t = λ log( λ ) we obtain W ( f + m , f − m ) ≤ C ( K, D, N ) r log λλ k f k L ( X, m ) , with C ( K, N, D ) growing linearly in D and as square root in N .Hence we can state one of the main results of this note. Theorem 5.7 (Nodal set
RCD -spaces) . Let
K, N ∈ R with N > . Let ( X, d , m ) be a m.m.s. satisfying RCD ( K, N ) . Assume moreover diam ( X ) = D < ∞ . Let f be an eigenfunction of the Laplacian ofeigenvalue λ > . Then the following estimate is valid: Per ( { x ∈ X : f ( x ) > } ) ≥ s λ log λ · k f k L ( X, m ) ¯ C K,D,N k f k L ∞ ( X, m ) , (5.2) where ¯ C K,D,N grows linearly in D if K ≥ and exponentially if K < and grows with power / in N .Proof. Since diam ( X ) < ∞ , it follows that m ( X ) < ∞ and therefore f ∈ L ( X, m ), it has zero meanand satisfies the growth conditions and regularity needed to invoke Theorem 4.1. Hence Theorem 4.1implies that W ( f + m , f − m ) · Per ( { x ∈ X : f ( x ) > } ) ≥ k f k L ( X, m ) C K,D k f k L ∞ ( X, m ) , that together with Proposition 5.6 implies that Per ( { x ∈ X : f ( x ) > } ) ≥ s λ log λ k f k L ( X, m ) C ( K, N, D ) C K,D k f k L ∞ ( X, m ) , giving therefore the claim.We are now in position of obtaining the explicit lower bound on the size of the nodal set of aneigenfunction stated in Theorem 1.4. Proof of Theorem 1.4.
It is a straightforward consequence of Theorem 5.7 and of the following ob-servation: given an eigenfunction f of eigenvalue λ , there exists a constant C = C ( K, N, D ) suchthat k f k L ∞ ( X, m ) ≤ Cλ N k f k L ( X, m ) , provided λ ≥ D − . Indeed from [8, Proposition 7.1] and assuming m ( X ) = 1, one has that k f k L ∞ ( X, m ) ≤ Cλ N k f k L ( X, m ) ≤ Cλ N k f k L ∞ ( X, m ) k f k L ( X, m ) , from which the claim follows dividing by the L ∞ norm and squaring both sides. We now consider functions obtained as linear combination of eigenfunctions. As expected, for thefollowing results it will be necessary to assume the linearity of the Laplacian, i.e. infinitesimal Hilber-tianity.We will however present two different upper bounds for the W distance between the positive andthe negative part of the function, one following the lines of Proposition 5.6 valid for RCD spaces andone following Lemma 5.1 valid for
MCP spaces. 23 roposition 6.1.
Let ( X, d , m ) be an essentially non-branching m.m.s. verifying MCP ( K, N ) with diam ( X ) = D < ∞ ; assume moreover ( X, d , m ) to be infinitesimally Hilbertian.Let f be a continuous function or, alternatively, f ∈ W , ( X, d , m ) , such that it satisfies in L sense f = P λ k ≥ λ a k f λ k , k ∈ N , where each f λ k is an eigenfunction with eigenvalue λ k .Then the following estimate on the size of the nodal set of f holds true: Per ( { x ∈ X : f ( x ) > } ) ≥ √ λ p m ( X ) C K,N,D · k f k L k f k L k f k L ∞ , where C K,N,D is given by Theorem 4.3.Proof.
From diam ( X ) < ∞ , it follows that m ( X ) < ∞ and therefore f ∈ L ( X, m ), it has zero meanand satisfies the growth conditions needed to apply Theorem 4.3. To prove the claim it will be thereforesufficient to obtain an upper bound for W ( f + m , f − m ).Using the Kantorovich formulation, there exists a 1-Lipschitz function such that W ( f + m , f − m ) = Z X f u m = X λ k ≥ λ a k Z X f λ k u m ≤ X λ k ≥ λ a k λ k k|∇ f λ k | w k L k|∇ u |k L ≤ p m ( X ) X λ k ≥ λ √ λ k k a k f λ k k L ≤ p m ( X ) √ λ X λ k ≥ λ k a k f λ k k L = p m ( X ) √ λ k f k L , where we used in the third identity k λ k |∇ f λ k | w k L = λ k k f λ k k L , and in the last one the orthogonalityof { f λ k } k ∈ N given by infinitesimally Hilbertianity. Lemma 6.2.
Let ( X, d , m ) be a m.m.s. verifying RCD ( K, N ) and such that diam ( X ) = D < ∞ and K ≥ . Let f : X → R be a continuous or, alternatively, f ∈ W , ( X, d , m ) , such that f = X λ k ≥ λ h f, f λ k i f λ k , { λ k } k ∈ N , where { f λ k } k ∈ N are eigenfunctions of the Laplacian of unitary L -norm with eigenvalue λ k , h f, f λ k i isthe scalar product of L ( X, m ) , h f λ j , f λ k i = δ j,k and convergence of the series is in L ( X, m ) . Then W ( f + m , f − m ) ≤ C ( K, N, D, m ( X )) (cid:18) λ log (cid:18) λ k f k L k f k L (cid:19)(cid:19) k f k L , with C ( K, N, D, m ( X )) an explicit constant, provided that λ ≥ p m ( X ) .Proof. Following the approach and the same notation of the proof of Proposition 5.6 we have W ( µ +0 , µ − ) ≤ W ( µ +0 , µ + t ) + W ( µ + t , µ − t ) + W ( µ − t , µ − ) , and deduce from Theorem 2.19 that W ( µ ± t , µ ± ) ≤ √ t k f k L ( X, m ) C ( t, K, N ) , where C ( t, K, N ) := (cid:16) N − e − K t/ K t/ (cid:17) / . Then to bound W ( µ + t , µ − t ), again using Lemma 5.5, byorthonormality of { f λ k } k it follows that k f t k L ( X, m ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ k ≥ λ e − λ k t h f, f λ k i f λ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( X, m ) ≤ m ( X ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ k ≥ λ e − λ k t h f, f λ k i f λ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( X, m ) = m ( X ) X λ k ≥ λ e − λ k t |h f, f λ k i| ≤ m ( X ) e − λt k f k L ( X, m ) . (6.1)24o finally W ( µ +0 , µ − ) ≤ √ t k f k L ( X, m ) C ( t, K, N ) + D p m ( X ) e − λt k f k L ( X, m ) . Using that K ≥ C ( t, K, N ) ≤ √ N ) and choosing t = λ log (cid:16) λ k f k L X, m ) k f k L X, m ) (cid:17) , it holds W ( µ +0 , µ − ) ≤ C ( K, N, D, m ( X )) (cid:18) λ log (cid:18) λ k f k L k f k L (cid:19)(cid:19) k f k L , proving the claim.The following result is then a straightforward consequence Corollary 6.3.
Let ( X, d , m ) be a m.m.s. verifying RCD ( K, N ) and such that diam ( X ) = D < ∞ .Let f : X → R be a continuous or, alternatively, f ∈ W , ( X, d , m ) such that f = X λ k ≥ λ h f, f λ k i f λ k , { λ k } k ∈ N , λ > , where { f λ k } k ∈ N are eigenfunctions of the Laplacian of unitary L -norm with eigenvalue λ k , h f, f λ k i isthe scalar product of L ( X, m ) , h f λ j , f λ k i = δ j,k and convergence of the series is in L ( X, m ) .Then the following estimate on the size of the nodal set of f holds true: Per ( { x ∈ X : f ( x ) > } ) ≥ √ λC ( K, N, D, m ( X )) log (cid:18) λ k f k L k f k L (cid:19) − / · k f k L k f k L ∞ , with C ( K, N, D, m ( X )) the same constant of Lemma 6.2, provided that λ ≥ p m ( X ) . References [1]
L. Ambrosio : Calculus and curvature-dimension bounds in metric measure spaces , Proceedingsof the ICM 2018.[2]
L. Ambrosio E. Bru´e and D. Semola : Rigidity of the 1-Bakry– ´Emery Inequality and Sets ofFinite Perimeter in RCD Spaces , Geom. Funct. Anal., (2019), 949–1001.[3] L. Ambrosio and S. Di Marino : Equivalent definitions of BV space and of total variation onmetric measure spaces , J. Funct. Anal., (2014), 4150–4188.[4]
L. Ambrosio, N. Gigli, and G. Savar´e : Calculus and heat flow in metric measure spaces andapplications to spaces with Ricci bounds from below , Invent. Math., , 2, (2014), 289–391.[5]
L. Ambrosio, N. Gigli, and G. Savar´e , Metric measure spaces with Riemannian Ricci curva-ture bounded from below : Duke Math. J., , (2014), 1405–1490.[6]
L. Ambrosio, N. Gigli, and G. Savar´e , Density of Lipschitz functions and equivalence of weakgradients in metric measure spaces , Rev. Mat. Iberoam., 29 (2013), pp. 969–996.[7]
L. Ambrosio, A. Mondino and G. Savar´e : Nonlinear diffusion equations and curvature con-ditions in metric measure spaces , Mem. Amer. Math. Soc., in press.[8]
L. Ambrosio, S. Honda, J. W. Portegies, D. Tewodrose : Embedding of
RCD ∗ ( K, N ) -spacesin L via eigenfunctions , Arxiv preprint 1812.03712.[9] P. B´erard and B. Helffer , Sturm’s theorem on zeros of linear combinations of eigenfunctions ,Expo. Math. 38 (2020), no. 1, 27–50, DOI 10.1016/j.exmath.2018.10.002.2510]
S. Bianchini and F. Cavalletti : The Monge problem for distance cost in geodesic spaces ,Commun. Math. Phys., (2013), 615 – 673 .[11]
A. Bjorn and J. Bjorn. : Nonlinear potential theory on metric spaces , volume 17 of EMS Tractsin Mathematics. European Mathematical Society (EMS), Zurich, 2011.[12]
J. Br¨uning : Uber Knoten von Eigenfunktionen des Laplace-Beltrami-Operators , Math. Z.,158(1):15–21, 1978.[13]
T. Carroll, X. Massaneda and J. Ortega-Cerd`a
An enhanced uncertainty principle forthe Vaserstein distance , Bull. London Math. Soc., (2020), doi:10.1112/blms.12390.[14]
F. Cavalletti : Monge problem in metric measure spaces with Riemannian curvature-dimensioncondition , Nonlinear Anal., (2014), 136–151.[15] F. Cavalletti : Decomposition of geodesics in the Wasserstein space and the globalization prop-erty , Geom. Funct. Anal., (2014), 493 – 551.[16] F. Cavalletti and E. Milman : The Globalization Theorem for the Curvature-Dimension Con-dition , Preprint arXiv:1612.07623.[17]
F. Cavalletti and A. Mondino : Sharp and rigid isoperimetric inequalities in metric-measurespaces with lower Ricci curvature bounds , Invent. Math., (2017), 803–849.[18] :
Sharp geometric and functional inequalities in metric measure spaces with lower Riccicurvature bounds , Geom. Topol., (2017), 603–645.[19] : Optimal maps in essentially non-branching spaces , Commun. Contemp. Math., , (6),(2017).[20] : Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds .Rend. Lincei Mat. Appl., (2018), 413–430.[21] : New formulas for the Laplacian of distance functions and applications , Anal. PDE, toappear.[22]
J. Cheeger : Differentiability of Lipschitz functions on metric measure spaces , Geom. Funct.Anal., 9 (1999), pp. 428–517.[23]
T. Colding and W. P. Minicozzi, II. : Lower bounds for nodal sets of eigenfunctions , Comm.Math. Phys., (2011), 777–784.[24]
D. Cordero-Erausquin, R.J. McCann and M. Schmuckenschl¨ager : A Riemannian in-terpolation inequality `a la Borell, Brascamp and Lieb . Invent. Math. (2001), 219–257.[25]
R.-T. Dong : Nodal sets of eigenfunctions on Riemann surfaces.
J. Differential Geom., (1992),493–506.[26] H. Donnelly and C. Fefferman : Nodal sets of eigenfunctions on Riemannian manifolds ,Invent. Math., (1988), 161–183.[27] H. Donnelly and C. Fefferman : Nodal sets for eigenfunctions of the Laplacian on surfaces ,J. Amer. Math. Soc., (1990), 333–353.[28] M. Erbar, K. Kuwada and K.T. Sturm:
On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner’s Inequality on Metric Measure Space , Invent. Math., ,(2015), no. 3, 993–1071.[29]
N. Gigli : On the differential structure of metric measure spaces and applications , Mem. Amer.Math. Soc., , no. 1113, (2015). 2630]
N. Gigli, B.-X. Han:
Independence on p of weak upper gradients on RCD spaces , J. Funct.Anal., (2016), 1–11.[31]
Q. Han, F. Lin : Elliptic Partial Differential Equations , New York University, Courant Instituteof Mathematical Sciences and American Mathematical Society, New York and Providence, 1997.[32]
R. Hardt and L. Simon : Nodal sets for solutions of elliptic equations , J. Differential Geom., (1989) 505–522.[33] B. Klartag:
Needle decomposition in Riemannian geometry , Mem. Amer. Math. Soc., ,(2017), no. 1180, v + 77 pp.[34]
A. Logunov : Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorffmeasure , Ann. of Math. (2), (2018), 221–239.[35]
A. Logunov : Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of thelower bound in Yau’s conjecture , Ann. of Math. (2), (2018), 241–262. .[36]
A. Logunov and E. Malinnikova : Review of Yau’s conjecture on zero sets of Laplace eigen-functions , Current Developments in Mathematics, Volume 2018, 179–212.[37]
J. Lott and C. Villani:
Ricci curvature for metric-measure spaces via optimal transport , Ann.of Math. (2), , (2009), 903–991.[38]
E. Milman:
The Quasi Curvature-Dimension Condition with applications to sub-Riemannianmanifolds , Comm. Pure Appl. Math., to appear.[39]
M. Miranda, Jr.: , Functions of bounded variation on “good” metric spaces , J. Math. Pures Appl.(9), 82 (2003), pp. 975–1004.[40]
N. S. Nadirashvili : The length of the nodal curve of an eigenfunction of the Laplace operator ,Uspekhi Mat. Nauk, (1988), 219–220.[41] S.I. Ohta:
On the measure contraction property of metric measure spaces , Comment. Math.Helv., (2007), 805–828.[42] : Finsler interpolation inequalities , Calc. Var., , (2009), 211–249.[43] A. Petrunin : Alexandrov meets Lott-Sturm-Villani , M¨unster J. Math., , (2011), 53–64.[44] T. Rajala:
Failure of the local-to-global property for CD ( K, N ) spaces , Ann. Sc. Norm. Super.Pisa Cl. Sci., , (2016), 45–68.[45] T. Rajala and K.T. Sturm:
Non-branching geodesics and optimal maps in strong CD ( K, ∞ ) -spaces , Calc. Var. Partial Differential Equations, , (2014), 831–846.[46] M.-K. von Renesse and K.T. Sturm:
Transport inequalities, gradient estimates, entropy andRicci curvature.
Comm. Pure Appl. Math. , (2005), 923–940.[47] A. Sagiv and S. Steinerberger : Transport and Interface: an Uncertainty Principle for theWasserstein distance , SIAM J. Math. Anal., accepted.[48]
C. Sogge and S. Zelditch : Lower bounds on the Hausdorff measure of nodal sets , Math. Res.Lett., (2011), 25–37.[49] C. Sogge and S. Zelditch : Lower bounds on the Hausdorff measure of nodal sets II.
Math.Res. Lett., (2012), 1361–1364.[50] S.M. Srivastava : A course on Borel sets , Graduate Texts in Mathematics, Springer 1998.2751]
S. Steinerberger : Lower bounds on nodal sets of eigenfunctions via the heat flow , Comm.Partial Differential Equations, (2014), 2240–2261.[52] S. Steinerberger : Oscillatory functions vanish on a large set , Asian J. Math., (2020), 177–190.[53] S. Steinerberger : A Metric Sturm-Liouville theory in Two Dimensions , Calc. Var. 59, (2020), https://doi.org/10.1007/s00526-019-1668-z.[54] S. Steinerberger : Wasserstein Distance, Fourier Series and Applications , preprintarXiv:1803.08011.[55]
K.T. Sturm:
On the geometry of metric measure spaces. I , Acta Math. , (2006), 65–131.[56]
K.T. Sturm:
On the geometry of metric measure spaces. II , Acta Math. , (2006), 133–177.[57]