Uniform Poincaré inequalities on measured metric spaces
aa r X i v : . [ m a t h . M G ] S e p UNIFORM POINCAR´E INEQUALITIES ON MEASURED METRIC SPACES
GAUTAM NEELAKANTAN MEMANA AND SOMA MAITY
Abstract.
Consider a proper geodesic metric space (
X, d ) equipped with a Borel measure µ. Weestablish a family of uniform Poincar´e inequalities on (
X, d, µ ) if it satisfies a local Poincar´e inequality( P loc ) and a condition on growth of volume. Consequently if µ is doubling and supports ( P loc ) thenit satisfies a ( σ, β, σ )-Poincar´e inequality. If ( X, d, µ ) is a δ -hyperbolic space then using the volumecomparison theorem in [3] we obtain a uniform Poincar´e inequality with exponential growth of thePoincar´e constant. Next we relate growth of Poincar´e constants to growth of discrete subgroups ofisometries of X which act on it properly. We show that if X is the universal cover of a compact CD ( K, ∞ ) space then it supports a uniform Poincar´e inequality and the Poincar´e constant dependson the growth of the fundamental group. Introduction
Poincar´e inequalities play an important role in studying analysis. Many important classical theoremsin analysis can be stated on a metric space with a doubling Borel measure if it supports a family ofPoincar´e inequalities [7]. In this paper, we will establish a family of uniform Poincar´e inequalities ona metric space equipped with a Borel measure which satisfies a local Poincar´e inequality along withcertain other geometric conditions.Let (
X, d ) be a proper geodesic metric space i.e. all closed balls in (
X, d ) are compact and any twopoints can be joined by a geodesic. Consider a Borel measure µ on X such that every closed ball hasa finite positive measure. We say ( X, d, µ ) a measured metric space as in [3]. A complete Riemannianmanifold with the volume measure induced from the metric is an example of a measured metric space.We also assume that there exists an increasing function f : (0 , ∞ ) → R such that µ ( B ( x, R )) µ ( B ( x, )) ≤ f ( R ) ∀ x ∈ X, ∀ R ≥ . (1.1)Note that the infimum of measures of balls of radius may be zero. If it is a positive constant c thenthe growth of volume is dominated by the function cf . Definition 1.
Let 1 ≤ σ < ∞ . ( X, d, µ ) is said to satisfy a local Poincar´e inequality ( P loc ) if thereexist positive constants C, r and L ≥ σ ≥ u ∈ C ( X ) and its upper gradientgradient g u : X → [0 , ∞ ], Z B ( x,R ) | u − u R | σ dµ ≤ C Z B ( x,LR ) | g u | σ dµ ∀ x ∈ X and 0 < R ≤ r . (1.2)where u R ( x ) is the mean of u on B ( x, R ) and C ( X ) is the space of continuous functions on X. Theorem 1.1.
Let (
X, d, µ ) be a measured metric space which satisfies the growth condition (1.1) and( P loc ) for r ≥ u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ C ( λR ) σ − f (4 λR ) Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) ∀ x ∈ X (1.3)for all R ≥ λ + L and σ ≥ λ = f (7 .
5) + 1 and C = 2 σ Cf σ +2 (3 . . The growth of volume of (
X, d, µ ) is polynomial if and only if µ is a doubling measure. If a measuredmetric space ( X, d, µ ) with a doubling measure µ also satisfies ( P loc ) then as a consequence of the above Mathematics Subject Classification.
Primary 53C21,53C23, 58J99.
Key words and phrases.
Poincar´e inequality, measured metric spaces, growth of volume. theorem it supports a uniform ( σ, β, σ )-Poincar´e inequality (see Corollary 3.1) i.e. there exist positiveconstants C , r, β and λ ≥ u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ C R β Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) ∀ x ∈ X, ∀ R ≥ r A complete Riemannian manifold (
M, g ) with Ricci curvature bounded from below supports a localPoincar´e inequality [4], [7], [2]. Using this fact Besson, Courtois and Hersonsky established a family of( σ, β, σ )-Poincar´e inequality on (
M, g ) when the growth of volume is polynomial and volumes of unitballs are bounded below by a positive constant in [2]. We observed that if Ricci curvature of (
M, g )is bounded below and it satisfies the growth condition (1.1) for some polynomial f ( R ) = vR α ( v > σ, β, σ )-Poincar´e inequality by Theorem 1.1.Next we study measured metric spaces with an exponential growth of volume. Riemannian manifoldswith negative sectional curvature are examples of such spaces. More generally we consider a δ -hyperbolicspace in the sense of Gromov equipped with a Borel measure. Using the volume comparison theoremproved in [3] and Theorem 1.1 we obtain the following theorem. Theorem 1.2.
Let (
X, d, µ ) be a measured δ -hyperbolic space which supports ( P loc ) for r ≥
1. Let Γbe a group acting on X isometrically and properly such that the diameter of the quotient space Γ \ X is bounded by D . Suppose the action of Γ is also measure preserving and the entropy of ( X, d, µ ) isbounded by H . Then there exist C ( δ, D, H, µ ) > λ ≥ u ∈ C ( X ) and itsupper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ C R σ + +6 HD e λHR Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) ∀ R ≥
52 (7 D + 4 δ ) , ∀ x ∈ X. For an explicit description of the constants we refer to Theorem 4.2 in section 4. In particular a δ -hyperbolic Cayley graph of a hyperbolic group G satisfies a strong Poincar´e inequality. Theorem 1.3.
Let X be a δ -hyperbolic Cayley graph of a finitely generated hyperbolic group equippedwith a measure µ. Suppose c ≤ µ ( x ) ≤ C for all x ∈ X for some c, C > X, µ ) isbounded by H . Then there exists C > R ≥ δ ) and u : X → R , Z B ( p,R ) | u ( x ) − u R | σ ≤ C R σ + e HR Z B ( p,R ) | g u | σ ( y ) dµ ( y ) ∀ p ∈ X where | g u | denotes the length of the gradient of u and σ ≥ . If a group Γ acts on X nicely then the growth of volume an be expressed in terms of the growthof Γ. Next we establish a relation between growth of Poincar´e constants and growth of Γ. Consider adiscrete subgroup Γ of isometries of ( X, d, µ ) acting on it properly such that the quotient space Γ \ X iscompact. Define, F Γ ( R ) = | Γ x ∩ B ( x, R ) | . (1.4)Here | . | denotes the cardinality of the set. F Γ ( R ) is independent of the choice of x and it determines thegrowth of Γ with respect to R . When Γ acts on a complete non-compact Riemannian manifold ( M, g )properly and Γ \ M is compact then section curvatures of ( M, g ) are bounded. Using Bishop-Gromovvolume comparison theorem we obtain the following theorem.
Theorem 1.4.
Let (
M, g ) be a complete non-compact Riemannian manifold with κ ≤ sec ≤ K anddimension n . Let Γ be a discrete subgroup of isometries of ( M, g ) acting on it properly such that thediameter of the quotient space Γ \ M is bounded by D. Then there exists positive constants C ( n, κ, K, σ ), r ( κ, n, D ) and λ ( n, κ, K ) ≥ u ∈ C ( M ) and R ≥ r , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ C R σ − F (2 λR ) Z B ( x,λR ) |∇ u ( z ) | σ dµ ( z ) ∀ x ∈ X where ∇ u denotes the gradient of u. Γ may not act on M freely, hence the quotient space may be an orbifold. Poincar´e inequality onRiemannian manifolds is an well studied topic. Theorem 1.4 relates growth of Poincar´e constant togrowth of groups. NIFORM POINCAR´E INEQUALITIES ON MEASURED METRIC SPACES 3
Now we assume that the action of Γ on a measured metric space (
X, d, µ ) is free and measurepreserving. If the volume and the diameter of Γ \ X are bounded above by V and D respectively then µ ( B ( x, R )) ≤ V F Γ ( R + D ) . Using Theorem 1.1 we prove that if Γ \ X supports a Poincar´e inequality then ( X, d, µ ) admits a uniformPoincar´e inequality (see Theorem 5.1). A lower bound on Ricci curvature plays a crucial role in obtainga local Poincar´e inequality on a Riemannian manifold. The concept of lower bounds on Ricci curvaturehas been generalized on metric measures spaces in the seminal papers by Sturm in [14],[15], by Lott andVillani in [10],[11]. They also proved Poincar´e inequality on metric measured spaces using lower boundson Ricci curvature and certain other conditions. Later Rajala proved a local Poincacar´e inequality onmore general metric measured spaces.
Theorem 1.5. [13] Suppose that (
X, d, µ ) is a CD ( K, ∞ ) space with K ≤
0. Then for any continuousfunction u on X and for any upper gradient g u of u Z B ( x,R ) | u − u B ( x,R ) | dµ ≤ Re | K | R Z B ( x, R ) | g u | dµ ∀ R > , ∀ x ∈ X. (1.5)We refer to [13] for the definition of CD ( K, ∞ ) spaces. From the proof of the above theorem weobserved the following result after applying Jensen’s inequality. Theorem 1.6.
Suppose that (
X, d, µ ) is a CD ( K, ∞ ) space with K ≤
0. Then there exists a positiveconstant C ( K, σ, R ) such that for any continuous function u on X and for any upper gradient g u of u Z B ( x,R ) | u − u B ( x,R ) | σ dµ ≤ C ( K, σ, R ) Z B ( x, R ) | g u | σ dµ ∀ R > , ∀ x ∈ X, σ ≥ . (1.6) C ( K, σ, R ) is continuous in R. Consequently,
Theorem 1.7.
Let Γ be a discrete subgroup of isometries of a measured metric space (
X, d, µ ) actingon it freely and properly such that the quotient space Γ \ X is compact. If the action of Γ is measurepreserving and (Γ \ X, ¯ d, ¯ µ ) is a CD ( K, ∞ ) space then there exist positive constants C , r and λ ≥ σ ≥ , u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ C R σ − F Γ (2 λR ) Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) ∀ R ≥ r, x ∈ X. (1.7)A δ -hyperbolic space may not have a local geometric structure required to support a local Poincar´einequality but if the quotient space Γ \ X in Theorem 1.2 is a CD ( K, ∞ ) space then X satisfies ( P loc ). Structure of the paper:
In this paper, the scheme of the proof of existence of such a uniformPoincar´e inequality will be similar to the one used in [2] and in [5]. In these papers the authors showedthat when the grow volume is polynomial a complete Riemannian manifold (
M, g ) satisfying a localdoubling condition( DV loc ) (see [5] for the definition) and a local Poincar´e inequality supports a uniformPoincar´e inequality if and only if a graph approximation of ( M, g ) supports a discrete version of Poincar´einequality [2], [5]. By replacing DV loc with an equivalent condition on the growth of volume given in(1 .
1) we obtain explicit values of constants in the Poincar´e inequality. The assumption on the lowerbound on r in Theorem 1.1 and Theorem 1.2 is required to choose a graph discretization of X withcanonical combinatorial distance one. This bound may be achieved by scaling the metric d suitably.Then the other geometric quantities will also change accordingly.In Section 2 we establish a strong Poincar´e inequality for a measured metric graph when it satisfies(1.1), which is an improvement of the Poincar´e inequality established in ([3]). In ([3]) a weak Poincar´einequality for a measured metric graph is established under the assumption of polynomial growth ofmeasure of balls and a uniform lower bound on the measure of vertices , whereas, with an improvementin the proof we obtained a strong Poincar´e inequality without the extra assumption of uniform lowerbound on the measure of vertices. Moreover, we also show that the constant appearing in the inequalitycan be further improved if we assume a uniform lower bound on the measure of vertices. GAUTAM NEELAKANTAN MEMANA AND SOMA MAITY
In Section 3 we first prove that an ǫ -discretization of ( X, d, µ ), which is a measured metric graph,is roughly isometric to (
X, d, µ ) under the assumption of growth condition( 1.1). Moreover, we alsoobtained a growth function for the ǫ -discretization satisfying (1.1) in terms of the growth function for( X, d, µ ) satisfying (1.1). In this method of approximation by a graph we try to emulate the foundationalwork of Kanai in [8],[9] and its later improvements made by Coulhon and Saloff-Coste in ([5]). Laterin this section, with the assumption of the existence of a local Poincar´e inequality on (
X, d, µ ) and thecondition on the growth of volume given in (1.1) we get a uniform Poincar´e inequality from the Poincar´einequality established on its ǫ -discretization.In section 4 we eshtablish a family of uniform Poincar´e inequalities on a Gromov hyperbolic spacesatisfying ( P loc ) with a bound on volume entropy. The inspiration to consider Gromov δ - hyperbolicspaces under such conditions is obtained from the volume comparison theorems proved in ([3]), whichwill give us an exponential growth function satisfying (1.1). Section 5 is devoted to developing arelation between growth of Poincar´e constants and growth of groups. The existence of uniform Poincar´einequality on a covering space is also discussed when its quotient space admits a Poincar´e inequality.2. Poincar´e Inequality on metric measured graphs
Let Y = ( V, E ) be a connected graph with a measure ν , where V, E denote the set of vertices andthe set of edges respectively. Define the distance ρ on Y as the canonical combinatorial distance andwe denote x ∼ y whenever x is adjacent to y . Let u : V → R be a function. Then the length of thegradient of u at a vertex x is defined as | δu | ( x ) = X x ∼ y | u ( x ) − u ( y ) | ! . (2.1)The integration with respect to a measure ν is defined as Z F u ( x ) dν ( x ) = X x ∈ F u ( x ) ν ( x ) for any F ⊂ Y. (2.2)The following theorem establishes a strong Poincar´e inequality for a measured metric graph ( Y, ρ, µ )that satisfies a growth condition (1.1). Even though the proof of the theorem is similar to the proof theTheorem 4.2 in [2], where the authors show weak local Poincar´e inequality for metric measured graphthat satisfies a polynomial growth of measure of balls and a uniform lower bound on ν ( x ), with the useof sharper inequalities we get a strong Poincar´e inequality and the condition on lower bound on ν ( x )can be dropped. Theorem 2.1.
Let (
Y, ρ, ν ) be a metric measured graph and f : (0 , ∞ ) → R be a function such that ν ( B ( x,R )) ν ( x ) ≤ f ( R ), ∀ x ∈ Y and R ≥ r > u : Y → R , σ ≥ R ≥ r > , Z B ( p,R ) | u ( x ) − u R | σ dν ( x ) ≤ σ R σ − f (2 R ) Z B ( p,R ) | δu | σ ( y ) dν ( y ) , ∀ p ∈ Y. Proof.
Consider u : Y → R and R ≥ r . By applying Jensen’s inequality we have, Z B ( p,R ) | u ( x ) − u R | σ dν ( x ) ≤ ν ( B ( p, R )) σ Z B ( p,R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( p,R ) | u ( x ) − u ( y ) | dν ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ dν ( x ) ≤ ν ( B ( p, R )) Z B ( p,R ) × B ( p,R ) | u ( x ) − u ( y ) | σ d ( ν ⊗ ν ) NIFORM POINCAR´E INEQUALITIES ON MEASURED METRIC SPACES 5
Minkowski inequality implies that Z B ( p,R ) × B ( p,R ) | u ( x ) − u ( y ) | σ d ( ν ⊗ ν ) ! σ ≤ Z B ( p,R ) × B ( p,R ) | u ( x ) − u ( p ) | σ d ( ν ⊗ ν ) ! σ + Z B ( p,R ) × B ( p,R ) | u ( y ) − u ( p ) | σ d ( ν ⊗ ν ) ! σ = 2 Z B ( p,R ) × B ( p,R ) | u ( x ) − u ( p ) | σ d ( ν ⊗ ν ) ! σ = σ ν ( B ( p, R )) Z B ( p,R ) | u ( x ) − u ( p ) | σ dν ( x ) ! σ Therefore, Z B ( p,R ) | u ( x ) − u R | σ dν ( x ) ≤ σ Z B ( p,R ) | u ( x ) − u ( p ) | σ dν ( x ) (2.3)Let γ p,x be a minimal geodesic joining p and x for x ∈ B ( p, R ) . Then l p,x = length( γ p,x ) ≤ R and | u ( x ) − u ( p ) | ≤ R γ p,x | δu | . Using Jensen’s inequality again we have, | u ( x ) − u ( p ) | σ ≤ l σ − p,x Z γ p,x | δu | σ ≤ R σ − Z γ p,x | δu | σ ≤ R σ − Z B ( p,R ) | δu | σ ( y ) dω ( y ) (2.4)where ω is the counting measure on Y . By the definition of | δu | , Z B ( p,R ) | δu | σ ( y ) dω ( y ) = X y ∈ B ( p,R ) | δu ( y ) | σ = 1 ν ( B ( p, R )) X y ∈ B ( p,R ) | δu ( y ) | σ ν ( y ) ν ( B ( p, R )) ν ( y ) (2.5)Since B ( p, R ) ⊂ B ( y, R ) for any y ∈ B ( p, R ) therefore, ν ( B ( p, R )) ν ( y ) ≤ ν ( B ( y, R )) ν ( y ) ≤ f (2 R ) . Hence Z B ( p,R ) | δu | σ ( y ) dω ( y ) ≤ f (2 R ) ν ( B ( p, R )) Z B ( p,R ) | δu | σ ( y ) dν ( y )Combining (2.3) and (2.4) we have Z B ( p,R ) | u ( x ) − u R | σ dν ( y ) ≤ Z B ( p,R ) σ R σ − f (2 R ) ν ( B ( p, R )) Z B ( p,R ) | δu | σ ( y ) dν ( y ) ! dν ( x ) ≤ σ R σ − f (2 R ) Z B ( p,R ) | δu | σ ( y ) dν ( y ) (cid:3) If µ ( x ) ≥ c for all x ∈ X for some c > Y, ρ, ν ) satisfies (1 .
1) then ν ( B ( x, R )) ≤ f ( R ) c . Theorem 2.2.
Let (
Y, ρ, ν ) be a metric measured. Suppose there exists a constant c > f : (0 , ∞ ) → R such that ν ( B ( x, R )) ≤ f ( R ) and ν ( x ) ≥ c , ∀ x ∈ Y and R ≥ r >
0. Then for any u : Y → R , σ ≥ R ≥ r > , Z B ( p,R ) | u ( x ) − u R | σ dν ( x ) ≤ σ cR σ − f ( R ) Z B ( p,R ) | δu | σ ( y ) dν ( y ) , ∀ p ∈ Y. GAUTAM NEELAKANTAN MEMANA AND SOMA MAITY
Proof.
From (2.5) we have, Z B ( p,R ) | δu | σ ( y ) dω ( y ) = X y ∈ B ( p,R ) | δu ( y ) | σ (2.6) ≤ c X y ∈ B ( p,R ) | δu ( y ) | σ ν ( y ) (2.7) ≤ c Z B ( p,R ) | δu | σ ( y ) dν ( y ) (2.8)From (2.3) and (2.4) we have, Z B ( p,R ) | u ( x ) − u R | σ dν ( y ) ≤ Z B ( p,R ) σ cR σ − Z B ( p,R ) | δu | σ ( y ) dν ( y ) ! dν ( x ) ≤ σ cR σ − ν ( B ( p, R )) Z B ( p,R ) | δu | σ ( y ) dν ( y ) ≤ σ cR σ − f ( R ) Z B ( p,R ) | δu | σ ( y ) dν ( y ) (cid:3) Since f is an increasing function f (2 R ) ≥ f ( R ) . Hence the constant is slightly better that the samein the previous theorem and in Theorem 4.2 in [2] for sufficiently large R. Uniform Poincar´e inequalities on metric spaces
Consider a measured metric space (
X, d, µ ) . In this section we prove the main theorem.
Definition 2. ( Y, ρ, ν ) is said to be an ǫ -discretization of ( X, d, µ ) for any ǫ > Y ⊂ X is greater than or equal to ǫ and the following holds.(i) X = ∪ y ∈ Y B X ( y, ǫ )(ii) ρ ( x, y ) = ǫ if d ( x, y ) < ǫ and x = y (iii) ν ( x ) = µ ( B ( x, ǫ ))( Y, ρ, ν ) may be considered as a metric measured graph where two vertices x, y ∈ Y are connectedby an edge of length ǫ if and only if d ( y, x ) < ǫ. The multiplicity of the covering { ( B X ( y, ǫ )) } y ∈ Y isthe supremum of the degree of vertices in Y. The following lemma is proved in [2] using a local doublingcondition. Following similar steps we obtained an estimate of the multiplicity of an ǫ -discretization of X in terms of the growth function f in (1.1). Lemma 3.1.
Let (
Y, ρ, ν ) be an ǫ -discretization of ( X, d, µ ) with ǫ ≥
1. If X satisfies (1.1) and M ( Y, Lǫ ) denotes the multiplicity of the covering { B ( x, Lǫ ) } x ∈ Y then M ( Y, Lǫ ) ≤ f (3 Lǫ + 12 ) , ∀ L ≥ . Proof.
Since ǫ ≥ { B ( x, ) } x ∈ Y forms a disjoint family of balls. Now, consider the set Z ⊂ Y such that for all z ∈ Z , B ( x, Lǫ ) ∩ B ( z, Lǫ ) = ∅ for some fixed x ∈ Y . Hence Z ⊂ B ( x, Lǫ ) and { B ( z, ) } z ∈ Z is a disjoint family of balls contained in B ( x, Lǫ + ). X z ∈ Z µ ( B ( z,
12 )) ≤ µ ( B ( x, Lǫ + 12 )) (3.1)Then, by using the condition on the growth of volume of balls µ ( B ( z, Lǫ + 12 )) ≤ f (3 Lǫ + 12 ) µ ( B ( z,
12 )) ∀ z ∈ Y. Now by using the above equation and and the fact that for z ∈ Z , B ( x, Lǫ + ) ⊂ B ( z, Lǫ + ) weget X z ∈ Z µ ( B ( z,
12 )) ≥ | Z | f (3 Lǫ + ) µ ( B ( x, Lǫ + 12 )) (3.2) NIFORM POINCAR´E INEQUALITIES ON MEASURED METRIC SPACES 7
Now, combining equations 3.1 and 3.2 we get | Z | ≤ f (3 Lǫ + ). Hence the lemma follows. (cid:3) Next two lemmas are proved in [9], [8] for Riemannian manifolds using a lower bound on Riccicurvature and injectivity radius. Later Coulhon and Saloff-Coste proved them on metric measuredspaces using the local doubling condition of the measure in [5]. Using (1.1) we expressed the constantsin terms of the growth function f. Lemma 3.2.
Let (
Y, ρ, ν ) be an ǫ -discretization of ( X, d, µ ) with ǫ ≥
1. If X satisfies (1.1)then d ( x, y ) ≤ ρ ( x, y ) ≤ f (6 ǫ + 12 )( d ( x, y ) + 2 ǫ ) ∀ x, y ∈ Y. Proof.
Let γ be a geodesic in X joining x and y . Let Y γ = { z ∈ Y : B ( z, ǫ ) ∩ γ = ∅} . Clearly, { B ( z, ǫ ) : z ∈ Y γ } covers γ and ρ ( x, y ) ≤ ǫ | Y γ | . Consider the positive integer k such that k − < d ( x, y ) /ǫ ≤ k .Let ( x = x , x , .., x k − , x k = y ) be points on γ such that d ( x j − , x j ) = d ( x, y ) /k for j = 1 , ...k . Since Y γ is contained in an ǫ neighbourhood of γ , Y γ ⊂ ∪ kj =0 { z ∈ Y : x j ∈ B ǫ ( z ) } . By Lemma 3.1, ρ ( x, y ) ≤ ǫ | Y γ | ≤ ǫ k X j =0 |{ z ∈ Y : x j ∈ B ǫ ( z ) }| ≤ ǫ ( k + 1) f (6 ǫ + 12 ) < f (6 ǫ + 12 ) (cid:0) d ( x, y ) + 2 ǫ (cid:1) Hence we have the required inequality. (cid:3)
Lemma 3.3.
Let (
Y, ρ, ν ) be an ǫ -discretization of ( X, d, µ ) with ǫ ≥
1. If X satisfies (1.1)then for all x ∈ Y ν ( B Y ( x, R )) ≤ f ( ǫ ) µ ( B X (2 R + 12 )) and ν ( B Y ( x, R )) ν ( B Y ( x, )) ≤ f ( ǫ ) f (2 R + 12 ) . (3.3)If R ′ = f (6 ǫ + )( R + 3 ǫ ) then µ ( B X ( x, R )) ≤ ν ( B Y ( x, R ′ )) and µ ( B X ( x, R )) µ ( B X ( x, )) ≤ f ( ǫ ) ν ( B Y ( x, R ′ )) ν ( B Y ( x, )) . Proof. ν ( B Y ( x, R )) = X y ∈ B Y ( x,R ) ν ( y ) = X y ∈ B Y ( x,R ) µ ( B X ( y, ǫ )) = f ( ǫ ) X y ∈ B Y ( x,R ) µ ( B X ( y,
12 )) . From Lemma (3.2) we obtain that y ∈ B X ( x, R ) for all y ∈ B Y ( x, R ). Observe that { B X ( y, ) } y ∈ Y are mutually disjoint and contained in B X ( x, R + ). Hence, ν ( B Y ( x, R )) ≤ f ( ǫ ) µ ( B X ( x, R + 1)) .ν ( B Y ( x, R )) ν ( x ) ≤ f ( ǫ ) µ ( B X ( x, R + )) µ ( B X ( x, ǫ )) ≤ f ( ǫ ) f (2 R + 12 )Let R ′ = f (6 ǫ + )( R + 3 ǫ ) and y ∈ B X ( x, R + ǫ ) ∩ Y. By Lemma (3.2) y ∈ B Y ( x, R ′ ). Therefore, B X ( x, R ) ⊂ ∪ y ∈ B X ( x,R + ǫ ) B X ( y, ǫ ) ⊂ ∪ y ∈ B Y ( x,R ′ ) B X ( y, ǫ ) .µ ( B X ( x, R )) ≤ X y ∈ B Y ( x,R ′ ) µ ( B X ( y, ǫ )) = ν ( B Y ( x, R ′ )) .µ ( B X ( x, R )) µ ( B X ( x, )) = f ( ǫ ) µ ( B X ( x, R )) µ ( B X ( x, ǫ )) ≤ f ( ǫ ) ν ( B Y ( x, R ′ )) ν ( B Y ( x, )) . (cid:3) Remark . As a consequence of the above lemmas we obtain a rough isometry between (
Y, ρ, µ ) and(
X, d, µ ) as defined in [8], [9]. This implies that the growth of volume of (
X, d, µ ) is polynomial (orexponential) if and only if the same holds for (
Y, ρ, ν ) . GAUTAM NEELAKANTAN MEMANA AND SOMA MAITY
Next we prove a uniform Poincar´e inequality on measured metric spaces assuming a local Poincar´einequality. An upper gradient of u ∈ C ( X ) is a Borel function g u : X → [0 , ∞ ) such that for each curve γ : [0 , → X with finite length l ( γ ) and constant speed, u ( γ (1)) − u ( γ (0)) ≤ l ( γ ) Z g ( γ ( t )) dt.u R denotes the mean of u on balls of radius R , u R ( x ) = 1 µ ( B ( x, R )) Z B ( x,R ) u dµ. Let (
Y, ρ, ν ) be an ǫ discretization of X . For any u ∈ C ( X ), ˜ u : Y → R is given by˜ u ( x ) = u B ( x,ǫ ) = 1 µ ( B ( x, ǫ )) Z B ( x,ǫ ) u ( z ) dµ ( z ) . || u || σ,E denotes the L σ -norm of a Borel function u on a Borel set E for any σ ≥ . Lemma 3.4. ([2] , Lemma 3.5 ) Let (
Y, ρ, ν ) be an ǫ -discretization of ( X, d, µ ) with ǫ ≥ g u be an upper gradient of u ∈ C ( X ). If X satisfies (1.1) and the ( P loc ) condition in (1.2) then || δ ˜ u || σ,B ( x,R ) ≤ C σ f (3 ǫ + 12 ) f σ (3 ǫ ) || g u || σ,B ( x,R +( L +2) ǫ ) where the constants L and C are same as in (1.2).This lemma is proved in ([2]) for Riemannian manifolds which satisfies the local doubling conditionand ( P loc ) condition (1.2). Since the local doubling condition and the growth condition (1.1) areequivalent, the same proof holds for ( X, d, µ ) as well for ǫ ≥
1. We will use this lemma to prove themain theorem.3.1.
Proof of the main theorem :
Proof.
We will follow the same idea that has been used by in ([2] , Theorem 3.1) for a Riemannianmanifold with Ricci curvature bounded below. Let (
Y, ρ, ν ) be a fixed ǫ -discretization of ( X, d, µ ) with ǫ = 1. Since B ( x, R ) ⊂ S y ∈ Y ∩ B ( x,R + ǫ ) B ( y, ǫ ), for any η ∈ R we have, Z B ( x,R ) | u ( z ) − η | σ dµ ( z ) ≤ X y ∈ Y ∩ B ( x,R + ǫ ) Z B ( y,ǫ ) | u ( z ) − η | σ dµ ( z ) . By applying Jensen’s inequality we have, Z B ( x,R ) | u ( z ) − η | σ dµ ( z ) ≤ σ − X y ∈ Y ∩ B ( x,R + ǫ ) Z B ( y,ǫ ) | u ( z ) − ˜ u ( y ) | σ dµ ( z )+2 σ − X y ∈ Y ∩ B ( x,R + ǫ ) ν ( y ) | ˜ u ( y ) − η | σ (3.4)Let us denote by (I) and (II), the first and the second term of the right hand side of the last inequality,respectively. One can bound (I) by local Poincar´e inequality (1.2) for radius ǫ since r ≥ I ) ≤ σ − C X y ∈ Y ∩ B ( x,R + ǫ ) Z B ( y,Lǫ ) | g u ( z ) | σ dµ ( z ) (3.5)By Lemma 3.1, the multiplicity of the covering { B ( y, Lǫ ) } y ∈ X is bounded by M ( Y, Lǫ ). For each y ∈ B ( x, R + ǫ ), B ( y, Lǫ ) ⊂ B ( x, R + ( L + 2 ǫ )). Putting ǫ = 1 we have,( I ) ≤ σ − C M ( Y, Lǫ ) Z B ( x,R +( L +2 ǫ )) | g u ( z ) | σ dµ ( z ) (3.6) ≤ σ Cf (3 L + 12 ) Z B ( x,R + L +2) | g u ( z ) | σ dµ ( z )Next we obtain a bound on (II) using the Poincar´e inequality on ( Y, ρ, ν ). We choose x ∈ Y such that d ( x, x ) < ǫ . Y ∩ B ( x, R + ǫ ) ⊂ B ( x , R + 2 ǫ ) NIFORM POINCAR´E INEQUALITIES ON MEASURED METRIC SPACES 9
To apply the Poincar´e inequality on Y , consider r = f (6 ǫ + )( R + 4 ǫ ), h ( r ) = f ( ǫ ) f (2 r + ). Thenusing Lemma (3.2) and Lemma (3.3) we have, B X ( x , R + 2 ǫ ) ∩ Y ⊂ B Y ( x , r ) and ν ( r ) ν ( ) ≤ h ( r ) . Wealso choose, η = ˜ u r = 1 ν ( B ( x , r )) X y ∈ B ( x ,r ) ˜ u ( y ) ν ( y ) . By Theorem 2.1 we obtain ( II ) ≤ σ − r σ − h (2 r ) X y ∈ B ( x ,r ) | δ ˜ u ( y ) | σ ν ( y ) (3.7)Now, by Lemma 3.4( II ) ≤ σ − r σ − h (2 r ) Cf σ (3 ǫ + 12 ) f (3 ǫ ) Z B ( x ,r +( L +2) ǫ ) | g u ( z ) | σ dµ ( z )We put ǫ = 1, λ = f (7 .
5) + 1 and let R ≥ λ + L . Then r = ( λ − R + 4) ≤ λR − L − . Therefore,( II ) ≤ σ − Cf σ +1 (3 . λR ) σ − f (1) f (4 λR ) Z B ( x ,λR − | g u ( z ) | σ dµ ( z ) ≤ σ − f (1) Cf σ +2 (3 . λR ) σ − f (4 λR ) Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) (3.8)for all R ≥ λ + L . From (3.6) we have,( I ) ≤ σ − Cf (3 L + 1) Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) (3.9)Therefore combining (3.9) and (3.8) we have, Z B ( x,R ) | u ( z ) − η | σ dµ ( z ) ≤ σ Cf σ +2 (3 . λR ) σ − f (4 λR ) Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) (3.10)We have the required uniform Poincar´e inequality from the following. Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ σ inf τ ∈ R Z B ( x,R ) | u ( z ) − τ | σ dµ ( z ) (3.11)We refer to [2] for a proof of the above inequality. (cid:3) A measure µ is called doubling for r ≥ r if there exists C ≥ µ ( B ( x, r )) µ ( B ( x, r )) ≤ C ∀ x ∈ X ∀ r ≥ r . When µ is doubling the growth of volume is polynomial. As a consequence ( X, d, µ ) satisfies a uniform( σ, β, σ )-Poincar´e inequality.
Corollary 3.1.
Let (
X, d, µ ) be a measured metric space. If (
X, d, µ ) satisfies ( P loc ) for r ≥ µ is doubling for R ≥ with the doubling constant C then there exists positive constants s , C ( σ, C , C ), λ ≥ r such that for any u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ C R σ + s − Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) ∀ R ≥ r. Proof. If µ is doubling then (see Theorem 5.2.2 [1]) µ ( B ( x, R )) µ ( B ( x, )) ≤ C R s with s = log C log 2 . for all x ∈ X and R ≥ . Hence we have the growth function f ( R ) = C R s . Now the result is animmediate consequence of Theorem 1.1. (cid:3) Corollary 3.2.
Let (
M, g ) be a complete Riemannian manifold with dimension n and Ric ≥ − kg for some k > . Let σ ≥
1. If
V ol ( B ( x,R )) V ol ( B ( x, )) ≤ V R α for all R ≥ r > C ( n, k, r, σ, V ) > λ ≥ u ∈ C ( M ), Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ C R α + σ − Z B ( x,λR ) |∇ u | σ dµ ( z ) ∀ R ≥ r. ∇ u denotes the gradient of u. Proof.
From Theorem 1.14 in [2] we have that if
Ric ≥ − kg then ( M, g ) satisfies the local Poincar´einequality (1.2) and the constant C ( n, k, R ) depends on k , n and R . Let C = sup R ≤ C ( n, k, R ) . Define, f ( R ) = V ∀ < R ≤ r = V R α ∀ R > r
Then
V ol ( B ( x,R )) V ol ( B ( x, )) ≤ f ( R ) . Now the result follows from Theorem 1.1. (cid:3) If µ ( B ( x, )) ≥ c for some c > X, d, µ ) satisfies the growth condition (1.1) then V ( R ) = sup x ∈ X µ ( B ( x, R )) ≤ cf ( R ) . With this additional assumption we can improve the constants appeared in Theorem 1.1.
Theorem 3.1.
Let (
X, d, µ ) be a measured metric which satisfies ( P loc ) for r ≥ c > V : (0 , ∞ ) → R such that µ ( x,
12 ) ≥ c and µ ( x, R ) ≤ V ( R ) ∀ x ∈ X then for any u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ ≤ σ c CV σ +1 (2 . λR ) σ − V (2 λR ) Z B ( x,λR ) | g u | σ ( z ) dµ ( z ) (3.12)for all R ≥ max { r , λ + L } and σ ≥ λ = cV (4 .
5) + 1 . Proof.
Let (
Y, ρ, ν ) be an ǫ -discretization of ( X, d, µ ) with ǫ ≥ . From (3.1) we obtain a new bound onthe multiplicity of the covering { B ( y, Lǫ ) } y ∈ Y . M ( Y, Lǫ ) ≤ cV (2 Lǫ + 12 ) (3.13)From the proof of Lemma (3.5) in [2] using this new bound on M ( ǫ ) we obtain || δ ˜ u || σ,B ( x,R ) ≤ Cc ) σ V (2 ǫ + 12 ) || g u || σ,B ( x,R +( L +2) ǫ ) (3.14)From the proof of Lemma (3.2) and Lemma (3.3) we also obtain, ρ ( x, y ) ≤ V (4 ǫ + 12 )( d ( x, y ) + 2 ǫ ) (3.15)and ν ( B Y ( x, R )) ≤ cV ( ǫ ) V (2 R ) . (3.16)For u ∈ C ( X ) let g u be an upper gradient of u. From (3.4) we have, Z B ( x,R ) | u ( z ) − η | σ ≤ I + II where I and II are the first term and the second term in (3.4) respectively. From (3.6) we have, I ≤ σ − CcV (2 L + 12 ) Z B ( x,R + L +2) | g u ( z ) | σ dµ ( z ) . (3.17)Next we choose x ∈ Y such that d ( x, x ) < . Hence, Y ∩ B ( x, R + ǫ ) ⊂ B ( x , R + 2 ǫ ) . NIFORM POINCAR´E INEQUALITIES ON MEASURED METRIC SPACES 11
To apply discrete Poincar´e inequality on Y we choose r = cV (4 . R + 4) and h ( r ) = cV (1) V (2 r ) .Therefore, using (3.15) and (3.16) we obtain, Y ∩ B X ( x , R + 2) ∈ B Y ( x , r ) and ν ( B Y ( x , r )) ≤ h ( r ) . We choose η = ˜ u r . Now by Theorem 2.2 we have, II ≤ σ − cr σ − h ( r ) k δu k σσ,B ( x ,r + L +2) . (3.14) implies that II ≤ σ − c CV σ (2 . r σ − h ( r ) Z B ( x ,r + L +2) | g u ( z ) | σ dµ ( z ) . Let λ = cV (4 .
5) + 1 and R ≥ λ + L . Then r = ( λ − R + 4) ≤ λR − L − . Putting values of r and h ( r ) we have, II ≤ σ − c CV σ (2 . V (1)( λR ) σ − V (2 λR ) Z B ( x,λR ) | g u | σ ( z ) dµ ( z )for all R ≥ λ + L . In the last step we used d ( x, x ) < . From (3.17) we have, I ≤ σ − CcV (2 L + 12 ) Z B ( x,λR ) | g u ( z ) | σ dµ ( z )Therefore, I + II ≤ σ c CV σ +1 (2 . λR ) σ − V (2 λR ) Z B ( x,λR ) | g u | σ ( z ) dµ ( z )Now the required result follows from the following inequality. Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ σ inf τ ∈ R Z B ( x,R ) | u ( z ) − τ | σ dµ ( z ) (3.18) (cid:3) Remark . Let ( M n , g ) be a complete Riemannian manifold with dimension n and Ric ≥ κ ( n − g .Suppose volume of unit balls are bounded from below c . Then ( M, g ) supports ( P loc ) as in [2]. UsingBishop-Gromov volume comparison theorem one may consider V ( R ) to be the volume of a ball of radius R in the space of constant curvature κ . Then λ can also be written in terms of n, c and κ. Remark . A family of uniform ( σ, β, σ )-Poincar´e inequality is established in [2] on a completeRiemannian manifold ( M n , g ) with Ric ≥ κg and volume of unit balls bounded below by c >
0. UsingTheorem 3.1 and Bishop-Gromov volume comparison theorem one may obtain the same theorem withexplicit values of the constants in terms of n, κ and c . Corollary 3.3.
Let (
X, d, µ ) be a measured metric space which satisfies ( P loc ) for r ≥ µ ( B ( x, )) ≥ c >
0. If µ is doubling for R ≥ r with the doubling constant C and µ ( B ( x, r )) ≤ V , ∀ x ∈ X then there exists positive constants s , C ( σ, C , C, c, r, V ), λ ≥ r such that for any u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ C R σ + s − Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) ∀ R ≥ r. Proof. If µ is doubling then for all R ≥ r and x ∈ X (see Theorem 5.2.2 [1]) µ ( B ( x, R )) µ ( B ( x, r )) ≤ C (cid:18) Rr (cid:19) s with s = log C log 2 . (3.19)Define , f ( R ) = V ∀ ≤ R ≤ r = V r s R s ∀ R > r Now the required result is an immediate consequence of Theorem 3.1. (cid:3) Poincar´e inequality on Gromov hyperbolic spaces
Let (
X, d ) be a δ hyperbolic space in the sense of Gromov and µ be a Borel measure on it. TheEntropy of a measured metric space ( X, d, µ ) is defined byEnt(
X, d, µ ) = lim inf R →∞ R ln ( µ ( B ( x, R )))It is independent of the choice of x. Let G be a finitely generated group with a finite set of generatorsΣ. ( G, Σ) is called a hyperbolic group if the Cayley graph Γ of ( G, Σ) is δ -hyperbolic as a metric space.Hyperbolic groups play an important role in geometric group theory. We prove a strong Poincar´einequality on Cayley graphs which are δ -hyperbolic. Theorem 4.1.
Let Γ be a δ hyperbolic Cayley graph of a hyperbolic group G equipped with a measure µ . Suppose c ≤ µ ( x ) ≤ C for all x ∈ Γ and the entropy of (Γ , µ ) is bounded by H . Then for R ≥ r = 10(1 + δ ), σ ≥ u : Γ → R Z B ( p,R ) | u ( x ) − u R | σ dµ ( x ) ≤ σ Cν ( r ) cr e H (1+ δ ) R σ + e HR Z B ( p,R ) | g u | σ ( x ) dµ ( x ) ∀ p ∈ X where | g u | denotes the length of the gradient of u and ν ( r ) is the number of elements of Γ in a ball ofradius r . Proof.
Let ν be the counting measure on Γ . Then ν is a G invariant and for any R > ν ( B ( x, R )) issame for all x ∈ Γ . We define ν ( R ) = ν ( B ( x, R )) . The diameter of G \ Γ is 1. Hence from Theorem 1.9part (ii) in [3] we have, ν ( R ) < ν ( r ) (cid:18) Rr (cid:19) e H ( R − r ) ∀ R ≥ r = 10(1 + δ ) . Now µ ( B ( x, R )) ≤ Cν ( R ) . Hence using Theorem 2.2 for all R ≥ r we have, Z B ( p,R ) | u ( x ) − u R | σ dµ ( x ) ≤ σ Cν ( r ) cr R σ + e H ( R − r ) Z B ( p,R ) | g u | σ ( x ) dµ ( x )= 2 σ Cν ( r ) cr e H (1+ δ ) R σ + e HR Z B ( p,R ) | g u | σ ( x ) dµ ( x )Hence the proof follows. (cid:3) Next we establish a uniform Poincar´e inequality on a measured δ -hyperbolic space which satisfies alocal Poincar´e inequality. Theorem 4.2.
Let (
X, d, µ ) be a measured δ -hyperbolic space which supports ( P loc ) for r ≥ X isometrically and properly such that the diameter of the quotientspace Γ \ X is bounded by D and r = (7 D + 4 δ ) ≥
2. Suppose the action of Γ is measure preserving andthe entropy of (
X, d, µ ) is bounded by H . Then there exist C ( δ, D, H, µ ) > λ ≥ u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ σ +3 HD ) c CV σ +20 HD r +6 HD e H (83 D +48 δ ) ( λR ) σ + +6 HD e λHR Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) ∀ R ≥ r and ∀ z ∈ X where λ = cV + 1, V = sup x ∈ X { µ ( B ( x, r )) } and c = inf ∈ X { µ ( B ( x, )) } . Proof.
Let V = sup { µ ( B ( x, r )) | x ∈ X } . From Theorem 1.9 part (i) in [3] we have, µ ( B ( x, R )) ≤ (cid:18) (cid:19) +6 HD V R +6 HD e HR r +6 HD e H (12 r − D ) V ( R ) = V ∀ R < r ≤ (cid:18) (cid:19) +6 HD V R +6 HD e HR r +6 HD e H (12 r − D ) ∀ R ≥ r NIFORM POINCAR´E INEQUALITIES ON MEASURED METRIC SPACES 13
Since r ≥ r ≥
5. Hence V (4 .
5) = V (2 .
5) = V . Now from Theorem 3.1 we have, for all R ≥ r , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ σ +3 HD ) c CV σ +20 HD r +6 HD e H (83 D +48 δ ) ( λR ) σ + +6 HD e λHR Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) (cid:3) Uniform Poincar´e inequalities and growth of groups
In this section we study how the growth of Poincar´e constants on a measured metric space (
X, d, µ )depend on growth of groups which act on it properly and isometrically. First we prove Theorem 1.4.
Proof of Theorem
M, g ) be a complete non-compact Riemannian manifold with dimension n. Let Γ be a discrete subgroup of isometries of (
M, g ) acting on it properly such that the diameter of thequotient space Γ \ M is bounded by D. Then for any x ∈ M , { B ( γ.x, D ) } γ ∈ Γ covers M . By continuityof Riemannian curvature, sectional curvatures of ( M, g ) are bounded on each B ( γ.x, D ). Since Γ actsisometrically sectional curvatures of ( M, g ) are bounded. Let κ ≤ sec ≤ K. Then Ricci curvature isbounded below by ( n − κ . From Theorem 1.14 in [2] there exists a constant C ( n, κ, R ) > u ∈ C ( M ), Z B ( x,R ) | u − u R | σ dv g ≤ C ( n, κ, R ) Z B ( x, R ) |∇ u | σ dv g ∀ R > , ∀ x ∈ M where dv g is the volume form induced from g. Let C = sup
R > ,V ol ( B ( x, R )) ≤ V ol ( B ( x, D )) F Γ ( R + D )Let V Kn ( R ) denote the volume of ball of radius R in the space of constant curvature K . From Bishop-Gromov volume comparison theorem V ol ( B ( x, D )) ≤ V κn ( D ) and V ol ( B ( x, )) ≥ V Kn ( ) for all x ∈ M. Hence,
V ol ( B ( x, R )) ≤ V κn ( D ) F Γ ( R + D )Now the required result follows from Theorem 3.1. Remark . If (
M, g ) is the universal cover of a compact Riemannian manifold with non-negative Riccicurvature then the growth of volume as well as the gorwth of the fundamental group Γ is polynomial [12].Consequently, (
M, g ) satisfies a ( σ, β, σ )-uniform Poincar´e inequality as in Corollary 3.2. In particularwhen a Riemannian manifold satisfies a (1 , , Remark . If (
M, g ) is the universal cover of a compact negatively curved Riemannian manifold thenthe growth of the fundament group Γ is exponential [12] i.e. F Γ ( R ) grows exponentially.Next we establish uniform Poincar´e inequalities on a covering space when the quotient space satisfiesa Poincar´e inequality. Let Γ be a discrete subgroup of isometries of ( X, d, µ ) acting on it freely andproperly such that Γ \ X is compact. The point-wise systole is defined as sys Γ ( x ) = inf γ = e d ( x, γ.x ) . The systole sys Γ of X is the infimum of sys Γ ( x ) over x ∈ X . Since Γ \ X is compact sys Γ is non-zero.Define the quotient metric ¯ d on Γ \ X as follows. For x, y ∈ Γ \ X choose ˜ x ∈ p − ( x ) and ˜ y ∈ p − ( y ) . Then ¯ d ( x, y ) = inf γ ∈ Γ d (˜ x, γ. ˜ y ) = inf γ ,γ ∈ Γ d ( γ . ˜ x, γ . ˜ y ) (5.1)Observe that the quotient map p is a covering map and p : B (˜ x, sys Γ ) → B ( x, sys Γ ) is an isometry forall x ∈ Γ \ X and ˜ x ∈ p − ( x ). Suppose the action of Γ is measure preserving. Define, the quotient Borelmeasure ¯ µ on X \ Γ such that p restricted to each B (˜ x, sys Γ ) is measure preserving for ˜ x ∈ X . Lemma 5.1.
If ( X \ Γ , ¯ d, ¯ µ ) satisfies a Poincar´e inequality, i.e. for any σ ≥ L ≥ C ( R ) > u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u − u R | σ d ¯ µ ≤ C ( R ) Z B ( x,LR ) | g u | σ d ¯ µ ∀ R >
X, d, µ ) satisfies ( P loc ) for R ≤ sys Γ L . Proof.
Consider a continuous function ˜ u : X → R and an upper gradient g ˜ u of ˜ u . Given R ≤ sys Γ L choose R ′ such that LR < R ′ < sys Γ and a bump function φ : X → R such that φ = 1 on B (˜ x, LR )and is compactly supported on B (˜ x, R ′ ). So, for a continuous function u : X → R we can define acorresponding continuous function u on B ( x, sys Γ ) for any x ∈ X \ Γ, namely u ( y ) = X ˜ y ∈ p − ( y ) ˜ u (˜ y ) φ (˜ y )which is well defined as the restriction map is bijective. Moreover, corresponding to the upper gradient g ˜ u we can define an upper gradient for u on B ( x, LR ) as g u ( y ) = X ˜ y ∈ p − ( y ) g ˜ u (˜ y ) φ (˜ y )To see that g u is indeed an upper gradient for u on B ( x, LR ) choose a unit speed curve α : [0 , → B ( x, LR ) and let ˜ α be its lift via the map p passing through B (˜ x, LR ). Then, | u ( α (1)) − u ( α (0)) | = | u (˜ α (1)) − u (˜ α (0)) | ≤ Z g ˜ u (˜ α ( t )) dt = Z g u ( α ( t )) dt Let C = sup { C ( R ) : R ≤ sys Γ } . Now we have the required local Poincar´e inequality as Z B (˜ x,R ) | ˜ u − ˜ u R | σ dµ = Z B ( x,R ) | u − u R | σ d ¯ µ ≤ C Z B ( x,LR ) | g u | σ d ¯ µ = C Z B (˜ x,LR ) | g ˜ u | σ dµ (cid:3) Theorem 5.1.
Consider a measured metric space (
X, d, µ ) . Let Γ be a discrete subgroup of isometriesof (
X, d, µ ) acting on it freely and properly such that Γ \ X is compact and sys Γ ≥ max { L, } . Letthe diameter and the volume of Γ \ X is bounded above by D and V respectively. If (Γ \ X, ¯ d, ¯ µ ) satisfiesa Poincar´e inequality (5.2) then for any u ∈ C ( X ) and its upper gradient g u , Z B ( x,R ) | u ( z ) − u R | σ dµ ( z ) ≤ σ c CV σ +20 ( λR ) σ − F Γ (2 λR ) Z B ( x,λR ) | g u ( z ) | σ dµ ( z ) (5.3)for all R ≥ λ + L and σ ≥
1, where λ = cV + D and c = inf x ∈ X µ ( B ( x, )). Proof.
The quotient space Γ \ X supports a Poincar´e inequality. By Lemma 5.2 X satisfies ( P loc ) definedin (1.2). Γ \ X is also compact. Therefore inf x ∈ X µ ( B ( x, )) is positive. Since the covering map is locallymeasure preserving for any R > µ ( B ( x, R )) ≤ V F Γ ( R + D ) ∀ x ∈ X (5.4)and for any R ≤ sys Γ , µ ( B ( x, R )) ≤ V for all x ∈ X. Define, f ( R ) = V ∀ R ≤ sys Γ V F Γ ( R + D ) ∀ R > sys Γ (cid:3) The required lower bound on systole is may be achieved by scaling ¯ d suitably. Then the othergeometric quantities will also change accordingly. If the quotient space Γ \ X is a CD ( K, ∞ ) spacethen by Theorem 1.6 Γ \ X supports a Poincar´e inequality. Now the proof of Theorem 1.7 follows fromTheorem 5.1. NIFORM POINCAR´E INEQUALITIES ON MEASURED METRIC SPACES 15
Remark . The function F Γ ( R ) determines the growth of the group Γ . In particular when X is simplyconnected the growth of the fundamental group of Γ \ X is determined by F Γ . As a consequence of theabove theorem the growth of the Poincar´e constant with respect to R is polynomial (or exponential) ifthe growth of Γ is polynomial (or exponential). References [1] L. Ambrosio and P. Tilli,
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Department of Mathematical Sciences, Indian Institute of Science Education and Research Mohali, India
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