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.
38. On the Dirichlet-to-Neumann Map for the $p$-Laplacian on a Metric Measure Space,
with Ryan Gibara (2024), on arXiv, to appear in Contemporary Math.
39. Neumann problems for p-harmonic functions, and induced nonlocal operators in metric measure spaces,
(with Luca Capogna, Josh Kline, Riikka Korte, and Marie Snipes),
to appear in Amer. J. Math. (Accepted 2024 October, available now on arXiv server (2022)).
——————————————————————————
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:
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. 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).
3. Homogeneous Newton-Sobolev spaces in metric measure spaces and their Banach space properties,
on arXiv (2023).
4. On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry,
with Ryan Gibara, Ilmari Kangasniemi (2024).
5. 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).
6. Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces,
with Takashi Kumagai, Ryosuke Shimizu (2024).
7. Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces,
with Luca Capogna, Ryan Gibara, Riikka Korte (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 26 October 2024.
Send questions, comments to shanmun "HAT" uc.edu.
Go to UC Math homepage.