Статті в журналах: "Measure metric space"

1

Wildrick, K., and T. Zürcher. "Space filling with metric measure spaces." Mathematische Zeitschrift 270, no. 1-2 (November 3, 2010): 103–31. http://dx.doi.org/10.1007/s00209-010-0787-1.

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2

Scheepers, Marion. "Finite powers of strong measure zero sets." Journal of Symbolic Logic 64, no. 3 (September 1999): 1295–306. http://dx.doi.org/10.2307/2586631.

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AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

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3

Heikkinen, Toni, Juha Lehrbäck, Juho Nuutinen, and Heli Tuominen. "Fractional Maximal Functions in Metric Measure Spaces." Analysis and Geometry in Metric Spaces 1 (May 28, 2013): 147–62. http://dx.doi.org/10.2478/agms-2013-0002.

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Abstract We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

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4

Edgar. "PACKING MEASURE IN GENERAL METRIC SPACE." Real Analysis Exchange 26, no. 2 (2000): 831. http://dx.doi.org/10.2307/44154081.

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5

Han, Bang-Xian, and Andrea Mondino. "Angles between Curves in Metric Measure Spaces." Analysis and Geometry in Metric Spaces 5, no. 1 (September 2, 2017): 47–68. http://dx.doi.org/10.1515/agms-2017-0003.

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Abstract The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD*(K, N) metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.

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6

Yaoyao, Han, and Zhao Kai. "Herz type Hardy spaces on non-homogeneous metric measure space." SCIENTIA SINICA Mathematica 48, no. 10 (October 1, 2018): 1315. http://dx.doi.org/10.1360/n012018-00118.

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7

Mazurenko, N., and M. Zarichnyi. "Invariant idempotent measures." Carpathian Mathematical Publications 10, no. 1 (July 3, 2018): 172–78. http://dx.doi.org/10.15330/cmp.10.1.172-178.

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The idempotent mathematics is a part of mathematics in which arithmetic operations in the reals are replaced by idempotent operations. In the idempotent mathematics, the notion of idempotent measure (Maslov measure) is a counterpart of the notion of probability measure. The idempotent measures found numerous applications in mathematics and related areas, in particular, the optimization theory, mathematical morphology, and game theory. In this note we introduce the notion of invariant idempotent measure for an iterated function system in a complete metric space. This is an idempotent counterpart of the notion of invariant probability measure defined by Hutchinson. Remark that the notion of invariant idempotent measure was previously considered by the authors for the class of ultrametric spaces. One of the main results is the existence and uniqueness theorem for the invariant idempotent measures in complete metric spaces. Unlikely to the corresponding Hutchinson's result for invariant probability measures, our proof does not rely on metrization of the space of idempotent measures. An analogous result can be also proved for the so-called in-homogeneous idempotent measures in complete metric spaces. Also, our considerations can be extended to the case of the max-min measures in complete metric spaces.

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8

Aïssaoui, Noureddine. "Strongly nonlinear potential theory on metric spaces." Abstract and Applied Analysis 7, no. 7 (2002): 357–74. http://dx.doi.org/10.1155/s1085337502203024.

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We define Orlicz-Sobolev spaces on an arbitrary metric space with a Borel regular outer measure, and we develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. We prove that each Orlicz-Sobolev function has a quasi-continuous representative. We give estimates for the capacity of balls when the measure is doubling. Under additional regularity assumption on the measure, we establish some relations between capacity and Hausdorff measures.

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9

Xie, Jialiang, Qingguo Li, Shuili Chen, and Huan Huang. "The fuzzy metric space based on fuzzy measure." Open Mathematics 14, no. 1 (January 1, 2016): 603–12. http://dx.doi.org/10.1515/math-2016-0051.

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Abstract In this paper, we study the relation between a fuzzy measure and a fuzzy metric which is induced by the fuzzy measure. We also discuss some basic properties of the constructed fuzzy metric space. In particular, we show that the nonatom of fuzzy measure space can be characterized in the constructed fuzzy metric space.

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10

Honda, Shouhei. "Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces." Analysis and Geometry in Metric Spaces 6, no. 1 (October 1, 2018): 129–45. http://dx.doi.org/10.1515/agms-2018-0007.

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Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.

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11

Rajala, Kai, Martti Rasimus, and Matthew Romney. "Uniformization with Infinitesimally Metric Measures." Journal of Geometric Analysis 31, no. 11 (May 26, 2021): 11445–70. http://dx.doi.org/10.1007/s12220-021-00689-y.

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AbstractWe consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to $${{\mathbb {R}}}^2$$ R 2 . Given a measure $$\mu $$ μ on such a space, we introduce $$\mu $$ μ -quasiconformal maps$$f:X \rightarrow {{\mathbb {R}}}^2$$ f : X → R 2 , whose definition involves deforming lengths of curves by $$\mu $$ μ . We show that if $$\mu $$ μ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $$\mu $$ μ -quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.

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12

Willerton, Simon. "Spread: A Measure of the Size of Metric Spaces." International Journal of Computational Geometry & Applications 25, no. 03 (September 2015): 207–25. http://dx.doi.org/10.1142/s0218195915500120.

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Motivated by Leinster-Cobbold measures of biodiversity, the notion of the spread of a finite metric space is introduced. This is related to Leinster’s magnitude of a metric space. Spread is generalized to infinite metric spaces equipped with a measure and is calculated for spheres and straight lines. For Riemannian manifolds the spread is related to the volume and total scalar curvature. A notion of scale-dependent dimension is introduced and seen for approximations to certain fractals to be numerically close to the Minkowski dimension of the original fractals.

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13

Lee, Keonhee, C. A. Morales, and Bomi Shin. "On the set of expansive measures." Communications in Contemporary Mathematics 20, no. 07 (October 14, 2018): 1750086. http://dx.doi.org/10.1142/s0219199717500869.

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We prove that the set of expansive measures of a homeomorphism of a compact metric space is a [Formula: see text] subset of the space of Borel probability measures equipped with the weak* topology. Next that every expansive measure of a homeomorphism of a compact metric space can be weak* approximated by expansive measures with invariant support. In addition, if the expansive measures of a homeomorphism of a compact metric space are dense in the space of Borel probability measures, then there is an expansive measure whose support is both invariant and close to the whole space with respect to the Hausdorff metric. Henceforth, if the expansive measures are dense in the space of Borel probability measures, the set of heteroclinic points has no interior and the space has no isolated points.

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14

Zhao, Fei, and Si Zong Guo. "L_p-Type of Weighted Fuzzy Number Metrics Induced by Fuzzy Structured Element." Advanced Materials Research 981 (July 2014): 279–86. http://dx.doi.org/10.4028/www.scientific.net/amr.981.279.

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For the objective fact that elements with different membership degrees should have different contribution to the metric measure between fuzzy numbers, this paper presents Lp-type of fuzzy number metrics weighted by structured element. Firstly, we define a metric weighted by structured element on the family (B[-1,1]) of all the same monotone and standard bounded functions on closed interval [-1,1] , and discuss the completeness and separability of those metric spaces; Next, using the fuzzy functional induced by normal fuzzy structured element, we give out a method that the metric of the closed bounded fuzzy number space is induced by the metric on function space B[-1,1]. Furthermore, a fuzzy number metric weighted by structured element which is induced by is presented ,and analyze completeness and separability of the induced fuzzy number metric spaces; Lastly, the difference and relationship between and the metric defined by traditional method are shown.

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15

Ambrosio, Luigi, Andrea Pinamonti, and Gareth Speight. "Weighted Sobolev spaces on metric measure spaces." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 746 (January 1, 2019): 39–65. http://dx.doi.org/10.1515/crelle-2016-0009.

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Abstract We investigate weighted Sobolev spaces on metric measure spaces {(X,\mathrm{d},\mathfrak{m})} . Denoting by ρ the weight function, we compare the space {W^{1,p}(X,\mathrm{d},\rho\mathfrak{m})} (which always coincides with the closure {H^{1,p}(X,\mathrm{d},\rho\mathfrak{m})} of Lipschitz functions) with the weighted Sobolev spaces {W^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})} and {H^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})} defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that {W^{1,p}(X,\mathrm{d},\rho\mathfrak{m})=H^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{% m})} . We also adapt the results in [23] and in the recent paper [27] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality {W^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})=H^{1,p}_{\rho}(X,\mathrm{d},% \mathfrak{m})} .

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16

Srivastava, Pramila, Mona Khare, and Y. K. Srivastava. "A fuzzy measure algebra as a metric space." Fuzzy Sets and Systems 79, no. 3 (May 1996): 395–400. http://dx.doi.org/10.1016/0165-0114(95)00177-8.

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17

Donatella Bongiorno and Giuseppa Corrao. "An Integral on a Complete Metric Measure Space." Real Analysis Exchange 40, no. 1 (2015): 157. http://dx.doi.org/10.14321/realanalexch.40.1.0157.

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18

Romanovskiĭ, N. N. "Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings." Siberian Mathematical Journal 54, no. 2 (March 2013): 353–67. http://dx.doi.org/10.1134/s0037446613020171.

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19

LU, GUANGHUI, and SHUANGPING TAO. "GENERALIZED MORREY SPACES OVER NONHOMOGENEOUS METRIC MEASURE SPACES." Journal of the Australian Mathematical Society 103, no. 2 (October 27, 2016): 268–78. http://dx.doi.org/10.1017/s1446788716000483.

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Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

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20

Björn, Anders, Jana Björn, and Nageswari Shanmugalingam. "Classification of metric measure spaces and their ends using p-harmonic functions." Annales Fennici Mathematici 47, no. 2 (July 16, 2022): 1025–52. http://dx.doi.org/10.54330/afm.120618.

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By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite $p$-energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local $p$-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the $p$-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant $p$-harmonic functions with finite $p$-energy as spaces having at least two well-separated $p$-hyperbolic sequences of sets towards infinity. We also show that every such space $X$ has a function $f \notin L^p(X) + \mathbf{R}$ with finite $p$-energy.

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21

Hernández, Kevin Alejandro, D. Cárdenas Peña, and Álvaro A. Orozco. "A space-structure based dissimilarity measure for categorical data." International Journal of Electrical and Computer Engineering (IJECE) 11, no. 1 (February 1, 2021): 620. http://dx.doi.org/10.11591/ijece.v11i1.pp620-627.

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The development of analysis methods for categorical data begun in 90's decade, and it has been booming in the last years. On the other hand, the performance of many of these methods depends on the used metric. Therefore, determining a dissimilarity measure for categorical data is one of the most attractive and recent challenges in data mining problems. However, several similarity/dissimilarity measures proposed in the literature have drawbacks due to high computational cost, or poor performance. For this reason, we propose a new distance metric for categorical data. We call it: Weighted pairing (W-P) based on feature space-structure, where the weights are understood like a degree of contribution of an attribute to the compact cluster structure. The performance of W-P metric was evaluated in the unsupervised learning framework in terms of cluster quality index. We test the W-P in six real categorical datasets downloaded from the public UCI repository, and we make a comparison with the distance metric (DM3) method and hamming metric (H-SBI). Results show that our proposal outperforms DM3 and H-SBI in different experimental configurations. Also, the W-P achieves highest rand index values and a better clustering discriminant than the other methods.

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22

EDALAT, ABBAS. "When Scott is weak on the top." Mathematical Structures in Computer Science 7, no. 5 (October 1997): 401–17. http://dx.doi.org/10.1017/s0960129597002338.

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We construct an approximating chain of simple valuations on the upper space of a compact metric space whose lub is a given probability measure on the metric space. We show that whenever a separable metric space is homeomorphic to a Gδ subset of an ω-continuous dcpo equipped with its Scott topology, the space of probability measures of the metric space equipped with the weak topology is homeomorphic with a subset of the maximal elements of the probabilistic power domain of the ω-continuous dcpo. Given an effective approximation of a probability measure by an increasing chain of normalised valuations on the upper space of a compact metric space, we show that the expected value of any Hölder continuous function on the space can be obtained up to any given accuracy. We present a novel application in computing integrals in dynamical systems. We obtain an algorithm to compute the expected value of any Hölder continuous function with respect to the unique invariant measure of the Feigenbaum map in the periodic doubling route to chaos.

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23

Yukich, J. E. "The convolution metric dg." Mathematical Proceedings of the Cambridge Philosophical Society 98, no. 3 (November 1985): 533–40. http://dx.doi.org/10.1017/s0305004100063738.

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SummaryWe introduce and study a new metric on denned bywhere is the space of probability measures on ℝk and where g: ℝk→ is a probability density satisfying certain mild conditions. The metric dg, relatively easy to compute, is shown to have useful and interesting properties not enjoyed by some other metrics on . In particular, letting pn denote the nth empirical measure for P, it is shown that under appropriate conditions satisfies a compact law of the iterated logarithm, converges in probability to the supremum of a Gaussian process, and has a useful stochastic integral representation.

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24

Perdue, Nicholas Adam. "The Vertical Space Problem." Cartographic Perspectives, no. 74 (October 7, 2013): 9–28. http://dx.doi.org/10.14714/cp74.83.

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Populations in contemporary cities are being measured, analyzed, or represented in less than optimal ways. Conventional methods of measuring density of populations in cities rely on calculating the number of people living within a bounded surface space. This approach fails to account for the multiple floor residential patterns of the contemporary urban landscape and exposes the vertical space problem in population analytics. To create an accurate representation of people in contemporary urban spaces, a move beyond the conventional conception of density is needed. This research aims to find a more appropriate solution to mapping humans in cities by employing a dasymetric method to represent the distribution of people in a city of vertical residential structures. The methodology creates an index to classify the amount of floor space for each person across the extent of the city, a metric called the personal space measure. The personal space measure is juxtaposed with the conventional population density measurements to provide a unique perspective on how population is concentrated across the urban space. The personal space metric demonstrates how improved metrics can be employed to better understand the social and structural landscape of cities. Chicago, with a large population and a high vertical extent, makes an ideal case study to develop a methodology to capture the phenomena of urban living in the 21st century and to explain alternative approaches to accurately and intelligently analyze the contemporary urban space.

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25

WANG, GUOFANG, and DELIANG XU. "HARMONIC MAPS FROM SMOOTH METRIC MEASURE SPACES." International Journal of Mathematics 23, no. 09 (July 31, 2012): 1250095. http://dx.doi.org/10.1142/s0129167x12500954.

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In this paper, we study a generalized harmonic map, ϕ-harmonic map, from a smooth metric measure space (M, g, e-ϕ dv) into a Riemannian manifold. We proved various rigidity results for the ϕ-harmonic maps under conditions in terms of the Bakry–Émery Ricci tensor.

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26

Soria, Javier, and Pedro Tradacete. "The least doubling constant of a metric measure space." Annales Academiae Scientiarum Fennicae Mathematica 44, no. 2 (June 2019): 1015–30. http://dx.doi.org/10.5186/aasfm.2019.4457.

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27

Gui, Yaoting. "A Singular Moser-Trudinger Inequality on Metric Measure Space." Journal of Partial Differential Equations 35, no. 4 (June 2022): 331–43. http://dx.doi.org/10.4208/jpde.v35.n4.3.

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28

Luukkainen, Jouni, and Eero Saksman. "Every complete doubling metric space carries a doubling measure." Proceedings of the American Mathematical Society 126, no. 2 (1998): 531–34. http://dx.doi.org/10.1090/s0002-9939-98-04201-4.

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29

Wen, Shengyou, and Min Wu. "Relations between packing premeasure and measure on metric space." Acta Mathematica Scientia 27, no. 1 (January 2007): 137–44. http://dx.doi.org/10.1016/s0252-9602(07)60012-5.

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30

Ozawa, Ryunosuke. "Concentration function for pyramid and quantum metric measure space." Proceedings of the American Mathematical Society 145, no. 3 (November 21, 2016): 1301–15. http://dx.doi.org/10.1090/proc/13282.

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31

Wang, Hailian, and Rulong Xie. "Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces." Open Mathematics 19, no. 1 (January 1, 2021): 1779–800. http://dx.doi.org/10.1515/math-2021-0139.

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Abstract Let ( X , d , μ ) \left(X,d,\mu ) denote nonhomogeneous metric measure space satisfying geometrically doubling and the upper doubling measure conditions. In this paper, the boundedness in Lebesgue spaces for two kinds of commutators, which are iterated commutators and commutators in summation form, generated by multilinear strongly singular integral operators with RBMO ( μ ) \left(\mu ) function on nonhomogeneous metric measure spaces ( X , d , μ ) \left(X,d,\mu ) is obtained.

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32

HYTÖNEN, TUOMAS, DACHUN YANG, and DONGYONG YANG. "The Hardy space H1 on non-homogeneous metric spaces." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 1 (December 8, 2011): 9–31. http://dx.doi.org/10.1017/s0305004111000776.

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AbstractLet (, d, μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. We introduce the atomic Hardy space H1(μ) and prove that its dual space is the known space RBMO(μ) in this context. Using this duality, we establish a criterion for the boundedness of linear operators from H1(μ) to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón–Zygmund operators from H1(μ) to L1(μ).

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33

Martio, Olli. "An alternative capacity in metric measure spaces." Ukrainian Mathematical Bulletin 18, no. 2 (July 9, 2021): 196–208. http://dx.doi.org/10.37069/1810-3200-2021-18-2-4.

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A new condenser capacity $\CMp(E,G)$ is introduced as an alternative to the classical Dirichlet capacity in a metric measure space $X$. For $p>1$, it coincides with the $M_p$-modulus of the curve family $\Gamma(E,G)$ joining $\partial G$ to an arbitrary set $E \subset G$ and, for $p = 1$, it lies between $AM_1(\Gamma(E,G))$ and $M_1(\Gamma(E,G))$. Moreover, the $\CMp(E,G)$-capacity has good measure theoretic regularity properties with respect to the set $E$. The $\CMp(E,G)$-capacity uses Lipschitz functions and their upper gradients. The doubling property of the measure $\mu$ and Poincar\'e inequalities in $X$ are not needed.

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34

Lee, K., and A. Rojas. "On Almost Shadowable Measures." Nelineinaya Dinamika 18, no. 2 (2022): 297–307. http://dx.doi.org/10.20537/nd220210.

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In this paper we study the almost shadowable measures for homeomorphisms on compact metric spaces. First, we give examples of measures that are not shadowable. Next, we show that almost shadowable measures are weakly shadowable, namely, that there are Borelians with a measure close to $1$ such that every pseudo-orbit through it can be shadowed. Afterwards, the set of weakly shadowable measures is shown to be an $F_{\sigma\delta}$ subset of the space of Borel probability measures. Also, we show that the weakly shadowable measures can be weakly* approximated by shadowable ones. Furthermore, the closure of the set of shadowable points has full measure with respect to any weakly shadowable measure. We show that the notions of shadowableness, almost shadowableness and weak shadowableness coincide for finitely supported measures, or, for every measure when the set of shadowable points is closed. We investigate the stability of weakly shadowable expansive measures for homeomorphisms on compact metric spaces.

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35

Kim, Yong-Hyuk, and Yourim Yoon. "Linkage-Based Distance Metric in the Search Space of Genetic Algorithms." Mathematical Problems in Engineering 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/680624.

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We propose a new distance metric, based on the linkage of genes, in the search space of genetic algorithms. This second-order distance measure is derived from the gene interaction graph and first-order distance, which is a natural distance in chromosomal spaces. We show that the proposed measure forms a metric space and can be computed efficiently. As an example application, we demonstrate how this measure can be used to estimate the extent to which gene rearrangement improves the performance of genetic algorithms.

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36

KLOECKNER, BENOÎT. "A GENERALIZATION OF HAUSDORFF DIMENSION APPLIED TO HILBERT CUBES AND WASSERSTEIN SPACES." Journal of Topology and Analysis 04, no. 02 (June 2012): 203–35. http://dx.doi.org/10.1142/s1793525312500094.

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A Wasserstein space is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be made precise.In the first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a natural set of spaces generalizing the usual Hilbert cube. These invariants are very similar to concepts initiated by Rogers, but our variant is specifically suited to tackle Lipschitz comparison.In the second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide uniform bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d. These arguments are very easily adapted to study the space of closed subsets of a compact metric space, partly generalizing results of Boardman, Goodey and McClure.

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37

CHEN, LI. "HARDY SPACES ON METRIC MEASURE SPACES WITH GENERALIZED SUB-GAUSSIAN HEAT KERNEL ESTIMATES." Journal of the Australian Mathematical Society 104, no. 2 (August 14, 2017): 162–94. http://dx.doi.org/10.1017/s144678871700012x.

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Hardy space theory has been studied on manifolds or metric measure spaces equipped with either Gaussian or sub-Gaussian heat kernel behaviour. However, there are natural examples where one finds a mix of both behaviours (locally Gaussian and at infinity sub-Gaussian), in which case the previous theory does not apply. Still we define molecular and square function Hardy spaces using appropriate scaling, and we show that they agree with Lebesgue spaces in some range. Besides, counterexamples are given in this setting that the $H^{p}$ space corresponding to Gaussian estimates may not coincide with $L^{p}$. As a motivation for this theory, we show that the Riesz transform maps our Hardy space $H^{1}$ into $L^{1}$.

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38

Buckley, Stephen M., and Simon L. Kokkendorff. "The Spherical Boundary and Volume Growth." ISRN Geometry 2012 (March 26, 2012): 1–13. http://dx.doi.org/10.5402/2012/484312.

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We consider the spherical boundary, a conformal boundary using a special class of conformal distortions. We prove that certain bounds on volume growth of suitable metric measure spaces imply that the spherical boundary is “small” (in cardinality or dimension) and give examples to show that the reverse implications fail. We also show that the spherical boundary of an annular convex proper length space consists of a single point. This result applies to l2-products of length spaces, since we prove that a natural metric, generalizing such “norm-like” product metrics on a (possibly infinite) product of unbounded length spaces, is annular convex.

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39

Yang, Dachun, and Yong Lin. "SPACES OF LIPSCHITZ TYPE ON METRIC SPACES AND THEIR APPLICATIONS." Proceedings of the Edinburgh Mathematical Society 47, no. 3 (October 2004): 709–52. http://dx.doi.org/10.1017/s0013091503000907.

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AbstractNew spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.AMS 2000 Mathematics subject classification: Primary 42B35. Secondary 46E35; 58J35; 43A99

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40

Martio, O. "The space of functions of bounded variation on curves in metric measure spaces." Conformal Geometry and Dynamics of the American Mathematical Society 20, no. 5 (April 19, 2016): 81–96. http://dx.doi.org/10.1090/ecgd/291.

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41

Ambrosio, Luigi, and Simone Di Marino. "Equivalent definitions of BV space and of total variation on metric measure spaces." Journal of Functional Analysis 266, no. 7 (April 2014): 4150–88. http://dx.doi.org/10.1016/j.jfa.2014.02.002.

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42

Guliyev, Vagif S., and Stefan G. Samko. "Maximal Operator in Variable Exponent Generalized Morrey Spaces on Quasi-metric Measure Space." Mediterranean Journal of Mathematics 13, no. 3 (April 16, 2015): 1151–65. http://dx.doi.org/10.1007/s00009-015-0561-z.

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43

Wang, Juan. "A Metric on the Space of Partly Reduced Phylogenetic Networks." BioMed Research International 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/7534258.

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Phylogenetic networks are a generalization of phylogenetic trees that allow for the representation of evolutionary events acting at the population level, such as recombination between genes, hybridization between lineages, and horizontal gene transfer. The researchers have designed several measures for computing the dissimilarity between two phylogenetic networks, and each measure has been proven to be a metric on a special kind of phylogenetic networks. However, none of the existing measures is a metric on the space of partly reduced phylogenetic networks. In this paper, we provide a metric,de-distance, on the space of partly reduced phylogenetic networks, which is polynomial-time computable.

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44

Li, Yanlin, Abimbola Abolarinwa, Ali H. Alkhaldi, and Akram Ali. "Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces." Mathematics 10, no. 23 (December 2, 2022): 4580. http://dx.doi.org/10.3390/math10234580.

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A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric measure space without any geometric constraint on the potential function. The adopted approach highlights some criteria for a smooth metric measure space to admit Hardy inequalities related to Witten and Witten p-Laplace operators. The results in this paper complement in several aspect to those obtained recently in the non-compact setting.

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45

LI, GANG. "A METRIC ON SPACE OF MEASURABLE FUNCTIONS AND THE RELATED CONVERGENCE." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 20, no. 02 (April 2012): 211–22. http://dx.doi.org/10.1142/s0218488512500109.

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A new metric is proposed on the space of measurable functions in the setting of non-additive measure theory. The convergence induced from the metric can be used to describe the convergence in measure for sequences of measurable functions. Furthermore, the space of measurable functions is complete under the metric.

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46

Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré. "Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space." EMS Newsletter 2017-3, no. 103 (2017): 19–28. http://dx.doi.org/10.4171/news/103/4.

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47

HORVÁTH, Á. G. "Normally Distributed Probability Measure on the Metric Space of Norms." Acta Mathematica Scientia 33, no. 5 (September 2013): 1231–42. http://dx.doi.org/10.1016/s0252-9602(13)60076-4.

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48

Case, Jeffrey S. "The weighted σk-curvature of a smooth metric measure space". Pacific Journal of Mathematics 299, № 2 (21 травня 2019): 339–99. http://dx.doi.org/10.2140/pjm.2019.299.339.

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49

Case, Jeffrey S. "The Energy of a Smooth Metric Measure Space and Applications." Journal of Geometric Analysis 25, no. 1 (July 27, 2013): 616–67. http://dx.doi.org/10.1007/s12220-013-9441-6.

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50

Roy, Kushal, Mantu Saha, Vahid Parvaneh, and Maryam Khorshidi. "A New Variant of Symmetric Distance Spaces and an Extension of the Banach Fixed-Point Theorem." Journal of Mathematics 2022 (March 9, 2022): 1–8. http://dx.doi.org/10.1155/2022/9282762.

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The notion of Δ -metric spaces has been proposed in this study as a generalization of b -metric spaces, extended b -metric spaces, and p -metric spaces. A number of topological characteristics of such spaces have been investigated in this paper. On such spaces, a noncompactness measure has been established, and some results in the framework of noncompactness measure have been achieved. We prove an analogous of the Banach contraction principle in such spaces based on this approach. In order to investigate the validity of the underlying space and our proven fixed-point theorems, supporting examples have been presented. Furthermore, the well-posedness of the fixed-point problem has been tested using our fixed-point result.

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