My research area is geometric function theory. I am
interested in the connections between inequalities in analysis and their
geometric

implications, and also in quasiconformal maps and the geometric properties of metric
measure spaces conserved by them. Currently I

also study interactions between geometry and the BV theory in
metric spaces, and the utility of sets of finite perimeter
in the study of

Sobolev spaces and quasiconformal
maps.

My thesis, 1999, at the University of Michigan.

The book

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

(with Juha Heinonen,
Pekka Koskela and Jeremy
Tyson),

published by Cambridge University Press,

New Mathematical Monographs series: February 2015 (ISBN:
9781107092341).

A preliminary version of this book can be found here.

Keep in mind that the downloads of the following
papers are earlier versions of the final articles and therefore may differ from
the

journal article. Moreover, University of Cincinnati is in the
process of converting its webpage server, and so this might not be updated

for a while.

——————————————————————————

**Newton-Sobolev spaces for metric measure spaces:**

**1.** Newtonian
spaces: An extension of Sobolev spaces to metric measure spaces

appeared in Rev. Mat. Iberoamericana, **16**
(2000) 243--279.

**2.** Removable
sets for the Poincaré inequality on metric spaces

(with Pekka Koskela and Heli
Tuominen)

appeared in Indiana Univ. Math. J., **49** (2000) 333--352.

**3.** Modulus
and continuous capacity

(with Sari kallunki)

appeared in Ann.
Acad. Sci. Fenn. Math., **26** (2001) 455--464.

**4.** Sobolev
classes of Banach space-valued functions and quasiconformal
mappings

(with Juha Heinonen, Pekka Koskela, and Jeremy Tyson)

appeared in J. Anal. Math., **85** (2001) 87--139.

**5.** Measurability
of equivalence classes and $MEC_p$-property in metric
spaces

(with Esa Järvenpää, Maarit Järvenpää,
Kevin Rogovin, and Sari Rogovin.)

appeared in Rev. Mat. Iberoamericana, **23**
no. 3 (2007) 811--830.

**6.** Sobolev
extensions of Hölder continuous and characteristic functions on metric spaces

(with Anders Björn and Jana Björn)

appeared in Canadian Journal of Mathematics, **59** (No. 6) (2007),
1135--1153.

**7.** Poincaré inequalities, uniform domains,
and extension properties for Newton-Sobolev spaces in metric spaces

(with Jana Björn)

appeared in J. Math. Anal. Appl., **332**, No. 1 (2007) 190--208.

**8.** Quasicontinuity of Newton--Sobolev
functions and density of Lipschitz functions in metric measure spaces

(with Anders Björn and Jana Björn)

appeared in Houston J. Math., **34** No. 4 (2008) 1197--1211.

**9.** Lebesgue
points and capacities via boxing inequalities in metric spaces

(with Juha Kinnunen, Riikka Korte, and Heli
Tuominen)

appeared in Indiana Univ. Math. J., **57** no. 1 (2008) 401-430.

**10.** Interpolation
properties of Besov spaces defined on metric spaces

(with Pekka Koskela and Amiran Gogatishvili)

appeared in Math. Nachr., **283** no.2
(2010) 215--231.

**11.** Geometric
properties of planar BV extension domains

(with Pekka Koskela and Michele Miranda)

appeared in Function Spaces; Topics Around the Research of Prof. Maz'ya I,

International
Mathematical Series (2010) (Springer collection).

**12.** The DeGiorgi measure and an obstacle problem related to minimal
surfaces in metric spaces,

(with Juha Kinnunen, Riikka Korte, and Heli Tuominen),

appeared in J.
Math. Pures Appl., **93** (2010)
599--622.

**13.** Non-conformal
Loewner type estimates for modulus of curve families

(with Tomasz
Adamowicz),

appeared in Ann. Acad. Sci. Fenn. Math., **35** (2010) 609--626.

**14.** The $\infty$-Poincaré inequality in
metric measure spaces

(with Estibalitz
Durand-Cartagena and Jesus
A. Jaramillo),

appeared in Michigan Math. J., **61** No.1 (2012) 63--85.

**15.** $p$-Poincaré inequality vs. $\infty$-Poincaré inequality; some counter-examples,

(with Estibalitz
Durand-Cartagena and Alex Williams),

appeared in Math.
Z., **271** No. 1--2 (2012) 447--467.

**16.** A
characterization of Newtonian functions with zero boundary values,

(with Juha Kinnunen, Riikka Korte, and Heli Tuominen),

Calc. Var. PDE, **43** (2012) 507--528.

**17.** Introduction
to $p$-modulus of path-families and Newtonian spaces,

appeared in J.
Anal., **18 **(2010), 349--360.

**18.** First order Poincar\'e inequalities in
metric measure spaces

(with Jesus Jaramillo and Estibalitz
Durand)

appeared in Ann. Acad. Scie. Fenn. Math., **38 **(2013), 287--308.

**19.** Pointwise
properties of functions of bounded variation in metric spaces,

(with Juha Kinnunen, Riikka Korte , and Heli
Tuominen)

Appeared in Revista Mat. Complutense,
**27 **(2014), 41--67.

**20.** Newton-Besov Spaces and Newton-Triebel-Lizorkin Spaces on Metric Measure Spaces,

(with Dachun Yang
and Wen Yuan),

appeared in Positivity **19** (2015) no. 2, 177--220.

**21.** Geometric analysis on Cantor sets and trees,

(with Anders Björn, Jana Björn, and Jim Gill),

appeared in Journal für die reine und angewandte Mathematik
(Crelle's journal) **725** (2017), 63--114.

**22.** Preservation
of bounded geometry under sphericalization and
flattening,

(with Xining Li),

appeared in Indiana
Univ. Math. J. **64** No. 5 (2015), 1303--1341.

**23.** Hajłasz Gradients Are
Upper Gradients,

(with Renjin Jiang, Dachun Yang, and Wen Yuan),

appeared in JMAA, **422** No. 1 (2015), 397--407.

**24.** Semmes family of curves and a characterization of functions of bounded
variation in terms of curves,

(with Riikka Korte
and Panu Lahti),

appeared in Calc. Var. PDE **54**, Issue 2, (2015) 1393--1424.

**25.** An elementary proof of Cheeger's theorem on
reflexivity of Newton-Sobolev spaces of functions in metric measure spaces,

(with Estibalitz Durand-Cartagena),

appeared in J.
Analysis , **21** (2013, published in 2015), 73--83.

**26.** Geometric characterizations of p-Poincaré
inequalities,

(with Estibalitz Durand Cartagena and
Jesus Jaramillo),

appeared in Pub.
Mat. **60** (2016), 81--111.

**27.** Infinity modulus and the essential metric,

(with Nathan Albin, Jared Hoppis, and
Pietro Poggi-Corradini),

appeared in JMAA, **467**, Issue 1 (2018), 570--584.

(2018).

**28.** Locally
$p$-admissible measures on $\R$,

(with Anders Björn and Jana Björn),

appeared in Journal of Functional Analysis, **278** Issue 4 (2020)

Link to the DOI.

**29.** Traces and extensions of certain weighted Sobolev spaces on $R^n$ and Besov functions on Ahlfors regular compact subsets of $R^n$.,

(with Jeff Lindquist) (2020).

appeared in Complex Analysis and its Synergies, Modern Geometric and Complex Analysis: A Celebration of Pekka Koskela’s 60th Birthday,

**7** (2021), no. 1, Paper No. 7, 12 pp.

**30**. Fine
properties of metric space-valued mappings of bounded variation in metric
measure spaces,

(with Iván Caamaño and Josh Kline),

to appear in Annali della Scuola
Normale Superiore di Pisa-Cl. Scienze
(2024), DOI: 10.2422/2036-2145.202309_017

**—————————————————————————**

**$p$-harmonic functions in metric measure spaces:**** **

**1.** Harmonic functions on
metric spaces

appeared in Illinois
J. Math., **45** no. 3 (2001) 1021--1050.

**2.** Regularity of quasi-minimizers on metric spaces

(with Juha Kinnunen)

appeared in Manuscripta Math.,
**105** (2001) 401--423.

**3.** Fat sets and
pointwise boundary estimates for $p$-harmonic functions in metric spaces

(with Jana Björn and Paul MacManus)

appeared in J. Anal. Math., **85** (2001) 339--369.

**4.** The Dirichlet
problem for $p$-harmonic functions on metric measure spaces

(with Anders Björn and Jana Björn)

appeared in J.
Reine Angew. Math. (Crelle)
**556** (2003) 173--203.

**5.** Some
convergence results for $p$-harmonic functions on metric measure spaces

appeared in the Proceedings of the London Math. Soc. **87** (2003)
226--246.

**6.** The Perron
method for $p$-harmonic functions in metric spaces

(with Anders Björn and Jana Björn)

appeared in J. Differential Equations **195** (2003) 398--429.

**7.** A problem of Baernstein on the equality of the $p$-harmonic measure of a
set and its closure

(with Anders Björn and Jana Björn)

appeared in Proc. Amer.
Math. Soc. **134** (2006) 509-519.

**8.** Hölder estimates
of $p$-harmonic extension operators

(PDF file) (with Hiroaki
Aikawa)

appeared in J. Differential Equations **220** (2006) No. 1, 18--45.

**9.** Polar
sets on metric spaces

(with Juha Kinnunen)

appeared in Transactions Amer. Math. Soc. **358** (2006) 11--37.

**10.** Equivalence of
AMLE, strong AMLE, and comparison with cones in metric measure spaces
(pdf file)

(with Petri Juutinen)

appeared in Math. Nachr. **279** (2006)
1083--1098.

**11.** Maximal
regularity via reverse Holder inequalities for elliptic systems of $n$-Laplace
type involving measures, (with Xiao Zhong
and Tero Kilpeläinen)

appeared in Ark. Mat. **46** (No. 1) (2008) 77--93.

**12.** Equivalence
and self-improvement of $p$-fatness and Hardy's inequality, and association
with uniform perfectness

(with Riikka Korte)

Math. Z. **264** (No. 1) (2010) 99--110.

**13.** Quasiconformality, homeomorphisms between
metric measure spaces preserving quasiminimizers, and
uniform density property

(with Niko
Marola and Riikka Korte)

appeared in Ark. Mat. **50** (2012) 111--134.

**14.** Comparisons
of relative BV-capacities and Sobolev capacity in metric spaces,

(with Heikki Hakkarainen),

appeared in Nonlinear
Anal., **74** (2011) 5525--5543.

**15.** Regularity
of sets with quasiminimal boundary surfaces in metric spaces,

(with Juha Kinnunen, Riikka Korte , and Andrew Lorent),

Appeared in J. Geom. Anal., **23** (2013) 1607--1640.

**16.** Quasiadditivity of
variational capacity (with Juha Lehrbäck),

appeared in Potential Anal. **40** (2014), no. 3, 247--265.

**17.** Boundary measures, generalized Gauss-Green formulas and the mean value
property in metric measure spaces,

(with Niko
Marola and Michele Miranda Jr.),

appeared in Revista Mat. Iberoamericana **31** (2015), no. 2,
497--530.

**18.** Stability and
continuity of functions of least gradient,

(with Heikki Hakkarainen,
Riikka Korte, Panu Lahti),

appeared in Analysis and Geometry in Metric Spaces **3** (July 2015)
no. 1, 123--139.

**19.** Fine properties and a notion of quasicontinuity
for BV functions on metric spaces,

(with Panu Lahti),

appeared in Journal
de Mathématiques Pures et Appliquées **107** (2017), no. 2, 150--182.

**20.** Trace and extension theorems for functions of bounded variation,

(with Lukáš Malý
and Marie Snipes),

appeared in Annali SNS,
**18** (2018) 313--341. (DOI Number: 10.2422/2036-2145.201511_007).

**21.** Brief survey on infinity-Poincaré
inequality and existence of infinity-harmonic functions,

in the book collection

Measure Theory in Non-Smooth Spaces, edited by Nicola Gigli (De
Gruyter Open) (2017) 243--287. ISBN -- 978-3-11-055083-2.

**22.** Dichotomy of global capacity density in metric measure space,

(with Hiroaki Aikawa, Anders Björn, Jana Björn),

appeared in Advances in Calculus of Variation, **11** (2018),
387--404.

**23.** An Analog of the Neumann Problem for the 1-Laplace Equation in the
Metric Setting: Existence, Boundary Regularity, and Stability,

(with Panu Lahti and Lukas Maly),

appeared in Analysis and Geometry in Metric Spaces **6** Issue 1
(2018) 1--31.

**24.** Trace theorems for functions of bounded variation in metric spaces,

(with Panu Lahti), on CVGMT preprint
server (2015),

appeared in J. Funct. Anal. **274**, Issue
10 (2018) 2754--2791.

**25.** Neumann problem for $p$-Laplace equation in metric spaces using a
variational approach: existence, boundedness, and boundary regularity,

(with Lukáš Malý)

on CVGMT
preprint server, appeared in J. Diff. Eq., **265**, Issue 6 (2018), 2431--2460.

**26.** Notions of Dirichlet problem for functions of least gradient in metric
measure spaces,

(with Riikka Korte, Panu Lahti, Xining Li), on CVGMT preprint server(2016).

Appeared in Revista Mat. Iberoamericana **35** Issue 6 (2019),
1603--1648.

**27.** Existence and uniqueness of infinity-harmonic functions under
assumption of infinity-Poincaré inequality,

(with Estibalitz Durand-Cartagena,
Jesús A. Jaramillo),

appeared in Math. Ann. **374** Issues 1--2 (2019) 881--906.

see also Online
First Math. Ann..

**28.** Domains in metric measure spaces with boundary of positive mean
curvature, and the Dirichlet problem for functions of least gradient,

(with Panu Lahti, Lukas Maly, Gareth Speight),

appeared in Journal of Geom. Anal., **29** Issue 4 (2019)
3176--3220.

**29.** The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy,

(with Anders Björn and Jana Björn),

appeared in Math.
Z. **297** (2021), no. 1-2, 827--854

**30.** Equivalence of solutions of eikonal equation
in metric spaces,

(with Qing Liu and Xiaodan Zhou),

appeared in J. Diff. Eq. **272** (2020), 979--1014.

**31.** Bounded geometry and p-harmonic functions under uniformization and hyperbolization,

(with Anders Björn and Jana Björn),

appeared in J. Geom. Anal., **31** (2021), no. 5, 5259--5308

**32.** Asymptotic behavior of BV functions and sets of finite perimeter in
metric measure spaces,

(with Sylvester Eriksson-Bique, James
T. Gill, and Panu Lahti),

arXiv preprint,

appeared in Trans. Amer. Math. Soc., **374** (2021), no. 11,
8201--8247.

**33.** Regularity of solutions to the fractional Cheeger-Laplacian
on Domains in metric spaces of bounded geometry,

(with Sylvester Eriksson-Bique,
Gianmarco Giovannardi, Riikka
Korte, and Gareth Speight),

CVGMT
preprint (2020).

appeared in J. Diff. Eq., **306** (2022), 590--632.

**34.** $p$-hyperbolicity of ends and families of paths in metric spaces.

Appeared in Fractal
geometry and stochastics. VI ,

191 -- 205, Progr. Probab.,
**76**, Birkhauser/Springer, Cham, (2021).

**35.** Classification of metric measure spaces and their ends using
$p$-harmonic functions,

(with Anders Björn and Jana Björn),

Appeared in Ann. Fenn. Math. **47 **No. 2 (2022), 1025--1052.

**36**. Solving a Dirichlet problem for unbounded domains via a conformal
transformation,

(with Ryan Gibara and Riikka Korte),

Appeared in Mathematische Annalen **389** (2024). 2857--2901.

**37**. Discontinuous eikonal equations in metric
measure spaces,

(with Qing Liu and Xiaodan Zhou), to
appear in Trans. AMS.

**——————————————————————————**

**Conformal Martin boundary and prime end boundary:**

**1.** Singular
functions on metric measure spaces

(with Ilkka Holopainen)

appeared in Collect. Math., **53** (2002) 313--332.

**2.** On the
conformal Martin boundary of domains in metric spaces

(with Ilkka Holopainen and Jeremy Tyson)

appeared in Papers on
Analysis: A volume dedicated to Olli Martio on the
occasion of his 60th birthday

Report. Univ. Jyväskylä 83 (2001) 147--168.

**3.** Singular
behavior of conformal Martin kernels, and non-tangential limits of conformal
mappings

appeared in Ann.
Acad. Sci. Fenn. Math. **29** (2004) 195--210.

**4.** Carleson type estimates for $p$-harmonic
functions and the conformal Martin boundary of John domains in metric measure
spaces.

(with Hiroaki
Aikawa)

appeared in Michigan Math. J. **53** (2005) 165--188.

**5.** Boundary Harnack principle for $p$-harmonic functions in smooth
Euclidean domains

(with Hiroaki Aikawa, Tero Kilpeläinen, and Xiao Zhong),

appeared in Potential Analysis **26** No.3 (2007) 281--301.

**6.** Uniformity
from Gromov hyperbolicity

(with David
Herron and Xiangdong Xie),

appeared in Illinois J. Math. **52** No. 4 (2008) 1065-1109
(published in November 2009).

**7.** A
rigidity property of some negatively curved solvable Lie groups

(with Xiangdong Xie),

appeared in Comment. Math. Helv. **87**
No. 4 (2012) 805--823.

**8.** Prime
ends for domains in metric spaces,

(with Tomasz Adamowicz, Anders Björn,
and Jana
Björn ),

appeared in Adv. Math., **238 **(2013) 459--505.

**9.** The Mazurkiewicz distance and sets that are
finitely connected at the boundary,

(with Anders Björn and Jana Björn),

appeared in J. Geom. Anal., **26** No. 2 (2016) 873--897.

**10.** Geometry
of prime end boundary and the Dirichlet problem for bounded domains in metric
measure spaces,

(with Dewey
Estep),

appeared in Potential Analysis, **42** No. 2 (2015) 335--363.

**11.** The Dirichlet problem for $p$-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities,

(with Anders Björn, and Jana Björn ),

appeared in J. Diff. Eq. **259** (2015), no. 7, 3078--3114.

**12.** The prime end capacity of inaccessible prime ends, resolutivity,
and the Kellogg property

(with Tomasz
Adamowicz),

appeared in J. Funct. Anal. **293**
(2019), 1633--1656.

**13.** Rough isometry between Gromov hyperbolic spaces and uniformization,

(with Jeff Lindquist),

appeared in Annales Fennici Mathematici
(formerly Ann. Acad. Sci. Fenn.),

**46** No. 1 (2021) 449--464.

**14.** Caratheodory-type
extension theorem with respect to prime end boundaries,

(with Joshua Kline and Jeff Lindquist),

appeared in J. Geom. Anal. (2020).

**15.** Conformal Transformation of Uniform Domains Under Weights that Depend
on Distance to the Boundary,

(with Ryan Gibara),

appeared in Anal.
Geom. Metric Spaces, **10 **(2022), no. 1, 297 – 312.

**16**. On Carrasco Piaggio's theorem characterizing quasisymmetric
maps from compact doubling spaces to Ahlfors regular
spaces,

appeared in Potentials and Partial Differential Equations (Suzanne
Lenhart and Jie Xiao Ed.), (2023) 23 -- 48.

**17.** Sharp Hausdorff content estimates for
accessible boundaries of domains in metric measure spaces of controlled
geometry,

with Sylvester Eriksson-Bique, Ryan Gibara, and Riikka Korte
(2023),

to appear in Trans. AMS.

**—————————————————————————**

**Dirichlet forms and other function spaces on
metric measure spaces:**

**1.** Lipschitz
continuity of Cheeger-harmonic functions in metric
measure spaces

(with Pekka Koskela and Kai Rajala)

appeared in J. Funct. Anal. **202** (2003) 147--173.

**2.** Dirichlet
forms, Poincaré inequalities, and the Sobolev spaces
of Korevaar-Schoen

(with Pekka Koskela and Jeremy Tyson)

appeared in Potential Analysis. **21** No.3 (2004) 241--262.

**3.** A
universality property of Sobolev spaces in metric measure spaces

appeared in the Springer collection Sobolev Spaces in Mathematics I, II, III (2009)

of the International Mathematical Series.

**4.** $L^\infty$-variational problem associated to Dirichlet forms

(with Pekka Koskela and Yuan Zhou)

Mathematical
Research Letters. **19** (2012) no. 6, 1263--1275. (Published in
2013.)

**5.** Intrinsic Geometry and Analysis of Diffusion Processes and $L^\infty$-Variational Problems,

(with Pekka Koskela
and Yuan Zhou),

appeared in Arch. Rat. Mech. Anal. (ARMA), **214** No. 1 (2014),
99--142.

**6.** Geometry
and Analysis of Dirichlet forms (II),

(with Pekka Koskela
and Yuan Zhou)

appeared in J. Funct. Anal., **267** No. 7
(2014), 2437--2477.

**7.** A characterization of BMO self-maps of a metric measure space,

(with Juha Kinnunen, Riikka Korte and Niko Marola),

appeared in Collectanea Math. **66**, Issue 3 (2015) 405--421.

**8.** Characterizations of sets of finite perimeter using heat kernels in
metric spaces,

(with Niko
Marola and Michele Miranda Jr.),

on CVGMT
preprint server (2014),

appeared in Potential
Analysis **45** (2016), Issue 4, pp 609--633.

**9.** Equivalence of two BV classes of functions in metric spaces, and
existence of a Semmes family of curves under a 1-Poincar\'e inequality,

(with Estibalitz
Durand-Cartagena, Sylvester Eriksson-Bique, and Riikka Korte),

CVGMT
preprint,

appeared in Advances in Calculus of Variations, **14** (2021), no.
2, 231--245.

**10.** Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally
$Q$-bounded geometry,

(with Rebekah Jones and Panu Lahti),

appeared in Indiana Univ. Math. J., **60** (2021), no. 5, Paper No.
170, 38 pp.

An older version with a weaker result can be found in arXiv Preprint Server, (2018).

**11.** Besov class via heat
semigroup on Dirichlet spaces I: Sobolev type inequalities,

(with Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, and Alexander Teplyaev),

appeared in J. Funct. Anal. **11** 108459,
48pp. (2020)

(or this link for online first article).

**12.** Besov class via heat
semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel
estimates,

(with Patricia Alonso-Ruiz, Fabrice Baudoin,
Li Chen, Luke Rogers, and Alexander Teplyaev),

appeared in Calc.
Var. PDE. **59** No. 3 (2020) Paper No. 103, 32 pp.

**13.** Besov class via heat
semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel
estimates,

(with Patricia Alonso-Ruiz, Fabrice Baudoin,
Li Chen, Luke Rogers, and Alexander Teplyaev),

appeared in Calc. Var. PDE. **60** (2021) no. 5 Paper no. 170, 38pp.

**14.** Extension and trace results for doubling metric measure spaces and
their hyperbolic fillings ,

(with Anders Björn and Jana Björn),

appeared in JMPA **159** (2022) 196—249.

**15.** Trace and extension theorems for homogeneous Sobolev and Besov spaces for unbounded uniform domains in metric
measure spaces,

(with Ryan Gibara),

appeared in Proc. Steklov Inst. Math. **323**
(2023) 101—119.

**16. **Potential
theory and quasisymmetric maps between compact Ahlfors regular metric measure spaces via Besov functions: preliminary,

(with Juha Lehrbäck),

appeared in *Pure Appl. Funct. Anal.*
9
(2024),
no.
1, 231–249.

**17**. Construction
of a Dirichlet form on metric measure spaces of controlled geometry,

(with Almaz Butaev,
and Liangbing Luo),

appeared in Potential
Anal. (2024).

**—————————————————————————**

——————————————————————————

**Preprints:**

**0.** Connections
between $\infty$-Poincare inequality, quasiconvexity,
and $N^{1,\infty}$,

(with Jesus Jaramillo and Estibalitz
Durand-Cartagena) in CRM preprint No. 895.

**1.** BV
functions and Besov spaces associated with Dirichlet
spaces,

(with Patricia Alonso-Ruiz, Fabrice Baudoin,
Li Chen, Luke Rogers, and Alexander Teplyaev),

This version contains many of the results from I--III of the
papers that have now appeared in journals (lised
above): On arXiv server, (2018).

**2.** Neumann problems for p-harmonic functions, and induced nonlocal
operators in metric measure spaces,

(with Luca Capogna, Josh Kline, Riikka
Korte, and Marie Snipes),

on arXiv server (2022).

**3.** Traces of Newton-Sobolev functions on the visible boundary of domains
in doubling metric measure space supporting a $p$-Poincaré
inequality,

with Sylvester Eriksson-Bique, Ryan Gibara, and Riikka Korte (2023).

**4.** Homogeneous Newton-Sobolev spaces in metric measure spaces and their
Banach space properties,

on arXiv (2023).

**5**. On the Dirichlet-to-Neumann Map for the $p$-Laplacian on
a Metric Measure Space,

with Ryan Gibara (2024), on arXiv.

**6**. On
homogeneous Newton-Sobolev spaces of functions in metric measure spaces of
uniformly locally controlled geometry,

with Ryan Gibara,
Ilmari Kangasniemi (2024).

**7**. Well-posedness of Dirichlet boundary value problems for
reflected fractional $p$-Laplace-type inhomogeneous equations in compact
doubling metric measure spaces,

with Josh Kline, Feng Li
(2024).

**8**. Finite
dimensionality of Besov spaces and potential-theoretic
decomposition of metric spaces,

with Takashi Kumagai, Ryosuke
Shimizu (2024).

**____________________________________________________________________**

Lipschitz and lower
semicontinuous functions, a paper by my colleague
Kevin Rogovin, placed on this website with his
permission.

This is a dvi file.

Lecture Notes on flat chains, from the first part of a graduate course given by my
advisor, Juha Heinonen, at
the University of

Michigan in Winter 1999.

Duality
of moduli and quasiconformal mappings in metric
spaces by Rebekah Jones and Panu
Lahti addresses the dual problem

to the one considered in my paper with these two authors, listed
above.

Last updated 11 September 2024.

Send questions, comments to **shanmun "HAT" uc.edu**.

Go to UC Math homepage.