2015, New mathematical monographs, ISBN 9781107092341, Volume 27, xii, 434

.... Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincaré inequality...

Sobolev spaces | Metric spaces

Sobolev spaces | Metric spaces

Book

2.
Geometry and dynamics in Gromov hyperbolic metric spaces

: with an emphasis on non-proper settings

2017, Mathematical surveys and monographs, ISBN 9781470434656, Volume 218, xxxv, 281 pages

Geometry, Hyperbolic | Ergodic theory | Measure and integration | Fuchsian groups and their generalizations | Infinite-dimensional Lie groups and their Lie algebras: general properties | Hyperbolic groups and nonpositively curved groups | Special aspects of infinite or finite groups | Semigroups of transformations, etc | Metric spaces | Conformal densities and Hausdorff dimension | Structure and classification of infinite or finite groups | Relations with number theory and harmonic analysis | Classical measure theory | Group theory and generalizations | Other groups of matrices | Complex dynamical systems | Lie groups | Hyperbolic spaces | Semigroups | Groups acting on trees | Topological groups, Lie groups | Dynamical systems and ergodic theory | Hausdorff and packing measures

Book

3.
Full Text
Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

Probability Theory and Related Fields, ISSN 0178-8051, 09/2009, Volume 145, Issue 1-2, pp. 285 - 322

We consider the space of complete and separable metric spaces which are equipped with a probability measure...

Gromov metric triple | Gromov-Hausdorff topology | Wasserstein metric | Weak topology | Λ-Coalescent | Prohorov metric | Metric measure spaces | ℝ-trees | weak topology | STATISTICS & PROBABILITY | R-trees | Lambda-Coalescent | Studies | Probability

Gromov metric triple | Gromov-Hausdorff topology | Wasserstein metric | Weak topology | Λ-Coalescent | Prohorov metric | Metric measure spaces | ℝ-trees | weak topology | STATISTICS & PROBABILITY | R-trees | Lambda-Coalescent | Studies | Probability

Journal Article

Duke mathematical journal, ISSN 0012-7094, 2014, Volume 163, Issue 7, pp. 1405 - 1490

In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d, m...

MATHEMATICS | SOBOLEV SPACES | TRANSPORT | INEQUALITIES | STABILITY | COMPACT ALEXANDROV SPACES | FINSLER MANIFOLDS | WASSERSTEIN SPACES | HEAT-FLOW | GRADIENT FLOWS | GEOMETRY | 60J65 | 35K05

MATHEMATICS | SOBOLEV SPACES | TRANSPORT | INEQUALITIES | STABILITY | COMPACT ALEXANDROV SPACES | FINSLER MANIFOLDS | WASSERSTEIN SPACES | HEAT-FLOW | GRADIENT FLOWS | GEOMETRY | 60J65 | 35K05

Journal Article

中国科学：数学英文版, ISSN 1674-7283, 2015, Volume 58, Issue 2, pp. 309 - 388

Let（X,d,）be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions.Let ρ∈（1,∞）,0〈p≤1≤q≤∞,p≠q,γ∈[1,∞）and ∈ ∈（0...

Weiss | Hardy空间 | 测度空间 | 幽门螺杆菌 | 分数次积分 | Campanato空间 | 应用 | ATB | molecular block | Campanato space | Lipschitz space | Mathematics | 30L99 | non-homogeneous metric measure space | ρ -weakly doubling measure | 42B35 | 42B30 | 42B20 | Calderón-Zygmund operator | Applications of Mathematics | atomic block | Hardy space | Non-homogeneous metric measure space | ρ-weakly doubling measure | Atomic block | Molecular block

Weiss | Hardy空间 | 测度空间 | 幽门螺杆菌 | 分数次积分 | Campanato空间 | 应用 | ATB | molecular block | Campanato space | Lipschitz space | Mathematics | 30L99 | non-homogeneous metric measure space | ρ -weakly doubling measure | 42B35 | 42B30 | 42B20 | Calderón-Zygmund operator | Applications of Mathematics | atomic block | Hardy space | Non-homogeneous metric measure space | ρ-weakly doubling measure | Atomic block | Molecular block

Journal Article

Inventiones mathematicae, ISSN 1432-1297, 2013, Volume 195, Issue 2, pp. 289 - 391

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m...

52C23 | 49J52 | 35K90 | Mathematics, general | Mathematics | 58J35 | 49Q20 | 31C25 | EXISTENCE | MATHEMATICS | EQUATIONS | CURVATURE CONDITIONS | GEOMETRY | Mathematical analysis | Dirichlet problem | Calculus | Heat transmission | Entropy | Density | Curvature | Heat transfer

52C23 | 49J52 | 35K90 | Mathematics, general | Mathematics | 58J35 | 49Q20 | 31C25 | EXISTENCE | MATHEMATICS | EQUATIONS | CURVATURE CONDITIONS | GEOMETRY | Mathematical analysis | Dirichlet problem | Calculus | Heat transmission | Entropy | Density | Curvature | Heat transfer

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 07/2015, Volume 367, Issue 7, pp. 4661 - 4701

é condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits...

MATHEMATICS | DIMENSION CONDITION | LIPSCHITZ FUNCTIONS | HEAT-FLOW | GEOMETRY | Differential Geometry | Metric Geometry | Mathematics

MATHEMATICS | DIMENSION CONDITION | LIPSCHITZ FUNCTIONS | HEAT-FLOW | GEOMETRY | Differential Geometry | Metric Geometry | Mathematics

Journal Article

Acta mathematica, ISSN 0001-5962, 2006, Volume 196, Issue 1, pp. 65 - 131

We introduce and analyze lower (Ricci) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾ K for metric measure spaces...

Mathematics, general | Mathematics | Manifolds | Metric space | Mathematical analysis | Transportation | Isomorphism | Entropy | Curvature | Estimates | Optimization

Mathematics, general | Mathematics | Manifolds | Metric space | Mathematical analysis | Transportation | Isomorphism | Entropy | Curvature | Estimates | Optimization

Journal Article

Czechoslovak Mathematical Journal, ISSN 0011-4642, 3/2019, Volume 69, Issue 1, pp. 207 - 223

... from Orlicz spaces L Φ(X) near L 1(X) to Orlicz spaces L Ψ(X) over metric measure spaces equipped with lower Ahlfors Q-regular measures, where Φ...

metric measure space | Mathematics | fractional integral | 31B15 | Ordinary Differential Equations | 46E35 | Analysis | Convex and Discrete Geometry | 46E30 | Mathematics, general | Orlicz space | Mathematical Modeling and Industrial Mathematics | lower Ahlfors regular | Riesz potential | MATHEMATICS | SOBOLEV EMBEDDINGS | INEQUALITIES | CALDERON-ZYGMUND OPERATORS | MORREY SPACES | RIESZ-POTENTIALS

metric measure space | Mathematics | fractional integral | 31B15 | Ordinary Differential Equations | 46E35 | Analysis | Convex and Discrete Geometry | 46E30 | Mathematics, general | Orlicz space | Mathematical Modeling and Industrial Mathematics | lower Ahlfors regular | Riesz potential | MATHEMATICS | SOBOLEV EMBEDDINGS | INEQUALITIES | CALDERON-ZYGMUND OPERATORS | MORREY SPACES | RIESZ-POTENTIALS

Journal Article

Discrete and Continuous Dynamical Systems- Series A, ISSN 1078-0947, 04/2014, Volume 34, Issue 4, pp. 1641 - 1661

We prove that the linear "heat" flow in a RCD(K, infinity) metric measure space (X, d, m) satisfies a contraction property with respect to every L-p-Kantorovich-Rubinstein-Wasserstein distance, p is an element...

Γ-calculus | Dirichlet forms | Ricci curvature | Metric-measure spaces | Optimal transport | EXISTENCE | MATHEMATICS, APPLIED | STABILITY | INEQUALITY | optimal transport | FOKKER-PLANCK EQUATIONS | MATHEMATICS | metric-measure spaces | MANIFOLDS | Gamma-calculus | GEOMETRY

Γ-calculus | Dirichlet forms | Ricci curvature | Metric-measure spaces | Optimal transport | EXISTENCE | MATHEMATICS, APPLIED | STABILITY | INEQUALITY | optimal transport | FOKKER-PLANCK EQUATIONS | MATHEMATICS | metric-measure spaces | MANIFOLDS | Gamma-calculus | GEOMETRY

Journal Article

Forum mathematicum, ISSN 1435-5337, 2011, Volume 25, Issue 4, pp. - - -

On a metric measure space satisfying the doubling property, we establish several optimal characterizations of Besov and Triebel...

Triebel–Lizorkin space | Hajłasz–Triebel–Lizorkin space | Hajłasz–Besov space | sharp maximal function | Besov space | metric measure space | MATHEMATICS | MATHEMATICS, APPLIED | SOBOLEV SPACES | MAXIMAL FUNCTIONS | Triebel-Lizorkin space | INEQUALITY | Hajlasz-Besov space | EXTENSION | Hajlasz-Triebel-Lizorkin space

Triebel–Lizorkin space | Hajłasz–Triebel–Lizorkin space | Hajłasz–Besov space | sharp maximal function | Besov space | metric measure space | MATHEMATICS | MATHEMATICS, APPLIED | SOBOLEV SPACES | MAXIMAL FUNCTIONS | Triebel-Lizorkin space | INEQUALITY | Hajlasz-Besov space | EXTENSION | Hajlasz-Triebel-Lizorkin space

Journal Article

Potential analysis, ISSN 1572-929X, 2018, Volume 51, Issue 1, pp. 127 - 146

We study several problems concerning conformal transformation on metric measure spaces, including the Sobolev space, the differential structure and the curvature-dimension condition under conformal transformations...

53C23 | 51F99 | Probability Theory and Stochastic Processes | 31E05 | Mathematics | 30L99 | Conformal transformation | Metric measure space | Ricci tensor | Geometry | Potential Theory | Functional Analysis | Bakry-Émery theory | Curvature-dimension condition | MATHEMATICS | Bakry-Emery theory | RICCI CURVATURE

53C23 | 51F99 | Probability Theory and Stochastic Processes | 31E05 | Mathematics | 30L99 | Conformal transformation | Metric measure space | Ricci tensor | Geometry | Potential Theory | Functional Analysis | Bakry-Émery theory | Curvature-dimension condition | MATHEMATICS | Bakry-Emery theory | RICCI CURVATURE

Journal Article

13.
Full Text
HARDY SPACES ON METRIC MEASURE SPACES WITH GENERALIZED SUB-GAUSSIAN HEAT KERNEL ESTIMATES

Journal of the Australian Mathematical Society (2001), ISSN 1446-7887, 04/2018, Volume 104, Issue 2, pp. 162 - 194

Hardy space theory has been studied on manifolds or metric measure spaces equipped with either Gaussian or sub-Gaussian heat kernel behaviour...

and phrases Hardy spaces | Riesz transforms | heat kernel estimates | metric measure spaces | MATHEMATICS | LOCAL DIRICHLET SPACES | BOUNDS | Hardy spaces | OPERATORS

and phrases Hardy spaces | Riesz transforms | heat kernel estimates | metric measure spaces | MATHEMATICS | LOCAL DIRICHLET SPACES | BOUNDS | Hardy spaces | OPERATORS

Journal Article

Revista Matematica Iberoamericana, ISSN 0213-2230, 2015, Volume 31, Issue 2, pp. 497 - 530

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1, 1...

Poincaré | Functions of bounded variation | Metric space | Newtonian space | Gauss-Green theorem | Green function | Perimeter measure | Harmonic function | Minkowski content | Singular function | Dirichlet form | Doubling measure | Inequality | doubling measure | MODULUS | THEOREM | perimeter measure | P-HARMONIC FUNCTIONS | harmonic function | metric space | MATHEMATICS | functions of bounded variation | singular function | SETS | Poincare inequality

Poincaré | Functions of bounded variation | Metric space | Newtonian space | Gauss-Green theorem | Green function | Perimeter measure | Harmonic function | Minkowski content | Singular function | Dirichlet form | Doubling measure | Inequality | doubling measure | MODULUS | THEOREM | perimeter measure | P-HARMONIC FUNCTIONS | harmonic function | metric space | MATHEMATICS | functions of bounded variation | singular function | SETS | Poincare inequality

Journal Article

Proceedings of the London Mathematical Society, ISSN 0024-6115, 11/2015, Volume 111, Issue 5, pp. 1071 - 1129

The aim of this paper is to discuss convergence of pointed metric measure spaces in the absence of any compactness condition...

EXISTENCE | MATHEMATICS | TANGENT-CONES | RIEMANNIAN-MANIFOLDS | UNIQUENESS | GEOMETRY | Equivalence | Mathematical analysis | Heat transmission | Spectra | Formulas (mathematics) | Curvature | Heat transfer | Convergence

EXISTENCE | MATHEMATICS | TANGENT-CONES | RIEMANNIAN-MANIFOLDS | UNIQUENESS | GEOMETRY | Equivalence | Mathematical analysis | Heat transmission | Spectra | Formulas (mathematics) | Curvature | Heat transfer | Convergence

Journal Article

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 05/2015, Volume 103, Issue 5, pp. 1228 - 1275

The main result of this article states that the (K,N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD⁎(KN,N+1...

Curvature-dimension condition | Cone | Metric measure space | Maximal diameter | LOCAL DIRICHLET SPACES | MATHEMATICS, APPLIED | LIPSCHITZ FUNCTIONS | RICCI CURVATURE | MATHEMATICS | SOBOLEV SPACES | PRODUCTS | BOUNDS | RIGIDITY | GEOMETRY

Curvature-dimension condition | Cone | Metric measure space | Maximal diameter | LOCAL DIRICHLET SPACES | MATHEMATICS, APPLIED | LIPSCHITZ FUNCTIONS | RICCI CURVATURE | MATHEMATICS | SOBOLEV SPACES | PRODUCTS | BOUNDS | RIGIDITY | GEOMETRY

Journal Article

Abstract and applied analysis, ISSN 1085-3375, 9/2013, Volume 2013, pp. 1 - 8

The authors give a definition of Morrey spaces for nonhomogeneous metric measure spaces and investigate the boundedness of some classical operators including maximal operator, fractional integral...

FRACTIONAL INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | THEOREM | HARDY-SPACES | BOUNDEDNESS | COMMUTATORS | CALDERON-ZYGMUND OPERATORS | POTENTIALS | NON-DOUBLING MEASURES | ANALYTIC CAPACITY | SEMIADDITIVITY | Metric spaces | Research | Integrals | Mathematical research | Operator theory | Operators

FRACTIONAL INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | THEOREM | HARDY-SPACES | BOUNDEDNESS | COMMUTATORS | CALDERON-ZYGMUND OPERATORS | POTENTIALS | NON-DOUBLING MEASURES | ANALYTIC CAPACITY | SEMIADDITIVITY | Metric spaces | Research | Integrals | Mathematical research | Operator theory | Operators

Journal Article

2011, ISBN 1405183691, xvi, 375

"A Probability Metrics Approach to Financial Risk Measures relates the field of probability metrics and risk measures to one another and applies them to finance for the first time...

Financial risk management | Probabilities

Financial risk management | Probabilities

Book

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, ISSN 1435-9855, 2019, Volume 21, Issue 6, pp. 1809 - 1854

We prove that a metric measure space (X, d, m) satisfying finite-dimensional lower Ricci curvature bounds and whose Sobolev space W(1,2...

MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES | rectifiable space | unique tangent space | Ricci curvature | optimal transport | GEOMETRY

MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES | rectifiable space | unique tangent space | Ricci curvature | optimal transport | GEOMETRY

Journal Article

Journal of mathematical analysis and applications, ISSN 0022-247X, 2018, Volume 464, Issue 1, pp. 707 - 724

.... Motivated by recent work on barycentric maps on spaces of probability measures on ordered Banach spaces, we introduce and study a stochastic order on the space of probability measures P(X...

Ordered metric space | Borel probability measure | AGH mean inequalities | Normal cone | Stochastic order | Wasserstein metric | MATHEMATICS | MATHEMATICS, APPLIED | BARYCENTRIC MAPS

Ordered metric space | Borel probability measure | AGH mean inequalities | Normal cone | Stochastic order | Wasserstein metric | MATHEMATICS | MATHEMATICS, APPLIED | BARYCENTRIC MAPS

Journal Article