� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 37, Num. 2 (2022), 211 – 221 doi:10.17398/2605-5686.37.2.211 Available online November 2, 2022 Topological Hausdorff dimension and Poincaré inequality C.A. DiMarco 1000 E. Henrietta Rd., Mathematics Department, Monroe Community College Rochester, NY 14623, USA cdimarco2@monroecc.edu Received February 26, 2022 Presented by G. Plebanek Accepted October 2, 2022 Abstract: A relationship between Poincaré inequalities and the topological Hausdorff dimension is exposed—a lower bound on the dimension of Ahlfors regular spaces satisfying a weak (1,p)-Poincaré inequality is given. Key words: Poincaré inequality, metric space, Cantor sets, topological dimension, Hausdorff dimen- sion, bi-Lipschitz map, Ahlfors regular. MSC (2020): Primary 28A80, 28A75; Secondary 28A78, 54F45. 1. Introduction Let (X,d) be a separable metric space. The subscript of dim indicates the type of dimension, and we set dim ∅ = −1 for every dimension. Poincaré inequalities are the forms of the Fundamental Theorem of Cal- culus that work in general metric spaces. Indeed, a one-dimensional Poincaré inequality is a direct consequence of the Fundamental Theorem of Calculus: Remark 1.1. Let f : [a,b] → R be differentiable. The Intermediate Value Theorem gives a point c ∈ [a,b] with f(c) = − ∫ b a f, the average of f on [a,b]. The Fundamental Theorem of Calculus then yields − ∫ b a ∣∣∣∣f(x) −− ∫ b a f ∣∣∣∣ dx ≤ (b−a)− ∫ b a |f ′| , which is inequality (1.1) found below, with p = λ = K = 1. There is an inherent connection between Poincaré inequalities and topo- logical Hausdorff dimension because both concepts take connectivity into ac- count. In order to discuss Poincaré inequalities, we include the following definition, which can be found in [4, p. 55]. ISSN: 0213-8743 (print), 2605-5686 (online) © The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.37.2.211 mailto:cdimarco2@monroecc.edu https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 212 c.a. dimarco Definition 1.2. Given a real valued function u in a metric space X, a Borel function ρ : X → [0,∞] is an upper gradient of u if |u(x) −u(y)| ≤ ∫ γ ρ ds for each rectifiable curve γ joining x and y in X. To prove the main result, we will use the upper pointwise dilation as a suitable upper gradient (see [2, p. 342]). Fact 1.3. If f : X → R is a locally Lipschitz function, the upper pointwise dilation Lip f(x) = lim sup r→0 sup y∈B(x,r) |f(x) −f(y)| r is an upper gradient of f. The following definition of a weak Poincaré inequality is from [4, p. 68], and a broader definition can be found in [2, p. 84]. Definition 1.4. Let (X,µ) be a metric measure space and let 1 ≤ p < ∞. Say that X admits a weak (1,p)-Poincaré inequality if there are constants 0 < λ ≤ 1 and K ≥ 1 so that − ∫ λB |u−uλB| dµ ≤ K(diam B) ( − ∫ B ρp dµ )1/p (1.1) for all balls B ⊂ X, for all bounded continuous functions u on B, and for all upper gradients ρ of u, where uλB is the average value of u on the set λB. Also assume µ(B(x,r)) > 0 whenever r > 0. It is not difficult to show that if a space supports a weak Poincaré inequality, then it is connected, and ∂B(x,r) 6= ∅ whenever r < 1 2 diam X [5, Proposition 8.1.6]. Such spaces are also quasiconvex, i.e., any two points can be connected by a curve of controlled length [5, Theorem 8.2.3]. Like the Hausdorff dimension, Poincaré inequalities are preserved by bi-Lipschitz maps, but the constants λ and K may change after application of a Lipschitz map. For a precise statement, see [2, Proposition 4.16]. Recently, results have surfaced that explain the relationship between Poin- caré inequalities and some particular fractals. Mackay, Tyson, and Wildrick investigated the potential presence of Poincaré inequalities on various car- pets —metric measure spaces that are homemorphic to the standard Sierpinsḱı topological hausdorff dimension 213 carpet. In short, a carpet of this kind is constructed in the same manner as the Sierpinsḱı carpet, except at each step the scaling factor need not be 1/3. Requiring that the sequence of scaling factors a = (a1,a2, . . . ) contain only re- ciprocals of odd integers that decrease to zero, one obtains a carpet (Sa, |·|,µ) with Euclidean metric | · | and measure µ, where µ arises as the weak limit of normalized Lebesgue measure on the precarpets. For the construction, see [8]. They provided a complete characterization of these carpets in terms of (1,p)-Poincaré inequalities as follows. Theorem 1.5. (Mackay, Tyson, Wildrick [8]) (i) The carpet (Sa, | · |,µ) supports a (1, 1)-Poincaré inequality if and only if a ∈ `1. (ii) The following are equivalent: (a) (Sa, | · |,µ) supports a (1,p)-Poincaré inequality for each p > 1. (b) (Sa, | · |,µ) supports a (1,p)-Poincaré inequality for some p > 1. (c) a ∈ `2. To see how topological Hausdorff dimension is related to connectivity, one need only consider Theorem 3.6 in [1]. That theorem gives an equivalent definition of topological Hausdorff dimension for separable metric spaces: dimtH X = min { d : ∃A ⊂ X such that dimH A ≤ d− 1 and dimt(X \A) ≤ 0 } . A significant advantage of imposing a Poincaré inequality like (1.1) is the flexibility that exists in choosing the function u and one of its upper gradients ρ. To apply (1.1) to the topological Hausdorff dimension of a given space X, one can apply the inequality to the boundary of an arbitrary open set U of X to determine a lower bound on dimH ∂U. If a non-trivial lower bound on dimH ∂U is achieved, then so is a lower bound on dimtH X. In the next section we apply this technique and exploit the Poincaré inequality to accomplish exactly that goal. A closely related concept was recently investigated by Lotfi in [7], which generalized the topological Hausdorff dimension by combining the definitions of topological dimension and µ-Hausdorff dimension. They presented upper and lower bounds for the so-called µ-topological Hausdorff dimension of the Sierpinsḱı carpet, and gave a large class of measures µ, where the associated µ- topological Hausdorff dimension of the Sierpinsḱı carpet coincides with these lower and upper bounds. 214 c.a. dimarco The main result requires that a space X satisfies a weak (1,p)-Poincaré inequality, and that it is Ahlfors regular. The following definition can be found in [4, p. 62]. Definition 1.6. If X is a metric space admitting a Borel regular measure µ such that C−1Rb ≤ µ(BR) ≤ CRb for some constant C ≥ 1, for some exponent b > 0, and for all closed balls BR of radius 0 < R < diam X, then X is called Ahlfors b-regular. An Ahlfors b-regular space has Hausdorff dimension b [4, p. 62], and is doubling : Definition 1.7. A metric measure space (X,d,µ) is doubling if there is C > 0 such that 0 < µ(B(x, 2r)) ≤ Cµ(B(x,r)) for all x ∈ X and for all r > 0. There is much interplay between Ahlfors regularity and weak(1,p)-Poincaré inequalities in metric spaces. For example, in [6], Lohvansuu and Rajala re- cently studied the duality of moduli in this context, where the Ahlfors reg- ularity constant is assumed to be greater than one. They proved that there is something of a dual relationship, with exponents p and p∗ = p p−1 , between the path modulus and the modulus of separating surfaces. It can be challenging to obtain nontrivial lower bounds on the topological Hausdorff dimension. In the presence of Ahlfors regularity, however, this prob- lem becomes more tractable. We now state the main result, which provides a lower bound in terms of the regularity and Poincaré constants. Theorem. Let (X,µ,d) be a complete, Ahlfors b-regular, (1,p)-Poincaré metric measure space. Then dimtH X ≥ b−p + 1. Due to Ahlfors regularity, equality is achieved if p = 1 because dimtH X ≤ dimH X = b. On the other hand, it is not clear whether a space exists that yields equality for any p > 1. 2. Preliminaries The symbol B(x,ε) denotes the open ball centered at x of radius ε. For x ∈ Rn, the Euclidean modulus of x is denoted |x|. Unless otherwise stated, distance in the metric space Y is denoted dY or simply d. We use the notation topological hausdorff dimension 215 fE = − ∫ E f dµ = 1 µ(E) ∫ f dµ for the average value of an integrable function f on E ⊂ X, where (X,d,µ) is a metric measure space. For any A ⊂ (X,d), the set Aδ is the δ-neighborhood of A in X. The symbol χU represents the characteristic function of any U ⊂ X. In order to define topological Hausdorff dimension, we include the defini- tion of Hausdorff dimension: Definition 2.1. The p-dimensional Hausdorff measure of X is Hp(X) = lim δ→0 inf { ∞∑ j=1 (diam Ej) p : X ⊂ ∞⋃ j=1 Ej and diam Ej ≤ δ for all j } ; the Hausdorff dimension of X is dimH X = inf{p : Hp(X) = 0}. An interesting combination of the Hausdorff and topological dimensions, called topological Hausdorff dimension, was introduced in [1]: dimtH X = inf{d : X has a basis U such that dimH ∂U ≤ d− 1 ∀U ∈U}. By Theorem 4.4 in [1], the topological Hausdorff dimension always falls between the topological dimension (dimt X) and the Hausdorff dimension (dimH X): Theorem 2.2. (Balka, Buczolich, Elekes[1])For any metric space X, dimt X ≤ dimtH X ≤ dimH X. (2.1) In certain favorable circumstances, the Hausdorff and topological Haus- dorff dimensions are additive under products. For any product space X ×Y , we use the metric d((x1,y1), (x2,y2)) = max(dX(x1,x2),dY (y1,y2)). For sake of completeness, we include Theorem 4.21 from [1] and several product formulas for Hausdorff dimension (see e.g. [3, Chapter 7]). Fact 2.3. If E ⊂ Rn, F ⊂ Rm are Borel sets, then dimH(E ×F) ≥ dimH E + dimH F. 216 c.a. dimarco Let dimHX be the upper box-counting dimension of X (see e.g. [3]). Fact 2.4. For any sets E ⊂ Rn and F ⊂ Rm dimH(E ×F) ≤ dimH E + dimBF. We call a Cantor set in [0, 1] uniform if it is constructed in the same way as the usual middle-thirds example, allowing for any scaling factor 0 < r < 1/2. Since uniform Cantor sets have equal Hausdorff and upper box dimensions, Facts 2.3 and 2.4 yield the following formula. Fact 2.5. If F ⊂ R is a uniform Cantor set, then for any E ⊂ Rn dimH(E ×F) = dimH E + dimH F. (2.2) In light of Facts 2.3 and 2.4, we observe the following convenient additivity property. Fact 2.6. If X ⊂ Rn and Y ⊂ Rm are Borel sets with dimH X = dimBX, dimH(X ×Y ) = dimH X + dimH Y. (2.3) The condition dimH X = dimBX holds for a wide variety of spaces. Theorem 2.7. If X is a nonempty separable metric space, then dimtH(X × [0, 1]) = dimH(X × [0, 1]) = dimH X + 1 . (2.4) In particular, for any value c > 2, R = X × [0, 1] can be chosen such that dimtH R = c. The first equality in (2.4) is due to Balka, Buczolich, and Elekes [1]. Because dimH[0, 1] = dimB[0, 1] = 1, the second equality in (2.7) is read- ily obtained considering Fact 2.6. Recall that the Hausdorff dimension is invariant under bi-Lipschitz maps. Definition 2.8. An embedding f is L-bi-Lipschitz if both f and f−1 are L-Lipschitz, and we say f is bi-Lipschitz if it is L-bi-Lipschitz for some L. topological hausdorff dimension 217 3. A lower bound on topological Hausdorff dimension for Poincaré Ahlfors regular spaces To provide a nontrivial lower bound on dimtH X, it suffices to consider an arbitrary bounded basis element U for the topology on X, and show that dimH ∂U ≥ b−p, where b and p are the regularity and Poincaré constants of X, respectively. Theorem 3.1. Let (X,µ,d) be a complete, Ahlfors b-regular, (1,p)- Poincaré metric measure space. Then dimtH X ≥ b−p + 1. Proof. Let U be basis for the topology on X, and consider a bounded element U ∈U, U 6= X. Choose δ > 0 small enough that δ < 1 2 diam(U), and both U \ (∂U)δ and Uδ c are nonempty. Let 0 < λ ≤ 1 and K ≥ 1 be as in Definition 1.4, and choose z0 ∈ U \ (∂U)δ. Choose R > 0 large enough that B(z0,R) ⊃ Uδ and B(z0,R) \ Uδ 6= ∅, and put B = B (z0,R/λ). Then R is large enough that Uδ ⊂ λB = B(z0,R). Fix an arbitrary finite covering D of ∂U by open balls as follows: D = {Di = B(xi, 2ri) : xi ∈ ∂U}, 2ri ≤ δ for all i . (3.1) We will show that there is a constant C > 0 such that ∑ i(diam Di) b−p ≥ C. Note that X is doubling because it is Ahlfors regular, and X is proper be- cause it is complete and doubling [5, Lemma 4.1.14]. Therefore ∂U is compact because it is closed and bounded. Given a finite covering D of ∂U satisfying (3.1), define the functions ui(x) = min { d(x,Dci ) ri , 1 } and u = max ( max i ui,χU ) . Notice that ui is 1 ri -Lipschitz, u is bounded, and u is continuous because D is a finite covering. Considering that 0 ≤ u ≤ 1, we have 0 ≤ uλB ≤ 1, and hence − ∫ λB |u−uλB| dµ ≥ 1 µ(λB) (∫ {x∈λB:u(x)=1} |u−uλB| dµ ) + 1 µ(λB) (∫ {x∈λB:u(x)=0} |u−uλB| dµ ) = 1 µ(λB) [ (1 −uλB)µ ( {u(x) = 1} ) + uλBµ ( {u(x) = 0} )] 218 c.a. dimarco ≥ 1 µ(λB) min { µ ( {u(x) = 1} ) ,µ ( {u(x) = 0} )} ≥ 1 µ(λB) min { µ(λB ∩U),µ ( λB ∩ (Uδ)c )} (3.2) ≥ 1 µ(λB) min { µ(U),µ ( λB \Uδ )} . The fact that X is b-regular provides a constant M ≥ 1 with M−1rb ≤ µ(Br) ≤ Mrb for any ball of radius r. In particular µ(λB) ≤ MRb, and µ(U) > 0 because U is open and non-empty. Also, recall that δ and R were chosen so that λB \Uδ = B(z0,R) \Uδ is open and nonempty. So there is a point z1 and an integer N > 0 such that B(z1, 1/N) ⊂ λB \Uδ . Applying regularity gives µ(λB \Uδ) ≥ µ(B(z1, 1/N)) ≥ 1 MNb . (3.3) In light of (3.2) and (3.3), we see that − ∫ λB |u−uλB| dµ ≥ 1 µ(λB) min{µ(U),µ ( λB \Uδ ) } ≥ 1 MRb min { µ(U), 1 MNb } = C′, (3.4) where the constant C′ > 0 is independent of the covering D. Next, we show that − ∫ λB |u−uλB|dµ ≤ C′′ ∑ i r b−p i for some C ′′ > 0. To this end, recall that the upper pointwise dilation of any locally Lipschitz function f is denoted Lip f, and note that lim sup y→x |f(x) −f(y)| d(x,y) = lim sup r→0 sup y∈B(x,r) |f(y) −f(x)| d(y,x) ≥ lim sup r→0 sup y∈B(x,r) |f(y) −f(x)| r = Lip f(x) . (3.5) topological hausdorff dimension 219 The fact that ui is 1 ri -Lipschitz, along with equation (3.5), show Lip ui(x) ≤ 1 ri for all x. Also Lip u ≤ maxi Lip ui, and Lip ui(x) = 0 for x /∈ Di. Ahlfors regularity implies µ(Di) ≤ M(2ri)b for all i, and therefore∫ B |Lip u|p dµ = ∫ B (Lip u)p dµ ≤ ∫ B [ max i (Lip ui) ]p dµ ≤ ∫ B ∑ i (Lip ui) p dµ ≤ ∑ i ∫ X (Lip ui) p dµ ≤ ∑ i µ(Di)r −p i ≤ 2 bM ∑ i r b−p i . (3.6) Finally, with the Poincaré inequality (1.1), (3.4), and (3.6), the regularity lower bound µ(B) ≥ M−1 (R/λ)b gives C′ ≤− ∫ λB |u−uλB| dµ ≤ K(diam B) ( − ∫ B |Lip u|p dµ )1/p ≤ K (2R/λ) µ(B)1/p (∫ B |Lip u|p dµ )1/p ≤ K (2R/λ) M−1/p (R/λ) b/p ( 2bM ∑ i r b−p i )1/p ≤ K (2R/λ) M−1/p (R/λ) b/p (2bM) 1/p (∑ i r b−p i )1/p = C′′ (∑ i r b−p i )1/p . (3.7) Therefore 0 < C ≤ ∑ i r b−p i , where C = ( C′/C′′) p is independent of the covering D. Suppose µ(X) < ∞. We will show that for any Di ∈ D, the radius ri is bounded above by a constant multiple of diam Di, where the constant depends only on X. To this end, consider the ball siDi, where si = (diam Di) −1. Then siDi has radius ri diam Di , and Ahlfors regularity provides 1 M ( ri diam Di )b ≤ µ(siDi) ≤ µ(X) < ∞ , ri ≤ M 1/bµ(X) 1/b diam Di . (3.8) 220 c.a. dimarco In light of (3.8) it is evident that 0 < C ≤ ∑ i r b−p i ≤ ∑ i ( M 1/bµ(X) 1/b )b−p (diam Di) b−p , and hence 0 < ∑ i(diam Di) b−p. Therefore dimH ∂U ≥ b−p for any such U, from which it follows that dimtH X ≥ b−p + 1. If µ(X) = ∞, put E = B(z0,a), 0 < a < diam X, and notice that E is complete and inherits both the Ahlfors b-regularity and (1,p)-Poincaré properties from X (with the same constants M,b,p, and λ). By Ahlfors regularity µ(E) ≤ Mab < ∞, so E satisfies the assumptions of the theorem in the case that has already been proven. Finally, monotonicity of tH-dimension shows that dimtH X ≥ dimtH E ≥ b−p + 1 . If p = 1, then equality holds in Theorem 3.1 because (2.1) guarantees that dimtH X ≤ dimH X = b, but whether equality can be achieved for some (1,p)-Poincaré space (X,µ) with p > 1 is a mystery. Question 3.2. Is there a number p > 1 with a space (X,µ) for which equality holds in Theorem 3.1? In order to answer Question 3.2, one needs a supply of spaces that support weak (1,p)-Poincaré inequalities for p > 1. Theorem 1.5 provides one source of potential examples. It is tempting to try to answer Question 3.2 with a carpet Sa = (Sa, |·|,µ) that supports a weak (1,p)-Poincaré inequality with p > 1. A problem arises, however, once one computes the tH-dimension of this space. Indeed, since Sa is Ahlfors 2-regular [8], dimH Sa = 2, and in order to have equality in Theorem 3.1, we would need dimtH Sa = 3 −p. Let Ca be the Cantor set in [0, 1] obtained from the sequence of scaling factors a. Since (Ca × [0, 1]) ⊂ Sa we see that dimtH Sa ≥ dimtH(Ca×[0, 1]) = 2 by monotonicity and additivity of tH-dimension. Therefore dimtH Sa = 2, and the equation dimtH Sa = 3−p is untenable because we assumed p > 1. Acknowledgements This paper is based on a part of a PhD thesis written by the author under the supervision of Leonid Kovalev at Syracuse University. topological hausdorff dimension 221 References [1] R. Balka, Z. Buczolich, M. Elekes, A new fractal dimension: the topological Hausdorff dimension, Adv. Math. 274 (2015), 881 – 927. [2] A. Björn, J. Björn, “ Nonlinear Potential Theory on Metric Spaces ”, EMS Tracts in Mathematics 17, European Mathematical Society (EMS), Zürich, 2011. [3] K. Falconer, “ Fractal Geometry ”, Second edition, Mathematical founda- tions and applications, John Wiley & Sons, Inc., Hoboken, NJ, 2003. [4] J. Heinonen, “ Lectures on Analysis on Metric Spaces ”, Universitext, Springer-Verlag, New York, 2001. [5] J. Heinonen, P. Koskela, N. Shanmugalingam, J.T. Tyson, “ Sobolev Spaces on Metric Measure Spaces. An Approach based on Up- per Gradients ”, New Mathematical Monographs 27, Cambridge University Press, Cambridge, 2015. [6] A. Lohvansuu, K. Rajala, Duality of moduli in regular metric spaces, Indiana Univ. Math. J. 70 (3) (2021), 1087 – 1102. [7] H. Lotfi, The µ-topological Hausdorff dimension, Extracta Math. 34 (2) (2019), 237 – 254. [8] J.M. Mackay, J.T. Tyson, K. Wildrick, Modulus and Poincaré in- equalities on non-self-similar Sierpiński carpets, Geom. Funct. Anal., 23 (3) (2013), 985 – 1034. Introduction Preliminaries A lower bound on topological Hausdorff dimension for Poincaré Ahlfors regular spaces