0#+1,&'+(&, · Last name First name Country Ihnatsyeva Lizaveta Finland Iseli Annina Switzerland...

Madrid, April–June 2015

D. Azagra

E. Durand

L. Guijarro

J. A. Jaramillo

E. Le Donne

M. Ritoré

Organizing Committee:

Workshop on

Analysis and Geometry

in Metric Spaces

ICMAT Madrid. June 1-5, 2015

in Metric Spacesin Metric Spaces

Bookof abstracts

Schedule

All lectures will take place in AULA NARANJA

Monday Tuesday Wednesday Thursday Friday

09:00–09:50 Registration

09:50–10:00 Opening

10:00–10:50 Kinnunen Yang Tyson Bjorn Coulhon

10:55–11:20 Coffee

11:20–12:10 Plaut Monti Rigot Galaz-Garcıa Hajłasz

12:20–12:40 Galli Wildrick Speight Lindquist

12:50–13:10 Maly Mitsuishi FranceschiShioya

Martınez

Lunch Special lunch Lunch

14:45–15:35 Shanmugalingam Serapioni Lang

Free afternoon

Korte

15:45–16:05 Adamowicz Azzam Lierl Zimmerman

16:10–16:35 Coffee Coffee

16:35–16:55 Ihnatsyeva David Guo

17:05–17:25 Solorzano Barbieri KostrzewaKleiner

17:30–17:50 Yepes Jiang Pavon

18:00 Bus Bus

List of Participants

Last name First name CountryAaltonen Martina FinlandAckermann Colleen USAAdamowicz Tomasz PolandAyato Mitsuishi JapanAzagra Daniel SpainAzzam Jonas SpainBarbieri Davide SpainBenedict Sita FinlandBjorn Jana SwedenBorondo Florentino SpainCanarecci Giovanni FinlandCanton Pire Alicia SpainCavallotto Edoardo FranceChen Li SpainChunaev Petr SpainCoulhon Thierry FranceDaniel Fox SpainDavid Guy C. USADurand Cartagena Estibalitz SpainFaessler Katrin SwitzerlandFranceschi Valentina ItalyFranchi Bruno ItalyGalaz-Garcıa Fernando GermanyGalli Matteo ItalyGhinassi Silvia USAGreco Elizabeth USAGuijarro Santamarıa Luis SpainGuo Changyu FinlandHajlasz Piotr USAHalverson Christopher USAHerzog Nicolas Switzerland

Last name First name CountryIhnatsyeva Lizaveta FinlandIseli Annina SwitzerlandJaramillo Jesus SpainJiang Renjin SpainJolany Hassan FranceKarak Nijjwal FinlandKauranen Aapo FinlandKinneberg Kyle USAKinnunen Juha FinlandKleiner Bruce USAKorte Riikka FinlandKostrzewa Tomasz PolandKuzmin Kirill ItalyLang Urs SwitzerlandLe Donne Enrico FinlandLierl Janna USALindquist Jeffrey USALuisto Rami FinlandLukyanenko Anton USAMackay John UKMaly Lukas FinlandMartınez Fernandez Antonio Roberto SpainMiesch Benjamin SwitzerlandMonti Roberto ItalyMoon Heather USAMudarra Carlos SpainNandi Debanjan FinlandNicolussi Golo Sebastiano FinlandNunez-Zimbron Jesus MexicoOjala Tuomo FinlandPalfia Miklos JapanPavon Mael SwitzerlandPlaut Conrad USARigot Severine FranceRitore Manuel SpainRomney Matthew USASantos Jaime MexicoSchumacher Carol USASeco Daniel UKSerapioni Raul ItalyShanmugalingam Nageswari USA

Last name First name CountryShioya Takashi JapanSnipes Marie USASolorzano Pedro BrazilSoultanis Elefterios FinlandSpeight Gareth ItalySzumanska Marta PolandTyson Jeremy USAVellis Vyron FinlandWildrick Kevin USAYang Paul USAYepes Nicolas Jesus SpainYoung Robert USAZhou Xiaodan USAZimmerman Scott USAZuercher Thomas UK

List of Abstracts

Plenary talks 1Jana BJORN, Boundary regularity for quasiminimizers . . . . . . . . . . . . . 3Thierry COULHON, Analysis on doubling metric measure spaces endowed with

a Dirichlet form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Fernando GALAZ GARCIA, Isometric Lie group actions on Alexandrov spaces . 3Piotr HAJŁASZ, Lipschitz homotopy groups of the Heisenberg groups . . . . . . 4Juha KINNUNEN, Parabolic weighted norm inequalities . . . . . . . . . . . . . 4Bruce KLEINER, New examples of PI spaces and some problems . . . . . . . . 4Riikka KORTE, A characterization of BV -functions via Semmes family of curves 5Urs LANG, Nonpositive curvature beyond Busemann spaces . . . . . . . . . . 5Roberto MONTI, Recent results on the regularity of H-minimal surfaces . . . . 6Conrad PLAUT, Gromov-Hausdorff Convergence of e-covers . . . . . . . . . . 6Severine RIGOT, Besicovitch covering property in Carnot groups . . . . . . . . 7Raul SERAPIONI, Intrinsic Lipschitz graphs in Carnot groups . . . . . . . . . . 7Nageswari SHANMUGALINGAM, Functions of bounded variation in metric set-

ting: three definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Takashi SHIOYA, Convergence of metric measure spaces . . . . . . . . . . . . 8Jeremy TYSON, Marstrand’s density theorem in the Heisenberg group . . . . . 8Paul YANG, CR geometry in 3-D . . . . . . . . . . . . . . . . . . . . . . . . . 8

Short talks 9Tomasz ADAMOWICZ, Prime ends and mappings . . . . . . . . . . . . . . . . 11Jonas AZZAM, A characterization of 1-rectifiable doubling measures with con-

nected supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Davide BARBIERI, On the Bourgain-Brezis-Mironescu characterization of Sobolev

spaces in certain metric spaces and related fractional norm inequalities . 12Guy C. DAVID, Bi-Lipschitz pieces between manifolds . . . . . . . . . . . . . 12Valentina FRANCESCHI, Isoperimetric problem in Grushin spaces and H-type

groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Matteo GALLI, Area-stationary and stable surfaces of class C1 in the sub-

Riemannian Heisenberg group H1 . . . . . . . . . . . . . . . . . . . . . 13Changyu GUO, The branch set of a quasiregular mapping: recent advances

and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Lizaveta IHNATSYEVA, Measure density and extension of Besov and Triebel–

Lizorkin functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Renjin JIANG, Li-Yau type inequalities for heat flows on metric measure spaces 14

Tomasz KOSTRZEWA, Sobolev spaces on metrizable groups . . . . . . . . . . 14Janna LIERL, Parabolic Harnack inequality for time-dependent non-symmetric

Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Jeff LINDQUIST, Modulus and Hyperbolic Fillings . . . . . . . . . . . . . . . 15Lukas MALY, Self-improvement of Poincare inequalities and lack thereof . . . 16Antonio Roberto MARTINEZ FERNANDEZ, Widths of convex bodies in the 2-

dimensional sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Ayato MITSUISHI, Locally Lipschitz contractibility of Alexandrov spaces and

its applications, orientability and metric currents of Alexandrov spacesand its applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Mael PAVON, Injective hulls of metric spaces . . . . . . . . . . . . . . . . . . 18Pedro SOLORZANO, Waning under Cheeger deformations . . . . . . . . . . . . 18Gareth SPEIGHT, Lusin Approximation and Horizontal Curves in Carnot Groups 19Kevin WILDRICK, Steiner’s formula in the Heisenberg group . . . . . . . . . . 19Jesus YEPES, On a linear refinement of the Prekopa-Leindler inequality . . . . 20Scott ZIMMERMAN, Whitney’s Extension Theorem for Curves in the Heisen-

berg Group Hn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Plenary TalksPlenary Talks

Research term ICMAT-AGMS Plenary talks

Boundary regularity for quasiminimizersJana BJORN (Linkoping University — Sweden)

A function u is a quasiminimizer in a domain W ⇢ Rn ifZ

j 6=0|—u|p dx Q

Z

j 6=0|—(u+j)|p dx

for all j 2 C•0 (W). Quasiminimizers provide a unified treatment of vari-

ational integrals, elliptic equations and quasiregular mappings, and sharemany (though not all) properties with p-harmonic functions. At the sametime, for example the comparison principle fails for them.

In the talk, we discuss some concrete pointwise estimates for quasimin-imizers and obtain a Wiener type condition for their boundary regularity. Itis also shown by examples that some of the estimates are sharp.

Analysis on doubling metric measure spacesendowed with a Dirichlet formThierry COULHON (Paris Sciences et Lettres ResearchUniversity — France)

In the above setting, we shall describe recent characterisations of pointwiseupper and lower bounds of the heat kernel. We shall give applications toanalysis on such spaces, by giving conditions under which the Riesz trans-form is bounded on Lp, or under which Sobolev spaces are algebras for thepointwise product. This relies on joint work of the author with FredericBernicot, Salahaddine Boutayeb, Dorothee Frey, and Adam Sikora.

Joint work with Frederic Bernicot, Salahaddine Boutayeb, Dorothee Frey, and AdamSikora.

Isometric Lie group actions on Alexandrov spacesFernando GALAZ GARCIA (Karlsruhe Institute of Technology(KIT) — Germany)

Alexandrov spaces (with curvature bounded below) are a natural syntheticgeneralization of Riemannian manifolds. In this talk I will discuss recentdevelopments on the geometry and topology of Alexandrov spaces with iso-metric actions of compact Lie groups.

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Book of abstracts Research term ICMAT-AGMS

Lipschitz homotopy groups of the Heisenberg groupsPiotr HAJŁASZ (University of Pittsburgh — USA)

The Heisenberg group is homeomorphic to the Euclidean space and henceall its homotopy groups are trivial. However, the situation is very different ifinstead of the classical homotopy groups one considers Lipschitz homotopygroups. They are defined as the classical homotopy groups but with therequirement that mappings and homotopies are Lipschitz continuous. In thetalk I will discuss recent results regarding the Lipschitz homotopy groups ofthe Heisenberg groups.

Parabolic weighted norm inequalitiesJuha KINNUNEN (Aalto University — Finland)

We discuss parabolic Muckenhoupt weights and functions of bounded meanoscillation (BMO) related to a doubly nonlinear parabolic partial differen-tial equation. In the natural geometry for the doubly nonlinear equationthe time variable scales as the modulus of the space variable raised to apower. Consequently the Euclidean balls and cubes have to be replacedby parabolic rectangles respecting this scaling in all estimates. An extrachallenge is given by the time lag appearing in the estimates. The mainresult gives a full characterization of weak and strong type weighted norminequalities for parabolic forward in time maximal operators. In addition,we give a Jones type factorization result for the parabolic Muckenhouptweights and a Coifman-Rochberg type characterization of the parabolicBMO through parabolic Muckenhoupt weights and maximal functions. Wealso discuss connections and applications of the results to regularity of non-linear parabolic partial differential equations. This is a joint work with OlliSaari at Aalto University.

Joint work with Olli Saari (Aalto University).

New examples of PI spaces and some problemsBruce KLEINER (Courant Institute for Mathematical Sciences — USA)

The lecture will be concerned with PI spaces —metric measure spaces thatare doubling and satisfy a Poincare inequality in the sense of Heinonen-Koskela. After reviewing some background, I will discuss some new exam-ples of PI spaces and related open problems.

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A characterization of BV -functions via Semmesfamily of curvesRiikka KORTE (University of Helsinki — Finland)

A doubling metric measure space supports a Semmes family of curves, iffor every pair of points, there exists a curve family with certain uniformityproperties. This condition implies (1,1)-Poincare inequality.

In [1], we define a geometric Semmes family of curves, i.e. a Semmesfamily of curves with some additional geometric properties for the curves. Inthis setting, we provide a Reshetnyak-type characterization of functions ofbounded variation in terms of the total variation on curves. We then use thischaracterization to prove a Federer-type characterization for sets of finiteperimeter: a set is of finite perimeter if and only if the Hausdorff measure ofits measure theoretic boundary is finite.

The plan of this talk is to first explain the idea of Federer’s proof andwhy it is difficult to extend to the setting of metric measure spaces. Then Iwill discuss Semmes family of curves, the extra conditions that we need forgeometric Semmes family and why these conditions are helpful in provingthe characterization. Lastly, I will explain how to construct the (geometric)Semmes family in Euclidean spaces, Heisenberg group and Gehring’s bow-tie.REFERENCES

[1] R. Korte, P. Lahti, and N. Shanmugalingam, Semmes family of curves and a character-ization of functions of bounded variation in terms of curves, to appear in Calc. Var. PartialDifferential Equations.

Nonpositive curvature beyond Busemann spacesUrs LANG (ETH Zurich — Switzerland)

A geodesic bicombing on a metric space selects for each pair of points ageodesic connecting them. The existence of a geodesic bicombing satisfy-ing a suitable convexity condition may be viewed as a weak (but non-coarse)notion of nonpositive curvature that allows for non-unique geodesics. Weshow that a number of constructions and results known for CAT(0) or Buse-mann spaces have analogs for spaces with convex bicombings: Boundaryat infinity, existence of a contracting barycenter map, fixed point theorems,existence of flat (normed) strips and planes, and the flat torus theorem. Thetalk is based on joint work with Dominic Descombes.

Joint work with Dominic Descombes.

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Recent results on the regularity of H-minimalsurfacesRoberto MONTI (Universita di Padova — Italy)

We review some recent results on the regularity of H-minimal boundaries inthe Heisenberg group: Lipschitz approximation, height estimate, harmonicapproximation, and also slicing formulas. This is part of a joint researchprogram with D. Vittone.

Joint work with Davide Vittone.

Gromov-Hausdorff Convergence of e-coversConrad PLAUT (University of Tennessee — USA)

The d -covers of Sormani-Wei, which have a number of interesting appli-cations in metric geometry, are known not to be “closed” with respect toGromov-Hasudorff convergence. In this paper we use the notion of essen-tial circles, introduced previously by the authors, to define a larger classof covering maps of compact geodesic spaces called “circle covers” thatare “closed” with respect to Gromov-Hausdorff convergence and include d -covers. In fact, we use circle covers to completely understand the limitingbehavior of d -covers. The proofs use the discrete homotopy methods devel-oped by Berestovskii, Plaut, and Wilkens, and in fact we show that whend = 3e

2 , the Sormani-Wei d -cover is isometric to the Berestovskii-Plaut-Wilkins e-cover. Of possible independent interest, our arguments involveshowing that “almost isometries” between compact geodesic spaces resultin explicitly controlled quasi-isometries between their e-covers. Finally, weuse essential circles to strengthen a theorem of E. Cartan by finding a newfinite set of generators of the fundamental group of a compact Riemannianmanifold. We conjecture that there is always a generating set of this sorthaving minimal cardinality among all generating sets.

Joint work with Jay Wilkins (University of North Carolina-Pembroke).

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Besicovitch covering property in Carnot groupsSeverine RIGOT (Nice University — France)

The Besicovitch covering property is an important and useful property inmetric measure spaces. It has been known for a long time that it fails in theHeisenberg groups for the most commonly used homogeneous distances.However it has been recently proved in a joint work with E. Le Donne thatit is valid for some bi-Lipschitz equivalent homogeneous distance. We willshow that this result extend to Carnot groups of step 2 whereas there is nohomogeneous distance for which the Besicovitch covering property holds assoon as the step of the group is 3 or higher (joint work with E. Le Donne). Iftime permits, we will give some consequences of these results and explainhow they extend to homogeneous distances and quasi-distances on moregeneral graded groups.

Joint work with Enrico Le Donne.

Intrinsic Lipschitz graphs in Carnot groupsRaul SERAPIONI (University of Trento — Italy)

We present the notions of intrinsic graphs and of intrinsic Lipschitz graphswithin Carnot groups. Intrinsic Lipschitz graphs are the natural local ana-logue inside Carnot groups of Lipschitz submanifolds inside Euclideanspaces. Intrinsic Lipschitz graphs unify different alternative approaches tosubmanifolds in a group, usually defined through Lipschitz parameteriza-tions or through level sets. We provide both geometric and analytic char-acterizations and a clarifying relation between these graphs and Rumin’scomplex of differential forms.

Functions of bounded variation in metric setting:three definitionsNageswari SHANMUGALINGAM (University of Cincinnati — USA)

In this talk we will give an introduction to the BV theory in the metric setting(due to Michele Miranda), and discuss a Ledoux type characterization and aheat semigroup characterization of sets of finite perimeter in this theory.

Joint work with Niko Marola (Aalto University, Finland) and Michele Miranda Jr.(University of Ferrara, Italy).

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Convergence of metric measure spacesTakashi SHIOYA (Mathematical Institute, Tohoku University — Japan)

Gromov introduced a new topology on the set of isomorphism classes ofmetric measure spaces, based on the idea of concentration of measurephenomenon due to Levy and Milman. This is a generalization of mea-sured Gromov-Hausdorff topology. Different from the measured Gromov-Hausdorff topology, Gromov’s topology is suitable to study a non-GH-precompact family of spaces. In this talk, I show the study of convergenceof spaces with unbounded dimension. One of our main theorems states thatthe n-dimensional sphere of radius

pn in Rn+1 converges to (R•,k ·k2,g•)

as n ! •, where k ·k2 is the l2 norm on R• which takes values in [0,+• ],and g• the infinite-dimensional standard Gaussian measure on R•.

Marstrand’s density theorem in the HeisenberggroupJeremy TYSON (University of Illinois at Urbana-Champaign — USA)

We establish a version of Marstrand’s density theorem in the Heisenberggroup Hn equipped with the Koranyi (gauge) metric dH . The proof is anadaptation of a method of Kirchheim and Preiss and relies on an analysisof uniform and uniformly distributed measures in (Hn,dH). We also dis-cuss ongoing work concerning a proposed classification of uniform mea-sures supported on smooth surfaces in (H1,dH). This talk is based on jointwork with Vasilis Chousionis and Valentino Magnani.

Joint work with Vasilis Chousionis (University of Helsinki, Finland) and ValentinoMagnani (University of Pisa, Italy).

CR geometry in 3-DPaul YANG (Princeton University — USA)

I will talk about several Sobolev inequalities for CR geometry in 3-D. Eachone is related with conformally covariant operators and its associated curva-ture. The most familiar one is the analogue of the Yamabe equation, and themost recent one has to do with the newly introduced Q-prime curvature.

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Short TalksShort Talks

Research term ICMAT-AGMS Short talks

Prime ends and mappingsTomasz ADAMOWICZ (Institute of Mathematics of the Polish Academy ofSciences, Warsaw — Poland)

The studies of prime ends have long history involving various approaches,for example due to Caratheodory, Nakki, Vaisala and Zorich. In the talk wepresent some of those theories, including recent developments, and discussproblems of continuous and homeomorphic extension of mappings to thetopological and prime end boundaries in the Euclidean setting and the settingof metric measure spaces. If time permits we will also present the extensionproblem in the setting of Heisenberg groups.

A characterization of 1-rectifiable doublingmeasures with connected supportsJonas AZZAM (Universitat Autonoma de Barcelona and Centre de RecercaMatematica — Spain)

Garnett, Killip, and Schul have exhibited a doubling measure µ with supportequal to Rd which is 1-rectifiable, meaning there are countably many curvesGi of finite length for which µ(Rd\

SGi) = 0. Motivated by this result, we

characterize when a doubling measure µ with support equal to a connectedmetric space X has a 1-rectifiable subset of positive measure by showing thisset coincides up to a set of µ-measure zero with the set of x 2 X for whichliminfr!0 µ(BX(x,r))/r > 0.

Joint work with Mihalis Mourgoglou (Universitat Autonoma de Barcelona and Centrede Recerca Matematica).

11


On the Bourgain-Brezis-Mironescu characterizationof Sobolev spaces in certain metric spaces andrelated fractional norm inequalitiesDavide BARBIERI (Universidad Autonoma de Madrid — Spain)

In a classical work of 2001, a new characterization of Sobolev spaces W 1,p,p > 1, was proposed by J.Bourgain, H.Brezis and P.Mironescu by show-ing an approximation property of gradient norms in terms of locally aver-aged finite differences. An analogous result can be obtained on stratifiedLie groups, which constitute the model setting for manifolds whose tangentbundle is generated by Hormander type vector fields. The obtained approx-imated norms also provide general families of Poincare-type inequalities,which can be proved by means of a constructive method based on a one-dimensional inequality. Fractional inequalities in terms of Gagliardo normscan be proved as a consequence of these Poincare inequalities and of self-improving properties of the approximated norms.

Bi-Lipschitz pieces between manifoldsGuy C. DAVID (Courant Institute, New York University — USA)

A well known class of questions asks the following: If X and Y are met-ric measure spaces, and f is a Lipschitz map between them whose im-age has positive measure, then must f have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present speaker!)and Semmes, we answer this question for a class of abstract Ahlfors s-regular topological d-manifolds.

Isoperimetric problem in Grushin spaces andH-type groupsValentina FRANCESCHI (Universita di Padova — Italy)

We prove existence, symmetry and regularity properties of isoperimetric setsin Grushin spaces under a symmetry assumption which depends on the di-mension. We also emphasize a relation with the isoperimetric problem inH-type groups.

Joint work with Roberto Monti (Universita di Padova).

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Area-stationary and stable surfaces of class C1 inthe sub-Riemannian Heisenberg group H1

Matteo GALLI (Universita di Bologna — Italy)

We consider surfaces of class C1 in the 3-dimensional sub-RiemannianHeisenberg group H1. Assuming the surface is area-stationary, i.e., a criti-cal point of the sub-Riemannian perimeter under compactly supported vari-ations, we show that its regular part is foliated by horizontal straight lines.In case the surface is complete and oriented, without singular points, andstable, i.e., a second order minimum of perimeter, we prove that the surfacemust be a vertical plane. This implies the following Bernstein type result: acomplete locally area-minimizing intrinsic graph of a C1 function in H1 is avertical plane.

Joint work with Manuel Ritore (Universidad de Granada).

The branch set of a quasiregular mapping: recentadvances and open problemsChangyu GUO (University of Jyvaskyla — Finland)

Let f : X ! Y be a continuous, discrete and open mapping between twolocally compact, locally doubling, locally linearly locally contrictible metricspaces, and let B⇤

f be the generalized branch set, i.e. points where f has nolocal homotopy inverse.

In his ICM 2002 address, Heinonen asked the following interesting ques-tion: Can we describe the geometry of allowable branch sets of quasiregularmappings between metric n-manifolds?

The study of the geometry of the branch set will lead to numerous impor-tant consequences in the theory of quasiregular mappings as well as analysison metric spaces. In this talk, I will discuss the porosity of the branch set ofa quasiregular mapping as well as its image in a very general setting, basedon a recent joint work with my friend Marshall Williams.

I will start the talk by giving an overview of the study related tothe branch set of quasiregular mappings in various settings and thenpresent/sketch the proof of the well-known Bonk-Heinonen theorem (2004)and Onninen-Rajala theorem (2009). After that, I will indicate the idea ofour approach in the Euclidean setting and put up some natural open prob-lems arised from our work.

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Measure density and extension of Besov andTriebel–Lizorkin functionsLizaveta IHNATSYEVA (University of Jyvaskyla — Finland)

We discuss extension properties of Besov-type functions and of Triebel-Lizorkin type functions in the setting of a metric measure space with adoubling measure. In particular, we give a characterization of extensiondomains for Hajłasz–Besov and for Hajłasz–Triebel–Lizorkin spaces.

Joint work with Toni Heikkinen (Aalto University) and Heli Tuominen (University ofJyvaskyla).

Li-Yau type inequalities for heat flows on metricmeasure spacesRenjin JIANG (Universitat Autonoma de Barcelona — Spain)

Let (X ,d,µ) be a RCDast(K,N) space with K 2 R and N 2 [1,•]. In thistalk, we will talk about some recent result on the functional inequalities(including Li-Yau inequality, Li-Yau-Hamilton inequality etc.) for the heatflow on (X ,d,µ). Some applications including heat kernel bounds are pre-sented.

Sobolev spaces on metrizable groupsTomasz KOSTRZEWA (Warsaw University of Technology — Poland)

We study Sobolev spaces on locally compact abelian groups. In my talk Iwill focus on embedding results when the dual group is metrizable. I willpresent the density of continuous functions and the trace theorem. Moreover,new embeddings of Sobolev spaces defined over p-adic numbers are shown.

Joint work with P. Gorka (Warsaw University of Technology).

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Parabolic Harnack inequality for time-dependentnon-symmetric Dirichlet formsJanna LIERL (University of Illinois at Urbana-Champaign — USA)

This talk is concerned with the parabolic Harnack inequality for appropriateweak solutions of the heat equation associated with non-symmetric, time-dependent, local Dirichlet forms. Assuming that the underlying metric mea-sure space has the volume doubling property and satisfies a Poincar’e in-equality, we show the local boundedness of weak solutions to the heat equa-tion. Complementing results of Sturm, Aronson, Serrin, and others, we givea proof of the parabolic Harnack inequality in the context of non-symmetricDirichlet forms.

Joint work with Laurent Saloff-Coste (Cornell University).

Modulus and Hyperbolic FillingsJeff LINDQUIST (University of California Los Angeles — USA)

Conformal modulus has proven to be be useful in analysis on metric spaces.There are some instances, however, where modulus techniques are desirablebut not applicable. By using hyperbolic fillings we develop the notion of theweak Q-capacity of two disjoint open sets. We prove that for disjoint openballs this generally controls the corresponding Q-modulus of the path familyconnecting the two balls. We also show that in a Q-dimensional Loewnerspace setting these two quantities are comparable.

15


Self-improvement of Poincare inequalities and lackthereofLukas MALY (University of Jyvaskyla — Finland)

Self-improvement and in particular open-endedness of Poincare inequalitiesin metric measure spaces will be discussed during the talk. Recently, DeJar-nette refined a deep result of Keith and Zhong by showing that an Orlicz-typePoincare inequality, whose governing Young function is in a certain senseclose to the pth power function, can be improved to a (p� e)-Poincare in-equality. The used method required a certain bound for the growth rate ofthe Young function. Optimality of such a bound was however left unsolved.

I will focus on improvement (in the sense of Keith and Zhong) of moregeneral Poincare inequalities that are in a way close to p-Poincare inequal-ities. Such a setting will allow us to find a fairly simple example of a com-plete Ahlfors regular metric space that supports a Lorentz-type Poincare in-equality, which however cannot be improved. In other words, a Lorentz-typePoincare inequality need not be an open-ended condition. Furthermore, thisparticular example can be used to show optimality of DeJarnette’s growthrate bound.

Widths of convex bodies in the 2-dimensional sphereAntonio Roberto MARTINEZ FERNANDEZ (Universidad deMurcia — Spain)

If K ✓ R2 is a planar convex body (i.e., compact convex subset with non-empty interior), the width of K in the direction u 2 S1, written w(K,u), isthe distance between two support lines of K orthogonal to u. This classicalnotion of width has an analog for convex bodies K in the 2-dimensionalsphere S2

k of curvature k > 0 (i.e., with radius 1/p

k). But in this case, thewidth function is defined over the boundary of the convex body K, w(K, ·) :bdK ! R.

If K ✓ S2k is a centrally symmetric convex body in the sphere, with center

cK , it is natural to define the notion of central width wc(K, ·) : TcKS2k !R, in

order to get a function defined on vectors instead of points of the boundarycurve of K.

Along this talk, we study these two notions of width of convex bodies inthe sphere, and we compare them with other geometric measures such as thearea, perimeter, diameter, inradius and circumradius.

Joint work with Marıa A. Hernandez-Cifre (Universidad de Murcia).

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Locally Lipschitz contractibility of Alexandrovspaces and its applications, orientability and metriccurrents of Alexandrov spaces and its applicationsAyato MITSUISHI (Tohoku University — Japan)

Alexandrov spaces are (finite-dimensional, complete, geodesic) metricspaces of curvature bounded from below in the sense that they satisfy tri-angle comparison condition of characterizing a lower curvature bound. Itis important that investigating geometry and topology of Alexandrov spacesthoroughly, to understand geometry and topology of collapsing Riemannianmanifolds, under a uniform lower curvature bound.

An important topological property of Alexandrov spaces was proved byPerelman, which states that any Alexandrov space is locally cone-like ([P],[P2]).

More precisely, for any point p in an Alexandrov space M, there is anr > 0 such that the closed metric ball B(p,r) centered at p of radius r ishomeomorphic to the closed cone over some compact metric space. Further,Perelman claimed that the homeomorphism here can be taken to be a bi-Lipschitz homeomorphism, but did not write a proof. In the present time,this problem is still open.

In this talk, I give you a partial answer to this claim. I will announce apartial answer to the claim, obtained by me and Yamaguchi (Kyoto Univ):Let M be a finite-dimensional Alexandrov space. For any p 2 M and r > 0, if thedistance function dp from p is regular on B(p,r)\{p}, then there is a Lipschitz mapH : B(p,r)⇥ [0,1] ! B(p,r) such that H(x,0) = x, H(x,1) = p and the functiont ! d(H(x, t), p) is monotone non-increasing for any x 2 B(p,r). In particular, forany p 2 M, there is an r > 0 such that the ball B(p,r) admits an H satisfying aproperty as above.

Precisely, we proved more strong property, that is called (strongly) lo-cally Lipschitz contractibility. I will also exhibit several applications of thisresult and related results obtained in [MY], [M3].[MY] A. Mitsuishi and T. Yamaguchi, Locally Lipschitz contractibility of Alexan-drov spaces and its applications, Pacific J. Math. 270 (2014), no. 2, 393–421.[M3] A. Mitsuishi, Orentations of Alexandrov spaces and a filling radiuis inequal-ity, work in progress.[P] G. Perelman, A. D. Alexandrov’s spaces of curvature bounded from below II,preprint, 1991.[P2] G. Perelman, Elements of Morse theory on Aleksandrov spaces, (Russian, withRussian summary), Algebra i Analiz 5 (1993), no. 1, 232–241; English transl., St.Petersburg Math. J. 5 (1994), no. 1, 205–213.

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Injective hulls of metric spacesMael PAVON (ETH Zurich — Switzerland)

The talk will start with a brief reminder of some basics on injective (or hy-perconvex) metric spaces and injective hulls (or tight spans). Well-knownexamples of injective metric spaces are the real line R, l•(I) for any indexset I, and all complete metric trees. The first goal of the talk is to present newclasses of examples. The second goal is related to the structure of injectivehulls. Namely, we will discuss conditions one can impose on a metric space(X ,d), in order to obtain geometric properties satisfied by the injective hullE(X ,d) like having restrictive isometry classes of cells.

Waning under Cheeger deformationsPedro SOLORZANO (UFSC — Brazil)

The pointed Gromov-Hausdorff convergence of the Sasaki metrics on thetangent bundles (both the total spaces and their projection maps, up to pass-ing to a subsequence) over a sequence of converging riemannian spaces inthe Gromov-Hausdorff sense. Furthermore, the structure of the limit is suchthat the fiber of the limit map at a point p is homeomorphic to a quotient ofthe form Rk/G and G is called em wane group at p of the sequence.

Waning in principal bundles undergoing a Cheeger deformation collaps-ing the total space to the base is entirely determined by the riemannianholonomy of the fibers. The Gromov-Hausdorff limit of the tangent bundlesover these principal bundles with their Sasaki metric is seen to be a locallytrivial fiber bundle containing the tangent space to the base as a subbundlein a natural way. Berger 3-spheres provide an example where the limit fibersare still of dimension 3.

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Lusin Approximation and Horizontal Curves inCarnot GroupsGareth SPEIGHT (Scuola Normale Superiore — Italy)

Carnot groups have strong geometric structure, including translations and di-lations, but a distinguished family of horizontal directions. We ask whethergeneral horizontal curves can be approximated by smooth horizontal curvesoutside a set of small measure.

Steiner’s formula in the Heisenberg groupKevin WILDRICK (Montana State University — USA)

Steiner’s formula states that the volume of an epsilon neighborhood of suf-ficiently smooth set in n- dimensional Euclidean space is a polynomial ofdegree n, whose coefficients carry information about the curvature of theboundary of the set. We will provide an analogous result for the Carnot-Caratheodory distance in the first Heisenberg group. Although the resultingfunction is not, in general, a polynomial, it is analytic and the coefficients inits series expansion are integrals of second order differential operators. Inparticular, this approach produces a candidate for the notion of horizontalGauss curvature in the Heisenberg group.

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On a linear refinement of the Prekopa-LeindlerinequalityJesus YEPES (ICMAT — Spain)

The Prekopa-Leindler inequality states that, given l 2 (0,1) and non-negative measurable functions f ,g,h : Rn �! R�0 such that, for any x,y 2Rn,

h�(1�l )x+ly

�� f (x)1�l g(y)l ,

then Z

Rnhdx �

✓Z

Rnf dx

◆1�l ✓Z

Rngdx

◆l.

This result is closely related to a number of important and classical in-equalities in Analysis such as Holder’s inequality or the reverse Young’s in-equality and to some geometric ones like the well-known Brunn-Minkowskiinequality.

In this talk we will show that under the sole assumption that f and ghave a common projection onto a hyperplane (which is the analytic coun-terpart of the projection onto a hyperplane of a set), the Prekopa-Leindlerinequality admits a linear refinement. That is, under such an assumption forthe functions f and g, the right-hand side in the above integral inequalitymay be exchanged by the convex combination of the integrals, which yieldsa stronger inequality. Moreover, the same inequality can be obtained whenassuming that both projections (not necessarily equal as functions) have thesame integral. We will explore the main idea of the proof of the latter result,which has a strong geometric flavor.

Joint work with A. Colesanti (Dipartimento di Matematica “U. Dini”), E. SaorınGomez (Institut fur Algebra und Geometrie, OvG Universitat Magdeburg).

Whitney’s Extension Theorem for Curves in theHeisenberg Group Hn

Scott ZIMMERMAN (University of Pittsburgh — USA)

For a compact set K ⇢Rn, Whitney’s extension theorem provides conditionsunder which a real valued function f : K ! R may be extended to a smoothF : Rn !R. If we instead consider a horizontal mapping f : K !Hn, Whit-ney’s conditions are not sufficient to guarantee the existence of a smooth,horizontal extension F : Rn ! Hn. In this talk, I will present an additionalcondition on f that guarantees the existence of a C1, horizontal extensionwhen K ⇢ R.

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Madrid, April–June 2015

Analysis and Geometry

in Metric Spaces

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