Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients

Preface 1. Introduction 2. Review of basic functional analysis 3. Lebesgue theory of Banach space-valued functions 4. Lipschitz functions and embeddings 5. Path integrals and modulus 6. Upper gradients 7. Sobolev spaces 8. Poincare inequalities 9. Consequences of Poincare inequalities 10. Other definitions of Sobolev-type spaces 11. Gromov-Hausdorff convergence and Poincare inequalities 12. Self-improvement of Poincare inequalities 13. An Introduction to Cheeger's differentiation theory 14. Examples, applications and further research directions References Notation index Subject index.

[1]  Beppo Levi,et al.  Sul principio di dirichlet , 1906 .

[2]  M. Fréchet Les dimensions d'un ensemble abstrait , 1910 .

[3]  H. Rademacher Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale , 1919 .

[4]  S. Banach,et al.  Théorie des opérations linéaires , 1932 .

[5]  E. J. McShane,et al.  Extension of range of functions , 1934 .

[6]  H. Whitney Analytic Extensions of Differentiable Functions Defined in Closed Sets , 1934 .

[7]  C. Kuratowski Quelques problèmes concernant les espaces métriques non-séparables , 1935 .

[8]  J. A. Clarkson Uniformly convex spaces , 1936 .

[9]  S. Sobolev On a theorem in functional analysis , 1938 .

[10]  S. Ulam,et al.  On the Existence of a Measure Invariant Under a Transformation , 1939 .

[11]  R. Richardson The International Congress of Mathematicians , 1932, Science.

[12]  M. Day Reflexive Banach spaces not isomorphic to uniformly convex spaces , 1941 .

[13]  Richard Courant,et al.  Studies and Essays presented to R Courant on his 60th Birthday, January 8, 1948 , 1948, Nature.

[14]  E. J. McShane Linear functionals on certain Banach spaces , 1950 .

[15]  L. Ahlfors,et al.  Conformal invariants and function-theoretic null-sets , 1950 .

[16]  T. H. Hildebrandt Integration in abstract spaces , 1953 .

[17]  J. Deny,et al.  Les espaces du type de Beppo Levi , 1954 .

[18]  Herbert Busemann,et al.  The geometry of geodesics , 1955 .

[19]  H. Rauch Harmonic and Analytic Functions Of Several Variables and The Maximal Theorem Of Hardy and Littlewood , 1956, Canadian Journal of Mathematics.

[20]  K. Smith A Generalization Of An Inequality Of Hardy and Littlewood , 1956, Canadian Journal of Mathematics.

[21]  B. Fuglede Extremal length and functional completion , 1957 .

[22]  Felix E. Browder,et al.  Functional analysis and partial differential equations. II , 1959 .

[23]  Formes et espaces de Dirichlet , 1960 .

[24]  H. Wallin Continuous functions and potential theory , 1963 .

[25]  Edwin Hewitt,et al.  Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable , 1965 .

[26]  C. B. Morrey Multiple Integrals in the Calculus of Variations , 1966 .

[27]  W. Rudin Real and complex analysis , 1968 .

[28]  W. Ziemer Extremal length and conformal capacity , 1967 .

[29]  Yu. G. Reshetnyak Space mappings with bounded distortion , 1967 .

[30]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[31]  W. Ziemer Extremal length and $p$-capacity. , 1969 .

[32]  R. Coifman,et al.  Singular integrals and multipliers on homogeneous spaces , 1970 .

[33]  M. Brelot On Topologies and Boundaries in Potential Theory , 1971 .

[34]  Jussi Väisälä,et al.  Lectures on n-Dimensional Quasiconformal Mappings , 1971 .

[35]  M. Schechter Principles of Functional Analysis , 1971 .

[36]  Ronald R. Coifman,et al.  Analyse Hamonique Non-Commutative sur Certains Espaces Homogenes , 1971 .

[37]  B. Muckenhoupt,et al.  Weighted norm inequalities for the Hardy maximal function , 1972 .

[38]  H. Fédérer,et al.  The Lebesgue Set of a Function whose Distribution Derivatives are p-th Power Summable , 1972 .

[39]  A. Calderón Estimates for singular integral operators in terms of maximal functions , 1972 .

[40]  L. Schwartz Radon measures on arbitrary topological spaces and cylindrical measures , 1973 .

[41]  J. Cooper SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[42]  Pertti Mattila,et al.  Integration in a space of measures , 1973 .

[43]  James R. Munkres,et al.  Topology; a first course , 1974 .

[44]  Joseph J. Hesse $p$-extremal length and $p$-measurable curve families , 1975 .

[45]  G. Weiss,et al.  Extensions of Hardy spaces and their use in analysis , 1977 .

[46]  Konrad Jacobs,et al.  Measure and integral , 1978 .

[47]  Enrico Giusti,et al.  On the regularity of the minima of variational integrals , 1982 .

[48]  Carlos E. Kenig,et al.  The local regularity of solutions of degenerate elliptic equations , 1982 .

[49]  P. Buser A note on the isoperimetric constant , 1982 .

[50]  Bruno Franchi,et al.  Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients , 1983 .

[51]  J. Doob Classical potential theory and its probabilistic counterpart , 1984 .

[52]  J. Diestel Sequences and series in Banach spaces , 1984 .

[53]  J. Mitchell On Carnot-Carathéodory metrics , 1985 .

[54]  J. Lindenstrauss,et al.  Extensions of lipschitz maps into Banach spaces , 1986 .

[55]  D. Jerison The Poincaré inequality for vector fields satisfying Hörmander’s condition , 1986 .

[56]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[57]  R. Strichartz Sub-Riemannian geometry , 1986 .

[58]  A. Kechris Classical descriptive set theory , 1987 .

[59]  S. Konyagin,et al.  On measures with the doubling condition , 1988 .

[60]  Matti Vuorinen,et al.  Conformal Geometry and Quasiregular Mappings , 1988 .

[61]  P. Pansu,et al.  Métriques de Carnot-Carthéodory et quasiisométries des espaces symétriques de rang un , 1989 .

[62]  H. M. Reimann An estimate for pseudoconformal capacities on the sphere , 1989 .

[63]  P. Pansu Dimension conforme et sphère à l'infini des variétés à courbure négative , 1989 .

[64]  W. Ziemer Weakly differentiable functions , 1989 .

[65]  L. Carleson,et al.  The Collected Works of Arne Beurling , 1989 .

[66]  Luigi Ambrosio,et al.  Metric space valued functions of bounded variation , 1990 .

[67]  É. Ghys,et al.  Sur Les Groupes Hyperboliques D'Apres Mikhael Gromov , 1990 .

[68]  I. Holopainen Nonlinear potential theory and quasiregular mappings on Riemannian manifolds , 1990 .

[69]  S. L. Sobolev,et al.  Some Applications of Functional Analysis in Mathematical Physics , 1991 .

[70]  P. Wojtaszczyk Banach Spaces For Analysts: Preface , 1991 .

[71]  M. Talagrand,et al.  Probability in Banach spaces , 1991 .

[72]  L. Evans Measure theory and fine properties of functions , 1992 .

[73]  L. Saloff-Coste,et al.  A note on Poincaré, Sobolev, and Harnack inequalities , 1992 .

[74]  T. Kumagai Regularity, closedness and spectral dimensions of the Dirichlet forms on P.C.F. self-similar sets , 1993 .

[75]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[76]  R. Schoen,et al.  Sobolev spaces and harmonic maps for metric space targets , 1993 .

[77]  J. Heinonen,et al.  Nonlinear Potential Theory of Degenerate Elliptic Equations , 1993 .

[78]  Richard L. Wheeden,et al.  Weighted Sobolev-Poincaré inequalities for Grushin type operators , 1994 .

[79]  B. Hambly,et al.  Transition density estimates for Brownian motion on affine nested fractals , 1994 .

[80]  Jun Kigami Effective resistances for harmonic structures on p.c.f. self-similar sets , 1994, Mathematical Proceedings of the Cambridge Philosophical Society.

[81]  M. Fukushima,et al.  Dirichlet forms and symmetric Markov processes , 1994 .

[82]  M. Ledoux,et al.  Sobolev inequalities in disguise , 1995 .

[83]  L. Hedberg,et al.  Function Spaces and Potential Theory , 1995 .

[84]  Pertti Mattila,et al.  Geometry of sets and measures in Euclidean spaces , 1995 .

[85]  U. Mosco,et al.  Sobolev inequalities on homogeneous spaces , 1995 .

[86]  P. Koskela,et al.  Sobolev meets Poincaré , 1995 .

[87]  Richard L. Wheeden,et al.  Representation formulas and weighted Poincar inequalities for Hrmander vector fields , 1995 .

[88]  J. Heinonen,et al.  Definitions of quasiconformality , 1995 .

[89]  Peter Li,et al.  Green's functions, harmonic functions, and volume comparison , 1995 .

[90]  Piotr Hajłasz,et al.  @ 1996 Kluwer Academic Publishers. Printed in the Netherlands. Sobolev Spaces on an Arbitrary Metric Space , 1994 .

[91]  S. Semmes,et al.  Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities , 1996 .

[92]  Hiroaki Aikawwa,et al.  Potential Theory - Selected Topics , 1996 .

[93]  Karl-Theodor Sturm,et al.  Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality , 1996 .

[94]  S. Semmes Good metric spaces without good parameterizations , 1996 .

[95]  Juha Kinnunen,et al.  THE SOBOLEV CAPACITY ON METRIC SPACES , 1996 .

[96]  M. Gromov Carnot-Carathéodory spaces seen from within , 1996 .

[97]  J. Heinonen,et al.  From local to global in quasiconformal structures. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[98]  Nicola Garofalo,et al.  ISOPERIMETRIC AND SOBOLEV INEQUALITIES FOR CARNOT-CARATHEODORY SPACES AND THE EXISTENCE OF MINIMAL SURFACES , 1996 .

[99]  Thierry Delmotte Inégalité de Harnack elliptique sur les graphes , 1997 .

[100]  S. Treil,et al.  Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces , 1997, math/9711210.

[101]  O. Martio,et al.  Traces of Sobolev Functions on Fractal Type Sets and Characterization of Extension Domains , 1997 .

[102]  Stephen Semmes,et al.  Fractured fractals and broken dreams : self-similar geometry through metric and measure , 1997 .

[103]  Marc Bourdon,et al.  Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings , 1997 .

[104]  T. Colding,et al.  HARMONIC FUNCTIONS ON MANIFOLDS , 1997 .

[105]  I. Holopainen,et al.  p-harmonic functions on graphs and manifolds , 1997 .

[106]  Jan Malý,et al.  Fine Regularity of Solutions of Elliptic Partial Differential Equations , 1997 .

[107]  Yu. G. Reshetnyak Sobolev-Type Classes of Functions with Values in a Metric Space. II , 1997 .

[108]  Christian Houdré,et al.  Some Connections Between Isoperimetric and Sobolev-Type Inequalities , 1997 .

[109]  N. Weaver Lipschitz algebras and derivations II: exterior differentiation , 1998, math/9807096.

[110]  J. Tyson Quasiconformality and quasisymmetry in metric measure spaces. , 1998 .

[111]  R. Strichartz Fractals in the Large , 1998, Canadian Journal of Mathematics.

[112]  Pawel Strzelecki,et al.  Subelliptic p-harmonic maps into spheres and the ghost of Hardy spaces , 1998 .

[113]  X. Tolsa COTLAR'S INEQUALITY WITHOUT THE DOUBLING CONDITION AND EXISTENCE OF PRINCIPAL VALUES FOR THE CAUCHY INTEGRAL OF MEASURES , 1998 .

[114]  J. Heinonen,et al.  Quasiconformal maps in metric spaces with controlled geometry , 1998 .

[115]  Robert E. Megginson An Introduction to Banach Space Theory , 1998 .

[116]  E. Saksman,et al.  Every complete doubling metric space carries a doubling measure , 1998 .

[117]  P. Koskela,et al.  Quasiconformal mappings and Sobolev spaces , 1998 .

[118]  Robert S. Strichartz,et al.  Some Properties of Laplacians on Fractals , 1999 .

[119]  E. Saksman REMARKS ON THE NONEXISTENCE OF DOUBLING MEASURES , 1999 .

[120]  Alexander Grigor'yan,et al.  Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds , 1999 .

[121]  S. Yau,et al.  Surveys in Differential Geometry , 1999 .

[122]  Stephen M. Buckley,et al.  IS THE MAXIMAL FUNCTION OF A LIPSCHITZ FUNCTION CONTINUOUS , 1999 .

[123]  X. Tolsa $L^2$-boundedness of the Cauchy integral operator for continuous measures , 1999 .

[124]  P. Koskela Removable sets for Sobolev spaces , 1999 .

[125]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[126]  Emmanuel Hebey Nonlinear analysis on manifolds: Sobolev spaces and inequalities , 1999 .

[127]  Jeff Cheeger,et al.  Differentiability of Lipschitz Functions on Metric Measure Spaces , 1999 .

[128]  M. Ohtsuka,et al.  EXTREMAL LENGTH OF VECTOR MEASURES , 1999 .

[129]  J. Lindenstrauss,et al.  Geometric Nonlinear Functional Analysis , 1999 .

[130]  M. Gromov Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[131]  B. Hambly,et al.  Transition Density Estimates for Diffusion Processes on Post Critically Finite Self‐Similar Fractals , 1999 .

[132]  DEFINITIONS OF SOBOLEV CLASSES ON METRIC SPACES , 1999 .

[133]  J. Tyson Geometric and analytic applications of a generalized definition of the conformal modulus. , 1999 .

[134]  R. Strichartz ANALYSIS ON FRACTALS , 1999 .

[135]  E. Lanconelli,et al.  On the Poincaré inequality for vector fields , 2000 .

[136]  Stephen J. Gardiner,et al.  Classical Potential Theory , 2000 .

[137]  T. Laakso Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality , 2000 .

[138]  Thomas de Quincey [C] , 2000, The Works of Thomas De Quincey, Vol. 1: Writings, 1799–1820.

[139]  N. Shanmugalingam Newtonian spaces: An extension of Sobolev spaces to metric measure spaces , 2000 .

[140]  W. Woess Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[141]  U. Lang,et al.  Extensions of Lipschitz maps into Hadamard spaces , 2000 .

[142]  Pekka Koskela,et al.  Sobolev met Poincaré , 2000 .

[143]  L. Ambrosio,et al.  Currents in metric spaces , 2000 .

[144]  J. Heinonen Lectures on Analysis on Metric Spaces , 2000 .

[145]  J. Heinonen,et al.  An n-dimensional space that admits a Poincare inequality but has no manifold points , 2000 .

[146]  N. Shanmugalingam,et al.  Regularity of quasi-minimizers on metric spaces , 2001 .

[147]  J. Heinonen,et al.  Sobolev classes of Banach space-valued functions and quasiconformal mappings , 2001 .

[148]  Arcwise Isometries,et al.  A Course in Metric Geometry , 2001 .

[149]  J. Kigami,et al.  Analysis on Fractals , 2001 .

[150]  R. Strichartz The Laplacian on the Sierpinski gasket via the method of averages , 2001 .

[151]  Laurent Saloff-Coste,et al.  Aspects of Sobolev-type inequalities , 2001 .

[152]  L. Saloff‐Coste RANDOM WALKS ON INFINITE GRAPHS AND GROUPS (Cambridge Tracts in Mathematics 138) , 2001 .

[153]  T. Shioya,et al.  Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces , 2001 .

[154]  G. Lu,et al.  Best constants for Moser-Trudinger inequalities on the Heisenberg group , 2001 .

[155]  Jun Kigami,et al.  Constructing a Laplacian on the Diamond Fractal , 2001, Exp. Math..

[156]  M. Troyanov,et al.  Axiomatic theory of Sobolev spaces , 2001 .

[157]  Luigi Ambrosio,et al.  Some Fine Properties of Sets of Finite Perimeter in Ahlfors Regular Metric Measure Spaces , 2001 .

[158]  Harnack inequality and hyperbolicity for subelliptic p-Laplacians with applications to Picard type theorems , 2001 .

[159]  N. Shanmugalingam,et al.  Fat sets and pointwise boundary estimates forp-harmonic functions in metric spaces , 2001 .

[160]  Bruno Franchi,et al.  Rectifiability and perimeter in the Heisenberg group , 2001 .

[161]  N. Shanmugalingam Harmonic functions on metric spaces , 2001 .

[162]  Robert S. Strichartz,et al.  Harmonic mappings of the Sierpinski gasket to the circle , 2001 .

[163]  T. O’Neil Geometric Measure Theory , 2002 .

[164]  Alexander Barvinok,et al.  A course in convexity , 2002, Graduate studies in mathematics.

[165]  A. Kufner,et al.  Axiomatic Sobolev spaces on metric spaces , 2002 .

[166]  G. Citti,et al.  Smoothness of Lipschitz-continuous graphs with nonvanishing Levi curvature , 2002 .

[167]  Quasisymmetric parametrizations of two-dimensional metric spheres , 2001, math/0107171.

[168]  J. Kinnunen,et al.  Lebesgue points for Sobolev functions on metric spaces , 2002 .

[169]  L. Capogna,et al.  Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type , 2002 .

[170]  M. Troyanov,et al.  Capacities in metric spaces , 2002 .

[171]  J. Verdera The fall of the doubling condition in Calderón-Zygmund theory , 2002 .

[172]  SOME NOVEL TYPES OF FRACTAL GEOMETRY (Oxford Mathematical Monographs) By STEPHEN SEMMES: 164 pp., £49.95, ISBN 0-19-850806-9 (Clarendon Press, Oxford, 2001). , 2002 .

[173]  R. Wheeden,et al.  Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces , 2002 .

[174]  P. Koskela,et al.  Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces☆ , 2003 .

[175]  信 大津賀 Extremal length and precise functions , 2003 .

[176]  Jiaxin Hu,et al.  Heat kernels on metric measure spaces and an application to semilinear elliptic equations , 2003 .

[177]  N. Shanmugalingam,et al.  The Dirichlet problem for p-harmonic functions on metric spaces , 2003 .

[178]  C. Villani Topics in Optimal Transportation , 2003 .

[179]  P. Hajłasz A new characterization of the Sobolev space , 2003 .

[180]  Robert S. Strichartz,et al.  Function spaces on fractals , 2003 .

[181]  Fedor Nazarov,et al.  TheTb-theorem on non-homogeneous spaces , 2003 .

[182]  L. Capogna,et al.  Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type , 2003 .

[183]  N. Shanmugalingam,et al.  The Perron method for p-harmonic functions in metric spaces , 2003 .

[184]  S. Keith Modulus and the Poincaré inequality on metric measure spaces , 2003 .

[185]  P. Assouad Plongements lipschitziens dans Rn , 2003 .

[186]  Michele Miranda,et al.  Functions of bounded variation on “good” metric spaces , 2003 .

[187]  S. Keith A differentiable structure for metric measure spaces , 2004 .

[188]  S. Keith Measurable differentiable structures and the Poincaré inequality , 2004 .

[189]  L. Ambrosio,et al.  Special Functions of Bounded Variation in Doubling Metric Measure Spaces , 2004 .

[190]  Diego Pallara,et al.  Calculus of variations : topics from the mathematical heritage of E. de Giorgi , 2004 .

[191]  Urs Lang,et al.  Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions , 2004, math/0410048.

[192]  Paolo Tilli,et al.  Topics on analysis in metric spaces , 2004 .

[193]  P. Koskela,et al.  Dirichlet Forms, Poincaré Inequalities, and the Sobolev Spaces of Korevaar and Schoen , 2004 .

[194]  R. Strichartz,et al.  p-Energy and p-Harmonic Functions on Sierpinski Gasket Type Fractals , 2004 .

[195]  C. Villani,et al.  Ricci curvature for metric-measure spaces via optimal transport , 2004, math/0412127.

[196]  Karl-Theodor Sturm,et al.  Transport inequalities, gradient estimates, entropy and Ricci curvature , 2005 .

[197]  Weak curvature conditions and functional inequalities , 2005, math/0506481.

[198]  N. Shanmugalingam,et al.  Polar sets on metric spaces , 2005 .

[199]  Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary , 2002, math/0208135.

[200]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[201]  James R. Lee,et al.  Extending Lipschitz functions via random metric partitions , 2005 .

[202]  S. Buckley,et al.  Sphericalization and flattening , 2005 .

[203]  Karl-Theodor Sturm Generalized Ricci bounds and convergence of metric measure spaces , 2005 .

[204]  Kai Rajala Surface families and boundary behavior of quasiregular mappings , 2005 .

[205]  Karl-Theodor Sturm A curvature-dimension condition for metric measure spaces , 2006 .

[206]  D. Danielli,et al.  Non-doubling Ahlfors Measures, Perimeter Measures, And the Characterization of the Trace Spaces of Sobolev Functions in Carnot-caratheodory Spaces , 2006 .

[207]  B. Kleiner The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity , 2006 .

[208]  Wiener criterion for Cheeger p-harmonic functions on metric spaces , 2006 .

[209]  Karl-Theodor Sturm,et al.  On the geometry of metric measure spaces. II , 2006 .

[210]  Pointwise characterizations of Hardy-Sobolev functions , 2006, math/0611901.

[211]  R. Montgomery A Tour of Subriemannian Geometries, Their Geodesics and Applications , 2006 .

[212]  J. Cheeger,et al.  Differentiating maps into L1, and the geometry of BV functions , 2006, math/0611954.

[213]  Bruce Kleiner,et al.  Generalized differentiation and bi-Lipschitz nonembedding in L1⁎ , 2006 .

[214]  P. Hajłasz Sobolev Mappings: Lipschitz Density is not a Bi-Lipschitz Invariant of the Target , 2006, math/0602029.

[215]  N. Shanmugalingam,et al.  Measurability of equivalence classes and MEC$_p$-property in metric spaces , 2007 .

[216]  Scott D. Pauls,et al.  An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem , 2007 .

[217]  J. M. Mackay Spaces and groups with conformal dimension greater than one , 2007, 0711.0417.

[218]  R. Korte Geometric Implications of the Poincaré Inequality , 2007 .

[219]  N. Shanmugalingam,et al.  Lebesgue points and capacities via the boxing inequality in metric spaces , 2008 .

[220]  P. Hajłasz Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces , 2009 .

[221]  R. Strichartz,et al.  Infinitesimal resistance metrics on Sierpinski gasket type fractals , 2008 .

[222]  D. Danielli,et al.  Local Behavior of p-harmonic Green’s Functions in Metric Spaces , 2008, 0807.1323.

[223]  N. Shanmugalingam,et al.  Uniformity from Gromov hyperbolicity , 2008 .

[224]  N. Shanmugalingam,et al.  Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces , 2008 .

[225]  Bruce Kleiner,et al.  Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property , 2008, 0808.3249.

[226]  X. Zhong,et al.  The Poincare inequality is an open ended condition , 2008 .

[227]  Xiangdong Xie,et al.  Metric space inversions, quasihyperbolic distance, and uniform spaces , 2008 .

[228]  Assaf Naor,et al.  Compression bounds for Lipschitz maps from the Heisenberg group to L1 , 2009, ArXiv.

[229]  Fabrice Baudoin,et al.  Perelman’s Entropy and Doubling Property on Riemannian Manifolds , 2009, 0911.1819.

[230]  Mathematische Annalen Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces , 2009 .

[231]  Piotr Hajłasz,et al.  Sobolev Mappings between Manifolds and Metric Spaces , 2009 .

[232]  A Universality Property of Sobolev Spaces in Metric Measure Spaces , 2009 .

[233]  Bruce Kleiner,et al.  Metric differentiation, monotonicity and maps to L1 , 2009, 0907.3295.

[234]  Hrant Hakobyan Conformal Dimension: Cantor Sets and Fuglede Modulus , 2009 .

[235]  B. Kleiner,et al.  Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups , 2010, 1002.1991.

[236]  Renjin Jiang Lipschitz Continuity of Solutions of Poisson Equations in Metric Measure Spaces , 2010, 1004.1101.

[237]  Marshall Williams Geometric and analytic quasiconformality in metric measure spaces , 2010, 1008.3588.

[238]  N. Shanmugalingam,et al.  The ∞-Poincaré Inequality in Metric Measure Spaces , 2010 .

[239]  O. Kharlampovich,et al.  Combinatorial and Geometric Group Theory , 2010 .

[240]  Assaf Naor,et al.  L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry , 2010, ArXiv.

[241]  R. Bass,et al.  Uniqueness of Brownian motion on Sierpinski carpets , 2010 .

[242]  Nicola Gigli,et al.  Heat Flow on Alexandrov Spaces , 2010, 1008.1319.

[243]  J. Heinonen,et al.  Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum , 2010 .

[244]  D. Herron Uniform metric spaces, annular quasiconvexity and pointed tangent spaces , 2011 .

[245]  J. Wu,et al.  Geometry and quasisymmetric parametrization of Semmes spaces , 2011, 1111.2197.

[246]  Bruce Kleiner,et al.  Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1 , 2011, 1110.2406.

[247]  B. Kleiner,et al.  DIFFERENTIABLE STRUCTURES ON METRIC MEASURE SPACES: A PRIMER , 2011, 1108.1324.

[248]  N. Shanmugalingam,et al.  Regularity of Sets with Quasiminimal Boundary Surfaces in Metric Spaces , 2011, 1105.3058.

[249]  S. Gersten Essays in Group Theory , 2011 .

[250]  Jasun Gong Rigidity of Derivations in the Plane and in Metric Measure Spaces , 2011, 1110.4282.

[251]  E. Durand-Cartagena,et al.  p-Poincaré inequality versus ∞-Poincaré inequality: some counterexamples , 2012 .

[252]  David Bate,et al.  Differentiability, porosity and doubling in metric measure spaces , 2011, 1108.0318.

[253]  Simone Di Marino,et al.  Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope , 2012, 1212.3779.

[254]  Naotaka Kajino,et al.  Heat Kernel Asymptotics for the Measurable Riemannian Structure on the Sierpinski Gasket , 2012 .

[255]  Anders Björn,et al.  Nonlinear Potential Theory on Metric Spaces , 2012 .

[256]  P. Koskela,et al.  Isoperimetric inequality from the poisson equation via curvature , 2012 .

[257]  Naotaka Kajino Time changes of local Dirichlet spaces by energy measures of harmonic functions , 2012 .

[258]  Jasun Gong The Lip-lip condition on metric measure spaces , 2012, 1208.2869.

[259]  P. Koskela,et al.  Geometry and Analysis of Dirichlet forms , 2012, 1208.4955.

[260]  C. Bishop,et al.  Frequency of dimension distortion under quasisymmetric mappings , 2012 .

[261]  L. Ambrosio,et al.  Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case , 2012, 1205.3288.

[262]  Alexander Brudnyi,et al.  Selected Topics in Analysis on Metric Spaces , 2012 .

[263]  S. Wenger,et al.  An upper gradient approach to weakly differentiable cochains , 2012, 1208.4350.

[264]  P. Koskela,et al.  $L^∞$-variational problem associated to Dirichlet forms , 2012 .

[265]  Sean Li,et al.  Coarse differentiation and quantitative nonembeddability for Carnot groups , 2013, 1304.6633.

[266]  Luk'avs Mal'y Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices , 2012, 1210.1442.

[267]  J. Tyson,et al.  Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets , 2012, 1201.3548.

[268]  Noel R. DeJarnette Self improving Orlicz-Poincare inequalities , 2013 .

[269]  L. Ambrosio,et al.  On the Bakry-\'Emery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces , 2013, 1309.4664.

[270]  Giuseppe Savaré Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces , 2013, Discrete & Continuous Dynamical Systems - A.

[271]  L. Ambrosio,et al.  Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. , 2011, 1111.3730.

[272]  Dachun Yang,et al.  Sobolev Spaces on Metric Measure Spaces , 2014 .

[273]  F. John Extremum Problems with Inequalities as Subsidiary Conditions , 2014 .

[274]  L. Ambrosio,et al.  Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds , 2012, 1209.5786.

[275]  Simone Di Marino,et al.  On the duality between $p$-modulus and probability measures , 2013, Journal of the European Mathematical Society (Print).

[276]  N. Shanmugalingam,et al.  Semmes family of curves and a characterization of functions of bounded variation in terms of curves , 2015 .

[277]  M. Miranda,et al.  Boundary measures, generalized Gauss–Green formulas, and mean value property in metric measure spaces , 2013, 1304.4352.

[278]  N. Shanmugalingam,et al.  Preservation of bounded geometry under sphericalization and flattening , 2015 .

[279]  D. Herron Gromov–Hausdorff distance for pointed metric spaces , 2016 .

[280]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[281]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[282]  H. Bateman Book Review: Ergebnisse der Mathematik und ihrer Grenzgebiete , 1933 .