aa r X i v : . [ m a t h . P R ] M a r Curvature dimension bounds on the deltoid model
Dominique Bakry, Olfa Zribi
Institut de Mathématiques de Toulouse,Université Paul Sabatier,118 route de Narbonne,31062 Toulouse,FRANCE
Abstract
The deltoid curve in R is the boundary of a domain on which there exist proba-bility measures and orthogonal polynomials for theses measures which are eigenvec-tors of diffusion operators. As such, they may be considered as a two dimensionalextension of the classical Jacobi operators. They belong to one of the 11 families ofsuch bounded domains in R . We study the curvature-dimension inequalities associ-ated to these operators, and deduce various bounds on the associated polynomials,together with Sobolev inequalities related to the associated Dirichlet forms The deltoid curve (also called Steiner’s hypocycloid), see figure page 7, is one ofthe bounded domains in R on which there exist a probability measure µ and asymmetric diffusion process, the eigenvectors of it being orthogonal polynomials for µ . These orthogonal polynomials have been introduced in [17, 18] and appear in theclassification of [19]. They appear in [5] as one of the eleven models in dimensiontwo for which such polynomials exist. Moreover, it seems that is one of the mostdifficult models to analyse, since there does not exist many geometric interpretationfor it. Beyond this, it is also interesting since it is deeply linked with the analysis ofthe A root system and of the spectral analysis of SU (3) matrices.This deltoid model and the associated generators provide an interesting objectto check various properties of diffusion operators, since one knows explicitly theeigenvalues, and has many informations on the eigenvectors. For example, theyhave satisfy recurrence formulas which allows for explicit computations, and in somecases generating functions, see [27].Since the associated measures and operators depend on a real parameter λ > (see equation (3.11)), one may try to understand how functional inequalities andcurvature properties depend on this parameter λ , and hence on geometric propertiesof the model. t turns out that for the specific cases λ = 1 and λ = 4 , one may producesimple geometric interpretations : in the first case from the Euclidean Laplace op-erator through the symmetries of the triangular lattice, in the second case from theCasimir operator on SU (3) acting on spectral measures. The SU (3) model providescurvature-dimension inequalities for the generic model for λ ≥ . It is not clear how-ever that these inequalities are optimal. It turns out that they indeed are. Quiteunexpectedly, the careful investigation of the CD ( ρ, n ) inequality for this model doesnot produce better results than the direct consequence of the SU (3) inequality. Incomparison with the classical case of Jacobi polynomials, which are orthogonal withrespect to the measure C a,b (1 − x ) a (1 + x ) b dx on ( − , , this situation is similar tothe symmetric case a = b , but differs from the dissymmetric case (see [1]).It seems worth to point out at least two interesting features of the computationsof curvature-dimension inequalities for this model. The first aspect concerns theexistence of an optimal dimension in the inequality. When one looks for curvature-dimension inequalities on a compact Riemannian manifold with dimension n , butfor a reversible measure which is not the Riemann measure, there is no optimal one.For any n > n , one may find some constant ρ ( n ) such that a CD ( ρ ( n ) , n ) inequalityholds. In general, ρ ( n ) goes to −∞ when n → n . It is only for Laplace operatorsthat one may expect some CD ( ρ, n ) inequality. This is not the case here. For any λ > , there is a bound n ( λ ) = 2 λ such that no CD ( ρ, n ) inequality may hold for n < n ( λ ) . However, for this limiting value n = 2 λ , the CD ( ( λ − , λ ) holds. Ofcourse, this phenomenon is due to the singularity of the density of the measure atthe boundaries of it’s support.The second aspect concerns the use of appropriate coordinate systems. Sincethe underlying metric is a flat metric in two dimensions, the curvature-dimensioninequality amounts to check for which values a and b one has an inequality of theform −∇∇ W ≥ a Id + b ∇ W ⊗ ∇ W, where W is the logarithm of the density measure with respect to the Riemannmeasure. It turns out that a proper choice of the coordinates leads to very simpleformulas, which is not the case if the computation is made through the use of thenaive usual coordinates in the Euclidean plane. This comes from the fact that wehave indeed at disposal a polynomial structure, expressed through the choice of thesecoordinates, and the function W satisfies nice relations with respect to this, namelya "boundary equation" described in (4.31). This is a good indication that if onewants to study such inequalities for higher dimensional models, these "polynomialcoordinates" should be used instead of the usual ones.Then, the curvature-dimension inequalities provide through Sobolev inequali-ties (2.6) various uniform bounds on the orthogonal polynomials, which turn out,up to some change in the parameters, to be equivalent to the Sobolev inequalityitself. They also provide bounds on kernels constructed from other spectral decom-positions, that is for operators which do not necessary commute with our startingoperator.Orthogonal polynomials on the interior of the deltoid curve belong to the largefamily of Heckman-Opdam polynomials associated with root systems [12, 13], and n the even larger class of MacDonald’s polynomials [21, 20, 22]. As such, they mayserve as a guide for more general models of diffusions associated with orthogonalpolynomials. One may find some extensive presentation of these pluri-dimensionalfamilies of orthogonal polynomials, for example in [9, 22].The associated diffusion generator is associated with some reflection group indimension 2. Operators associated with reflection groups in R d , known as Dunkloperators, are extensively studied in the literature, from the points of view of specialfunctions related to Lie group analysis or Hecke algebras, as well as from the pointof view of the associated heat equations, or in probability and statistics [8, 15, 14,24, 23].Most of the language and notations related to diffusion operators, and in par-ticular the links between curvature-dimension inequalities and Sobolev inequalities,together with the bounds one may deduce for eigenvectors, are borrowed from [3].The fact that Sobolev inequalities are equivalent to bounds on the heat kernel goesback to [7, 26], and the relations between curvature dimension inequalities go backto [2] and are exposed in [3], among others. It is classical that one may deduce fromthem bounds on the spectral projectors. However, the fact that these bounds may inturn provide Sobolev inequalities, which is the content of Theorem 5.4, seems quitenew, at least to our knowledge, although a similar result concerning logarithmicSobolev inequalities is exposed in [3]. Notice that this recovers a Sobolev inequalitywith a weaker exponent, not to talk about the optimal constants, which are alwaysout of reach with this kind of techniques.Many of the properties concerning the spectral decomposition of the operatoron the deltoid, recurrence formulas for the associated orthogonal polynomials, gen-erating functions, etc., may be found in [27]. We shall not use the results of thispaper here, apart the representations coming from symmetry groups in R and from SU (3) , that we recall briefly in Section 3.The paper is organized as follows. In Section 2, we present the general curvature-dimension inequalities and the associated Sobolev inequalities. We show how thisprovides bounds on the eigenvectors. Section 3 is a short introduction to the modelassociated with the deltoid curve, where we explain the two geometric specific cases.Section 4 provides the associated curvature-dimension inequalities, first from the SU (3) -model, then from direct computation. We provide two approaches for thisgeneral case, first in subsection 4.2 using the naive system of coordinates, and then insubsection 4.3 with the adapted system of coordinates. The choice to present thesetwo approaches, and the striking difference in the complexity of the computation,aims at underlining the efficiency of the good "polynomial coordinates". Finally, inSection 5, we give the various bounds on polynomials and operators we are lookingfor. We briefly recall in this Section the context of symmetric diffusion operators, follow-ing [6], in a specific context adapted to our setting. For a given probability space ( X, X , µ ) , we suppose given an algebra A of functions such that A ⊂ ∩ ≤ p< ∞ L p ( µ ) , is dense in L ( µ ) , and which is stable under composition with smooth functions Φ . In our case, A may be chosen as the class of restrictions to the domain of smoothfunctions defined in a neighborhood of it, without any boundary condition. Thisparticular choice for A is made possible thanks to a special property of the oper-ator, which satisfies a "boundary equation", which is our model has the specificform (4.31). It is valid as soon one deals with operators having polynomial eigen-vectors on a bounded domain, see [4]. In most of the cases, we may as well restrictour attention to polynomials, although this would not be appropriate for Sobolevinequalities. A bilinear application Γ :
A × A 7→ A is given such that, ∀ f ∈ A , Γ( f, f ) ≥ , which satisfies Γ(Φ( f , · · · , f k ) , g ) = P i ∂ i ΦΓ( f i , g ) for any smoothfunction Φ . A linear operator L is defined through(2.1) Z X f L ( g ) dµ = − Z X Γ( f, g ) dµ and we assume that L maps A into A . In this context, all the properties of themodel are described by Γ and µ , and the model is then entirely described by thetriple ( X, Γ , µ ) (see [3]). We then extend L into a self adjoint operator and wesuppose that A is dense in the domain of L .Then, for f = ( f , · · · , f k ) and for any smooth function Φ (2.2) L (Φ( f )) = k X ∂ i Φ( f ) L ( f i ) + k X i,j =1 ∂ ij Φ( f )Γ( f i , f j ) . We have Γ( f, g ) = 12 (cid:16) L ( f g ) − f L ( g ) − g L ( f ) (cid:17) . We moreover define the Γ operator as(2.3) Γ ( f, g ) = 12 (cid:16) L Γ( f, g ) − Γ( f, L g ) − Γ( g, L f ) (cid:17) . Then, for any parameters ρ ∈ R and n ∈ [1 , ∞ ] , we say that L satisfies acurvature-dimension inequality CD ( ρ, n ) if and only if ∀ f ∈ A , Γ ( f, f ) ≥ ρ Γ( f, f ) + 1 n ( L f ) . It is worth to observe that the CD ( ρ, n ) inequality is local. For a general ellipticoperator on a smooth manifold M with dimension n , one may always decompose L into ∆ g + ∇ log V , where ∆ g is the Laplace operator associated with the co-metric ( g ) and V is the density of µ with respect to the Riemann measure. In which case,the operator Γ may be decomposed as(2.4) Γ ( f, f ) = |∇∇ f | + (Ric g − ∇∇ log V )( ∇ f, ∇ f ) , where Ric g denotes the Ricci curvature computed for the Riemannian metric asso-ciated with g , ∇∇ log V is the Hessian of log V , also computed in this metric, and |∇∇ f | is the Hilbert-Schmidt norm of the Hessian of f . n this case, the CD ( ρ, n ) inequality holds if and only if n ≥ n and, when V isnot constant, when n > n and(2.5) Ric g − ∇∇ log V ≥ ρg + 1 n − n ∇ log V ⊗ ∇ log V. Of course, when L = ∆ g , this amounts to n ≥ n and Ric g ≥ ρg . In thiscase, there exists a best choice for both ρ and n , namely for n the dimension of themanifold and for ρ a lower bound on the Ricci tensor, that is the infimum over M of the lowest eigenvalue of this tensor.In this paper, we are mainly mainly interested the case where L = ∆ + ∇ log( V ) ,where ∆ is the Euclidean Laplace operator in some open set of R n . In which casethe measure µ is V dx , and the CD ( ρ, n ) inequality holds if and only if n ≥ n and −∇∇ log( V ) ≥ ρ + 1 n − n ∇ log( V ) ⊗ ∇ log( V ) . In order for it to be satisfied, we may look at local inequalities CD ( ρ ( x ) , n ( x )) and try to find such a pair ( ρ ( x ) , n ( x )) for which ρ ( x ) is bounded below and n ( x ) bounded above. For a generic function V , there is no "best" local inequality ingeneral. The CD ( ρ, n ) inequality requires that the symmetric tensor −∇∇ log V isbounded below by ρ Id . If ρ is the best real number such that −∇∇ log V ≥ ρ Id (that is ρ is the lowest eigenvalue at the point x of −∇∇ log V ), then the inequalityholds as soon as ρ ≥ ρ and ( ρ − ρ )( n − n ) ≥ |∇ log V | .But in our case, as already mentioned in the introduction, we are not in thissituation. There is a lower bound on the admissible dimension, which is strictlybigger than n . To understand this phenomenon, one may analyse a bit further this CD ( ρ, n ) inequality at a given point on the manifold.It may happen that, at some point x , the eigenvector of −∇∇ log V correspondingto some eigenvalue ρ ( x ) > ρ ( x ) is parallel to ∇ log V . Let then in this case, thereis a best choice for both ρ ( x ) and n ( x ) , which is ( ρ ( x ) = ρ ( x ) ,n ( x ) = n + ρ ( x ) − ρ ( x ) |∇ log V | . In the model that we shall consider later, we shall see that this happens asymp-totically on the boundary of the set we are working on, and the constants n and ρ computed at this boundary points are valid for all other points x .When some CD ( ρ, n ) inequality holds, with ρ and n constant, and whenever ρ > , and < n < ∞ , then ( X, Γ , µ ) satisfies a tight Sobolev inequality. For p = nn − , and for any f ∈ A , we have(2.6) (cid:16) Z X f p dµ (cid:17) /p ≤ Z X f dµ + 4 n ( n − n − ρ Z X Γ( f, f ) dµ. More generally, an n - dimensional Sobolev inequality is an inequality of the form(2.7) k f k n/ ( n − ≤ A k f k + C Z Γ( f, f ) dµ. hen µ is a probability measure, we say that the inequality is tight when theconstant A is , and provided some Sobolev inequality holds, tightness is equivalentto the fact that a Poincaré inequality occurs, which is automatic in our case sincethe spectrum is discrete (see [3]).When a Sobolev inequality (2.7) holds, then the associated semigroup P t =exp( t L ) is ultracontractive, that is, for any q ∈ [2 , ∞ ] (2.8) k P t f k q ≤ C t n ( − q ) k f k , < t ≤ , with C = (cid:16) Cn A (cid:17) n/ . This last constant C is not sharp however. The bound is valid for q = ∞ andindeed, the result for any q ∈ (2 , ∞ ) is a consequence of the case q = ∞ through aninterpolation argument. It turns out that his last ultra contractive bound (for somegiven q , but for any t ∈ (0 , ) is in turn equivalent to the Sobolev inequality.There is another equivalent version(2.9) k P t f k ∞ ≤ Ct − n/ k f k , < t ≤ , for which one deduces immediately that the semigroup P t has a density which isbounded above by Ct − n/ .When ≤ n ≤ , one may replace Sobolev inequalities by Nash inequalities,which play the same rôle, see remark 5.6. However, the best constants that one maydeduce from curvature-dimension inequalities for Nash inequalities are not known(see [6]).As a consequence of ultracontractive bounds, whenever f is an eigenvector for L with eigenvalue − λ , and provided that R f dµ = 1 , one has(2.10) k f k q ≤ C inf t> e λt t n ( − q ) = C C n,q λ n ( − q ) , with C n,q = inf s> e s s − n ( − q ) , which follows immediately from the fact that P t f = e − λt f . This applies in particular for q = ∞ , and produces uniform bounds on theeigenvectors from the knowledge of their L ( µ ) norms. We describe first the operator associated with the deltoid curve associated with afamily of orthogonal polynomials. Most of the details may be found in [27]. Thedeltoid curve is a degree 4 algebraic plane curve which may be parametrized as x ( t ) = 2 cos t + cos 2 t, y ( t ) = 2 sin t − sin 2 t igure 1: The deltoid domain. The connected component of the complementary of the curve which contains is a bounded open set, that we refer to as the deltoid domain D . Indeed, in whatfollows, we shall work on this domain scaled by the factor / , which will producemuch more convenient formulas. It turns out that there exist on this domain a oneparameter family L ( λ ) of symmetric diffusion operator which may be diagonalized ina basis of orthogonal polynomials. It was introduced in [17, 18], and further studiedin [27]. This is one of the 11 families of sets in dimension 2 carrying such diffusionoperators, as described in [5].In order to describe the operator, and thanks to the diffusion property (2.2), itis enough to describe Γ( x, x ) , Γ( x, y ) , Γ( y, y ) , L ( λ ) ( x ) and L ( λ ) ( y ) (the Γ operatordoes not depend on λ here).The symmetric matrix (cid:18) Γ( x, x ) Γ( x, y )Γ( y, x ) Γ( y, y ) (cid:19) is referred to in what follows as themetric associated with the operator, although properly speaking it is in fact a co-metric. It is indeed easier to use the complex structure of R ≃ C , and use thecomplex variables Z = x + iy , ¯ Z = x − iy , with L ( Z ) = ( x ) + i L ( y ) , L ( ¯ Z ) = L ( x ) − i L ( y ) , and Γ( Z, Z ) = Γ( x, x ) − Γ( y, y ) + 2 i Γ( x, y ) , Γ( Z, ¯ Z ) = Γ( x, y ) + Γ( y, y ) . he formulas are much simpler with these variables, and L ( λ ) is then described as(3.11) Γ( Z, Z ) = ¯ Z − Z , Γ( ¯ Z, ¯ Z ) = Z − ¯ Z , Γ( ¯
Z, Z ) = 1 / − Z ¯ Z ) , L ( Z ) = − λZ, L ( ¯ Z ) = − λ ¯ Z, where λ > is a real parameter.The boundary of this domain turns out to the curve with equation(3.12) P ( Z, ¯ Z ) = Γ( Z, ¯ Z ) − Γ( Z, Z )Γ( ¯ Z, ¯ Z ) = 0 , and inside the domain, the associated metric is positive definite, so that it corre-sponds to some elliptic operator on it. The reversible measure associated with it hasdensity C α P ( Z, ¯ Z ) α with respect to the Lebsegue measure, where λ = (6 α + 5) / ,and is a probability measure exactly when λ > (we refer to [27] for more details).There are two particular cases which are worth understanding, namely λ = 1 and λ = 4 , corresponding to the parameters α = ± / . We briefly present thosetwo models, referring to [27] for more details, since we shall make a strong use ofthem in what follows.In the first case, one sees that this operator is nothing else that the image of theEuclidean Laplace operator on R acting on the functions which are invariant underthe symmetries around the lines of a regular triangular lattice.For the first one, one considers the three unit third roots of identity in C , say ( e , e , e ) = (1 , j, ¯ j ) . Then, consider the functions Z and z k : C C which aredefined as(3.13) z k ( z ) = e i ℜ ( z ¯ e k ) , Z = 13 ( z + z + z ) They satisfy | z k | = 1 and z z z = 1 .It is easily seen that, for the Euclidean Laplace operator on R , Z and ¯ Z satisfythe relations (3.11) with λ = 1 . Moreover, the function Z : C C is a diffeomor-phism between the interior T of the triangle T and the deltoid domain D , where T isone of the equilateral triangles with containing the two edges and π/ . The func-tions which are invariant under the symmetries of the triangular lattice generatedby this triangle T are exactly functions of Z . Therefore, the image of L (1) through Z − : D 7→ T is nothing else that the Laplace operator on T . We may as well lookat the image of the operator L ( λ ) and it is then simply(3.14) L ( λ ) = ∆( f ) + ( α + 1 / ∇ log W ∇ f = ∆( f ) + λ − ∇ log W ∇ f where ∆ is the usual Laplace operator in R and the function W is defined form thefunctions z j described in equation (3.13) as(3.15) W = − ( z − z ) ( z − z ) ( z − z ) . ne should be aware here that thanks to the properties of the functions z j , − ( z − z ) ( z − z ) ( z − z ) is a real valued function taking values in (0 , ∞ ) (andvanishes only at the boundaries of T ).This representation provides a way of computing CD ( ρ, n ) inequalities for L ( λ ) ,following the description of Section 2.The second description follows from the Casimir operator on SU (3) . This lat-ter group is a semi-simple compact Lie group, and as such as a canonical Laplace(Casimir) operator which commutes (both from left and right) to the group ac-tion [16, 10] . Namely, in any such semi simple compact Lie group G , one considersit’s Lie algebra G , naturally endowed with a Lie algebra structure G × G 7→ G , ( X, Y ) [ X, Y ] . The Lie algebra structure provides on G a natural quadraticform K (the Killing form) as follows : for any element X ∈ G , one considers ad ( X ) : G 7→ G , Y [ X, Y ] , and K ( X, Y ) = − trace (cid:16) ad ( X ) ad ( Y ) (cid:17) . It turnsout that this quadratic form is positive definite exactly when the group is compactand semi-simple. If one considers, for this Killing form, any orthonormal basis ( X i ) in the Lie algebra, the quantity P i X i , computed in the enveloping algebra, does notdepend on the choice of the basis, and commutes with any element on the Lie algebraitself (this means that this commutation property depends only on the Lie-algebrastructure and not on the way that the elements of this Lie algebra are effectivelyrepresented as linear operators).Now, to any X ∈ G is associated a first order operator D X on G defined asfollows(3.16) D X ( f )( g ) = ∂ t f ( ge tX ) | t =0 . The application X D X is a representation of the Lie algebra into the linear spaceof vector fields ( [ D X , D Y ] = D [ X,Y ] ), and any identity in the Lie algebra (on onit’s enveloping algebra) translates to an identity on those differential operators. Wework here with the right action g ge tX but we could as well work with the leftaction g e tX g . For any orthonormal basis X i of G for the Killing form K , onedefines the Casimir operator(3.17) L = X i D X i . It does not depend of the choice of the basis and commutes with the group action,that is [ L , D X ] = 0 for any X ∈ G .This Killing form provides an Euclidean quadratic form in the tangent plane atidentity in G (the Lie algebra G ), which may be translated to the tangent plane atany point g ∈ G through the group action, and endows G with a natural Riemanianstructure. It turns out that the Casimir operator L is also the Laplace operator forthis structure. For the group SU ( d ) that we are interested in, one wants to preciselydescribe the action of this Casimir operator on the entries of the matrix g = ( z ij ) in SU ( d ) . That is, writing the entries z pq = x pq + iy pq , we consider x pq and y pq asfunctions G R , and, for any i, j, k, l , we want to compute L SU ( d ) ( x ij ) , L SU ( d ) ( y ij ) , Γ SU ( d ) ( x ij , x k,l ) , Γ SU ( d ) ( x ij , y k,l ) , Γ SU ( d ) ( y ij , x k,l ) , here Γ SU ( d ) is the square field operator associated with L SU ( d ) . In order to getsimpler formulae, it is once again better to work with the complex valued functions z pq , writing for such a function z = x + iy , L SU ( d ) ( z ) = L SU ( d ) ( x ) + iL SU ( d ) ( y ) , Γ SU ( d ) ( z, z ) = Γ( x, x ) − Γ( y, y ) + 2 i Γ( x, y ) , Γ SU ( d ) ( z, ¯ z ) = Γ( x, x ) + Γ( y, y ) . If one denotes by E p,q the matrix with entries ( E p,q ) i,j = δ ip δ jq , a base of the Liealgebra of SU ( d ) is given by R k,l = ( E k,l − E l,k ) k 94 Id + 16 ∇ log W ⊗ ∇ log W. For λ > , multiplying the previous inequality by λ − , this in turns gives for thegeneral case provides Corollary 4.3. For any λ ≥ , L ( λ ) satisfies a CD ( λ − , λ ) inequality. Observe that indeed the limiting case λ = 1 corresponds to the Laplace operatoron R which satisfies a CD (0 , inequality.It is not clear however that this inequality is sharp. Indeed, going from the CD inequality on SU (3) to the same CD inequality for L (4) , we may as well have lostinformation. In general, as we already mentioned, on a smooth compact manifold(with no boundaries) with dimension n , there is no optimal CD ( ρ, n ) inequality,and for any n > n one may find some ρ ( n ) such that the operator ∆ + ∇ log V satisfies a CD ( ρ, n ) inequality. Moreover, this does not tell us anything about thecase where λ < .As mentioned in the introduction, the optimal computation for this inequalityfor a generic λ is not elementary. In this section, we shall perform directly thecomputations of ∇∇ log W and ∇ log W ⊗ ∇ log W to observe that the SU (3) caseindeed gives the optimal answer, which is quite surprising. Of course, on SU (3) the CD ( ρ, n ) inequality is optimal at every point x ∈ SU (3) , while on the projectedmodel, it is optimal only at some point on the boundary of the deltoid domain D .We shall show the following roposition 4.4. 1. For λ < , the operator L ( λ ) does not satisfy any CD ( ρ, ∞ ) inequality.2. For λ > , the operator satisfy no CD ( ρ, n ) inequality for any n < λ . More-over, the best constant ρ in the CD ( ρ, λ ) inequality is ρ = λ − .Proof. — Everything boils down, for λ > , to check inequalities of the form −∇∇ log W ≥ c Id + c ∇ log W ⊗ ∇ log W, and for λ < to check the the tensor ∇∇ log W is not bounded below.To perform the computations, we shall use the triangle model, that is move backeverything of T through the map Z − , since on T the metric is the identity and theHessian is computed in the usual way.In what follows, we shall use the functions log( z ) for a complex variable z = 0 without any precaution about which determination of the argument we chose for thelogarithm, since indeed we are only concerned with the one form d log( z ) and it’sderivative.Let us recall that W = − ( z − z ) ( z − z ) ( z − z ) , with z k = e i ( E k .z ) , where(4.24) E = (cid:18) (cid:19) , E = − √ ! , E = − − √ ! . Then, setting σ = log( z − z ) + log( z − z ) + log( z − z ) , log W = 2 σ , up tosome additive (eventually complex) constant, and we are looking for(4.25) − ∇∇ σ ≥ bI + a ∇ σ ⊗ ∇ σ. What we want to show is first that ∇∇ σ is not bounded above, and then thatthe former inequality (4.25) may not hold if a > / . Moreover, we want to checkthat for a = 1 / , the best lower bound for b is b = 9 / . It turns out that this isquite technical.We shall need a few intermediate steps to check this inequality. Lemma 4.5. We have ∇ z k = iz k E k , ∇ log( z p − z q ) = i z p E p − z q E q z p − z q , ∇∇ log( z p − z q ) = z p z q ( z p − z q ) ( E p − E q ) ⊙ . Proof. — (Of Lemma 4.5)The two first identities are immediate. For the last one, we write ∇∇ log( z p − z q ) = i ∇ ( 1 z p − z q ( z p E p − z q E q ))= − z p − z q ( z p E ⊗ p − z q E ⊗ q ) + 1( z p − z q ) ( z p E p − z q E q ) ⊗ As a consequence, one has orollary 4.6. (4.26) ∇∇ log σ = [ z z ( z − z ) V ⊙ + z z ( z − z ) V ⊙ + z z ( z − z ) V ⊙ ] . (4.27) ∇ log σ ⊗ ∇ log σ = − [ z z − z V + z z − z V + z z − z V ] ⊙ . where V = E − E , V = E − E , V = E − E and E i , i = 1 , , are defined in (4.24) . Remark 4.7. Observe that | V i | = 3 and that the complex valued vector U = z z − z V + z z − z V + z z − z V , has purely imaginary components: thanks to the fact that V + V + V = 0 , andusing ¯ z i = z − i , one sees that ¯ U = − U . Therefore − U ⊗ U ≥ .On the other hand, the tensor z z ( z − z ) V ⊙ + z z ( z − z ) V ⊙ + z z ( z − z ) V ⊙ is real.Proof. — (Of Corollary 4.6). Equation (4.26) is a direct consequence of Lemma 4.5,while (4.27) follows from ∇ σ ⊗ ∇ σ = − [ z E − z E z − z + z E − z E z − z + z E − z E z − z ] ⊗ = − [ E + z z − z ( E − E ) + E + z z − z ( E − E ) + E + z z − z ( E − E )] ⊗ and the fact that E + E + E = 0 .First, observe that when z = e iφ z with φ → , with z = 1 , j, ¯ j , the tensor ∇∇ σ is equivalent to − φ/ V ⊗ V , and therefore is not bounded below. Thisshows that there cannot exist any CD ( ρ, ∞ ) inequality for λ < .From now on, the parameter a ∈ R being fixed, let us call b ( a ) the best constant b in inequality (4.25) at a given point in the interior of the triangle. It is obtained thelowest eigenvalue of the symmetric tensor ∇∇ σ − a ∇ σ ⊗∇ σ . We want to understandwhen this function is bounded below.It is then better to change coordinates and consider a basis W = V / √ and W which is orthogonal to W and norm , such that V = √ − W + √ W ) , V = √ − W − √ W ) . Moreover, we shall set z = z u , | u | = 1 , so that z = 1 / ( z u ) , and z = z .Observe that the image of ( x, y ) ( z, u ) , is S , where S is the set complex numberswith modulus 1. ith those notations, inequality (4.25) becomes AW ⊙ + BW ⊙ + CW ⊙ W − u ) (1 − zu ) (1 − zu ) ≥ b ( W ⊙ + W ⊙ ) , where A = 3 (cid:16)(cid:0) a ( u + 1) + (2 a − u (cid:1) (1 + u z ) − uz ( u + 1) (cid:0) a ( u + 1) + u − u + 1 (cid:1) (1 + z u )+2 u z (cid:0) a ( u + 1) + a ( u + u ) + (6 a − u (cid:1)(cid:17) B = 9( u − (cid:16) a ( u z + 1) − ( u z + uz ) − ( u z + u z ) + ( − a + 4) u z (cid:17) C = 6 √ u − − z u ) (cid:0) a ( u z + 1) + a ( u z + u ) + (1 − a )( u z + uz ) − u z (cid:1) Setting z = e iθ and u = e iφ we have A = 12 u z (cid:16) θ + 3 φ )( a cos φ/ − − φ/ 2) cos( θ + 3 / φ )((4 a + 1) cos φ − a cos(2 φ ) + a cos( φ ) + 3( a − (cid:17) B = − u z (sin ( φ/ (cid:16) a cos(2 θ + 3 φ ) − cos( θ + 2 φ ) − cos( θ + φ ) + (2 − a ) (cid:17) C = 48 √ z u sin( φ/ 2) sin( θ + 3 / φ ) (cid:16) a cos( θ + 2 φ ) + a cos( θ + φ ) + (1 − a ) cos( φ ) − (cid:17) while the denominator may be written as − z u sin ( φ/ 2) sin ( θ/ φ/ 2) sin ( θ/ φ ) . Simplfying everything by u z , and letting A = − Az − u − , B = − Bz − u − , C = Cz − u − , and N = 4 sin ( φ/ 2) sin ( θ/ φ/ 2) sin ( θ/ φ ) , we see that thebest constant b ( a ) , at some point ( z, u ) is then b ( a ) = A + B − p ( A − B ) + C N . We now may prove the following Lemma 4.8. The function b ( a ) is unbounded below on the set | z | = | u | = 1 if a > / . roof. — (Of Lemma 4.8) We shall see in Lemma 4.10 that the function b ( a ) isbounded below for a = 1 / . This of course shows that it is also bounded below forany a < / . What we have to prove then is that the function b ( a ) is unboundedbelow when a > / .Let us concentrate on the case a < . It is enough to observe the asymptotics of b ( a ) around θ = φ = 0 . The result is obtained when choosing φ = λθ . Then, onehas A ≃ − a ) θ = αθ B ≃ λ (1 − a ) θ = βθ C ≃ − √ λaθ = γθ Then, b ( a ) ≃ c/θ , where the constant c has the sign of βα − γ , that is of (1 − a ) . When a > / , this converges to −∞ when θ → . Remark 4.9. When choosing in the previous argument φ = λθ , one sees that b ( a ) is unbounded below as soon as a > / . This asymptotics is not enough to capturethe optimal bound. We now concentrate on the case a = 1 / . We are able to compute explicitly thelower bound for a = 1 / , which corresponds and fits with the SU (3) computation,although the explicit computation of the lower bound is not explicit (and not reallyof interest) for the other values of a < / .We will study the function b ( a ) in case a = . Lemma 4.10. The function b (1 / is bounded below and it’s lower bound is / Proof. — (Of Lemma 4.10)In the case a = 1 / the function b (1 / have the following form in (z,u): b (1 / 3) = 12 1( u − ( zu − ( zu − (cid:16) P ( z, u ) − p Q ( z, u ) (cid:17) where P ( z, u ) = ( u − u + 1)(1 + u z ) − zu ( u + 1)( u − u + 1)(1 + z u )+ u z ( u + 8 u − u + 8 u + 1) .Q ( z, u ) = [( z u − zu − zu + u − u + 1)( z u − z u + z u − zu − zu + 1)( z u + z u + zu − zu + zu + u + 1) ] . Using the same notations than in Lemma 4.8, b (1 / can be written also in ( θ, φ ) in this form(4.28) b (1 / 3) = 12 N ( θ, φ ) D ( θ, φ ) here N ( θ, φ ) = 2 (cid:16) θ + 3 φ )(cos( φ ) − − φ/ 2) cos( θ + 3 / φ )(2 cos( φ ) − φ ) + 8 cos( φ ) − − | T || φ/ 2) cos( θ + 3 / φ ) + cos( φ ) − | (cid:17) D ( θ, φ ) = − sin ( φ/ 2) sin ( θ/ φ ) sin ( θ/ φ/ where T = z u − φ zu + 2 cos( φ ) − and | T | = √ T ¯ T Setting x = cos( φ/ , y = cos( θ + 3 / φ ) , and y = x + w we may rewrite this as(4.29) b (1 / 3) = 14 (2(1 − x ) − xw ) + 3 w (1 − x ) − (2(1 − x ) − xw ) p (2(1 − x ) − xw ) + 3 w ( x − − x ) w To see this, we just replacing in (4.28) cos( φ ) = 2 x − , cos(2 φ ) = 8 x − x + 1 , cos(2 θ + 3 φ ) = 2 y − ( φ − x , sin( φ + θ θ + φ x − y ) we obtain N ( x, y ) = 8 (cid:16) x − x y + 2 x y + x + 10 xy − y − −| x + xy − | p x + 4 x y − x y − y − x + 2 xy + 4 (cid:17) D ( x, y ) = − (1 − x )( x − y ) then if we set y = x + w we have the resultNow, in formula (4.29), we set t = | − x ) − xw ) w √ − x | , and then b (1 / becomes b (1 / 3) = 14 ( t + 3 − t p t − , t ≥ √ . It is an easy exercise to check that the lower bound of this last function of t is / .We now collect the results of Lemmas 4.8 and 4.10 to get Proposition 4.4. .3 A simpler proof of the curvature-dimension inequal-ity As mentioned in the introduction, we shall show that the use of the complex coor-dinates ( Z, ¯ Z ) provide a much simpler proof of Proposition 4.4. Everything relieson the boundary equation (4.31), which is nothing else than a particular case of ageneral equation which is valid as soon as orthogonal polynomials come into play(see [4]). In the coordinates ( Z, ¯ Z ) ∈ D , the operator L ( λ ) takes a simpler form,even if the metric looks more complicated. This illustrates the use of the appro-priate coordinates whenever one has a polynomial structure such as this deltoidmodel.Once again, our aim is to compute the Hessian of the function log P , where P isdefined in equation (3.12).Following [3], the Hessian of f , applied to dh, dk , that is in a local system ofcoordinates ∇∇ ij ( f ) ∂ i h∂ j k , may be defined as(4.30) H [ f ]( h, k ) = 12 (cid:0) Γ( h, Γ( f, k )) + Γ( k, Γ( f, h )) − Γ( f, Γ( k, h ) (cid:1) . We want to apply this with f = log P and h, k = Z, ¯ Z . For this, one may use theboundary equation which takes in this context the particular form(4.31) Γ( Z, log( P )) = − Z, Γ( Z, log P ) = − Z, and is easily checked from formulae (3.11)From this, we deduce that, for any function G ( Z, Z ) , Γ(log P, G ) = − D ( G ) ,where D is the Euler operator Z∂ Z + Z∂ Z Let us write H = H [log P ]( Z, Z ) , H = H [log P ]( Z, Z ) , H = H [log P ]( Z, Z ) . From the previous remarks, we get H = − Z, Z ) + 32 D (Γ( Z, Z )) .H = − Z, Z ) + 32 D (Γ( Z, Z )) , and H = − Z, Z ) + 32 D (Γ( Z, Z )) . In other words, with the obvious notations, H = − 3Γ + D Γ In the same way, the tensor ∇ log P ⊗ ∇ log P may be computed in this systemof coordinates as M = 9 Z ZZZZ Z ! . In the end, the inequality(4.32) − ∇∇ log P ≥ b Γ + a ∇ log P ⊗ ∇ log P mounts to (3 − b )Γ − D Γ − a M ≥ , where Γ denotes the matrix (cid:18) Γ( Z, Z ) Γ( Z, Z )Γ( Z, Z ) Γ( Z, Z ) (cid:19) . For such a tensor ( R ij ) in complex coordinates, to represent a non negative realtensor amounts to ask that R ≥ and ( R ) ≥ R R .R ≥ reads − b + ( b − a ) ZZ ≥ , and for this to be true on Ω amountsto ask b ≤ , a ≤ / , since ZZ varies from to on Ω .The second one writes(4.33) [(3 − b ) / b / − a ZZ ] ≥ (3 / − b ) ZZ +( b − a ) Z Z +( Z + Z )( b − a )(3 / − b ) . Writing everything in polar coordinates Z = ρe iθ , this writes as P ( ρ ) ≥ ρ cos(3 θ )( b − a )(3 / − b ) , where P is a degree 2 polynomial.Observe that this requires to be true for any ρ ∈ [0 , when cos(3 θ ) = 0 (whichcorresponds to the cusps of the deltoid curve).But, with the explicit computation of P , one gets P ( ρ ) − ρ ( b − a )(3 / − b ) = 14 (1 − ρ )(3 − b + b ρ ) (cid:0) − b + ρ (3 − b )+3 ρ ( b − a ) (cid:1) . For the maximal value a = 1 / , P ( ρ ) − ρ ( b − a )(3 / − b ) = (1 − ρ ) (3 − b + 3 ρ (2 − b )) , and we get a bound b ≤ / .For these values a = 1 / and b = 9 / , equation (4.33) writes − ρ ) ≥ ρ + ρ − ρ cos(3 θ ) , while the condition Γ( Z, Z ) ≥ Γ( Z, Z )Γ( Z, Z ) , which characterizes the points in ¯ D ,writes 14 (1 − ρ ) ≥ ρ + ρ − ρ cos(3 θ ) , so the the inequality is satisfied everywhere in D . Observe that the critical points forthe curvature-dimension inequality for the critical values are attained at the cusps. emark 4.11. Observe that the values b = 9 / and a = 1 / in inequality (4.32) are once again exactly the bounds obtained in equation (4.23) . Moreover, we knowthat even with a = 0 (corresponding to a CD ( ρ, ∞ ) inequality), if we look forthe optimal value for b , it is clear from this method that the best constant b is b max < , so that whatever the constant a ∈ [0 , / , the optimal value for b liesin the interval [9 / , . (The optimal constant b ( a ) may be explicitly computed buthas no real interest.) As described in Section 2, from the curvature dimension inequality, we may obtainbounds on the supremum of the associated eigenvectors. More precisely, whenevera CD ( ρ, n ) inequality holds with ρ > and n < ∞ , there exists a constant C suchthat for any eigenvector P satisfying L ( P ) = − µP , then k P k ∞ ≤ Cµ n/ k P k , where the L norm is computed with respect to the invariant measure of the operator L . Turning to the case of the operator L ( λ ) on the deltoid, we recall from [27]that the associated eigenvectors which are polynomials with total degree n haveeigenvalues µ p,q = ( λ − p + q ) + p + q + pq , with p + q = n . More precisely,of any n ≥ such that p + q = n , when p = q , there is a dimension associatedeigenspace. In complex variables, for such value µ p,q , there is a unique degree n polynomial P p,q ( Z, ¯ Z ) with highest degree term Z p ¯ Z q and another one which is ¯ P p,q ( Z, ¯ Z ) = P p,q ( ¯ Z, Z ) = P q,p ( Z, ¯ Z ) eigenvector (the polynomial P p,q having realcoefficients). For p = q however, the associated eigenspace is one dimensional. Thereal forms are S p,q = ( P p,q + P q,p ) and A p,q = − i ( P p,q − P q,p ) , which form a realbasis for this eigenspace.When λ > , for L ( λ ) , for any µ p,q and of any polynomial P in the associatedeigenspace, one gets from the CD ( λ − , λ ) inequality(5.34) k P k ∞ ≤ C ( λ ) µ λ/ . Looking at the constants, this does not produce any estimates for λ = 1 or < λ < . However, for λ = 1 , one may consider the following. The operator L (1) is nothing else than the usual Laplace operator acting on functions f ( Z ) , where thefunction Z is given in (3.13). As functions of ( x, y ) in the real plane, those functionsare periodic in x with period π and in y with period π/ √ . As such, the associatedsemigroup P (1) t is an image of the product semigroup of the associated dimensionaltorus (that is the semigroup on the real line acting on periodic functions). Moreprecisely, when considering a function on the deltoid as a function of ( x, y ) ∈ R , P (1) t (( x, y, dx ′ , dy ′ ) = P S (4 π ) t ( x, dx ′ ) P S (4 π/ √ t ( y, dy ′ ) , here P S ( τ ) t ( x, dx ′ ) is the semigroup of the torus with radius τ , that is the semigroupof the one-dimensional Brownian motion acing on τ -periodic functions. Since bothsemigroups have a density which is bounded above by C/ √ t for some constant C and for < t ≤ , it turns out that P (1) t has a density which is bounded above by C ′ /t . This is enough to get the bound on the associated eigenvectors. In the end,we get Proposition 5.1. For any λ ≥ , there exists a constant C ( λ ) depending on λ only,such that for any polynomial P eigenvector of L ( λ ) with eigenvalue µ = 0 , one has (5.35) k P k ∞ ≤ C ( λ ) µ λ/ k f k . Remark 5.2. Looking at the constants, whenever λ → , the constant C ( λ ) in (5.35) goes to ∞ , and there is an unexpected discontinuity in the constants. Indeed, ourcomputations are not the best possible. One may sharpen them with the help ofspectral gaps, that is the knowledge of the lowest non eigenvalue, which here is λ .More precisely, one may reinforce the constants in the ultracontractive bounds undera CD ( ρ, n ) inequality and the knowledge of this lowest eigenvalue (see remark page313 in [6]). But the argument in this modified estimate produces a Sobolev inequalitywith any dimensional parameter m > n , and a constant which is not improved when m → n . There is therefore a balance in the optimal bound on P between the value of µ (for µ large one wants m to be the lowest possible), and for λ → (when λ → , onewants C ( λ ) not too big). We could such have produced a better bound. But indeed,the remark in [6] as it stands is not really valid for ρ = 0 which corresponds in ourcase to λ = 1 . One would have to sharpen this estimate, both for the case ρ = 0 andfor the value of m . It is indeed true that one may obtain a n -dimensional Sobolevinequality (under it’s entropic form) under a estimate on the lowest eigenvalue andsome CD ( ρ, n ) inequality, even for ρ < , but the argument in [6] is clearly notsufficient for that and requires further analysis. From the point of view of the invariant measure µ ( λ ) of L ( λ ) , what is relevant isthe decomposition of L ( µ ( λ ) ) into spaces of orthogonal polynomials. More precisely,when denoting P k the space of polynomials with total degree less than or equal to k , one considers the subspace H k of P k which is orthogonal to P k , such that one hasthe orthogonal decomposition L (cid:0) µ ( λ ) (cid:1) = ⊕ ∞ k =1 H k , where H is the space of constant functions.One has Proposition 5.3. There exists a constant C ( λ ) such that, for any k ≥ and any P ∈ H k (5.36) k P k ∞ ≤ C ( λ ) k λ +1 / k P k . Proof. — One may decompose H k into the eigenspaces associated to L ( λ ) . Thereare r k = [ k/ 2] + 1 such eigenspaces, and all the eigenvalues belong to the interval [ k ( λ + k − / , k ( λ + k − , or, when k ≥ , in the interval [3 / k , λk ] . riting P ∈ H k as P = P r k i =1 a i P i where P i is an eigenvector with k P i k = 1 and k P k = P r k a i , one has from (5.35) and the bound on the eigenvalues in H k k P k ∞ ≤ r k X | a i |k P i k ∞ ≤ C ( λ ) λ (1+ λ/ r k X | a i | k λ ≤ C ( λ ) λ (1+ λ/ k λ r / k vuut r k X a i , from which the bound follows immediately.One may wonder how far inequality (5.36) is from the Sobolev inequality westarted from. Observe first that, for the heat kernel P ( λ ) t , for any function P ∈ H k ,one has kP ( λ ) t P k ≤ exp( − / tk ) k P k , since all eigenvalues of P t on H k are bounded below by exp( − / tk ) .Therefore, we have, for any P ∈ H k kP ( λ ) t P k ∞ ≤ C ( λ ) exp( − / tk ) k λ +1 / k P k . Observe that this relies only on the bound (5.35) together with the knowledge of theeigenvalues. Theorem 5.4. Let P t be a symmetric Markov semigroup with reversible probabilitymeasure µ and generator L . Assume that L satisfies a Poincaré inequality and thatone has a decomposition into orthogonal spaces L ( µ ) = ⊕ k H k , where H k is a linearspace, with the property that, for some real number a > and for any f ∈ H k , k P t f k ∞ ≤ Ck p e − atk k f k . Then, L satisfies a tight Sobolev inequality with dimension m = 2 p + 1 .Proof. — Following [6], and from the existence of a Poincaré inequality, it is enoughto prove that, for t ∈ (0 , and for some constant C , k P t f k ∞ ≤ Ct − m/ k f k . Wemay restrict to the case where k f k = 1 . For f ∈ L ( µ ) , let us write f = P k f k ,where f k ∈ H k and P k k f k k = 1 . k P t f k ∞ ≤ X k k P t f k k ∞ ≤ X k k p e − atk k f k k ≤ (cid:16) X k k p e − atk (cid:17) / . One may compare the sum P k k p e − atk with R ∞ x p exp( − atx ) dx , where thefunction x p exp( − atx ) is increasing on (0 , p p/ (2 at ) and decreasing on ( p p/ (2 at ) , ∞ ) ,and we see that, for < t ≤ , X k k p e − atk ≤ C ( a, p ) t − ( p +1) / , < t ≤ . herefore, following the results exposed in Section 2, we get a Sobolev inequalitywith dimension m = 2( p + 1) . The existence of a Poincaré inequality (that is of astrictly positive first non zero eigenvalue for −L ) insures that we may get a tightSobolev inequality (2.7). This gives the result.Looking at the values for the deltoid model, we see that the estimate provides aSobolev inequality with dimension m = 2 λ + 3 , whereas we started from a Sobolevinequality with dimension λ . One may wonder if this lost in dimension (from n to n + 3 ) is due to too crude estimates on both the eigenvalues and the summation inthe series, or from the fact that the spaces H k are k dimensional. Indeed, even in thecase or one dimensional Jacobi operators, where the eigenspaces are one dimensional,where the eigenvalues for the associated operator are k ( k + c ) for polynomials withdegree k , one would pass with the same method from a Sobolev inequality withdimension n to a Sobolev inequality with dimension n + 1 . This is in big contrastwith the case of logarithmic Sobolev inequalities, where estimates on the L p boundson the eigenvectors are indeed equivalent to logarithmic Sobolev inequalities (see [3]).Finally, we directly get from this a criterium for a symmetric operator constructedfrom orthogonal polynomial would have a bounded density. Proposition 5.5. Let K be a symmetric operator in L ( µ ( λ ) ) which maps H k into H k and is such that, for any P ∈ H k , k K ( P ) k ≤ ν k k P k . If A = P k ν k k λ +1 < ∞ ,then K may be represented by a bounded kernel, K ( f )( x ) = Z f ( y ) k ( x, y ) dµ ( y ) , where | k | ≤ A .Proof. — Arguing as in the proof of Theorem (5.4), we may write f = P k f k with f k ∈ H k . Then, k K ( f ) k ∞ ≤ X k k K ( f k ) k ∞ ≤ X k k λ/ / k K ( f ) k ≤ X k ν k k λ/ / k f k k ≤ ( X k ν k k λ/ / ) / k f k . Therefore, the operator K is bounded from L into L ∞ with norm A / . By sym-metry and duality, the same is true from L into L , and by composition, K isbounded from L into L ∞ with norm A . It therefore may be represented by a kernel k bounded by A . Remark 5.6. The method presented here says nothing about the case where <λ < . Indeed, in this case, one may expect to have a two-dimensional behavior forthe heat kernel, that is k P k , ∞ ≤ Ct − / , < t ≤ . In this context, it is better toreplace Sobolev inequalities by Nash inequalities, that is inequalities of the form k f k ≤ k f k θ (cid:16) k f k + C Z Γ( f, f ) dµ (cid:17) − θ , here θ = n +2 is a dimensional parameter. When n > , this is equivalent to aSobolev inequality with dimension n , but for n ∈ (1 , , this is still equivalent toa bound k P t k , ∞ ≤ C ′ t − n/ (see [3]). As mentioned, we may expect when <λ < some Nash inequality with dimensional parameter n = 2 . We cannot expectany smaller value for n since, applied to any function compactly supported in theinterior of D , this would contradict the classical two dimensional Nash inequality inan open domain of R . However, the singularity of the measure at the cusps of thedeltoid make things a bit hard to analyze. Following the method developed in [3],pages 370-371, we are able to prove Nash inequalities with dimension n = 5 / ,with however a constant C ( λ ) which goes to infinity when λ → . 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