Book Review
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MathSciNet review:
2139304
Full text of review:
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Book Information:
Authors:
Luca Capogna,
Carlos E. Kenig and
Loredana Lanzani
Title:
Harmonic measure: Geometric and analytic points of view
Additional book information:
University Lecture Series, vol. 35,
Amer. Math. Soc., Providence, RI,
2005,
x+155 pp.,
ISBN 978-0-8218-3920-1,
US$28.00$
H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144. MR 618549
Luca Capogna, Carlos E. Kenig, and Loredana Lanzani, Harmonic measure, University Lecture Series, vol. 35, American Mathematical Society, Providence, RI, 2005. Geometric and analytic points of view. MR 2139304, DOI 10.1090/ulect/035
C. J. Bishop, L. Carleson, J. B. Garnett, and P. W. Jones, Harmonic measures supported on curves, Pacific J. Math. 138 (1989), no. 2, 233–236. MR 996199
Christopher J. Bishop and Peter W. Jones, Harmonic measure, $L^2$ estimates and the Schwarzian derivative, J. Anal. Math. 62 (1994), 77–113. MR 1269200, DOI 10.1007/BF02835949
G. David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals, Indiana Univ. Math. J. 39 (1990), no. 3, 831–845. MR 1078740, DOI 10.1512/iumj.1990.39.39040
Guy David, Carlos Kenig, and Tatiana Toro, Asymptotically optimally doubling measures and Reifenberg flat sets with vanishing constant, Comm. Pure Appl. Math. 54 (2001), no. 4, 385–449. MR 1808649, DOI 10.1002/1097-0312(200104)54:4<385::AID-CPA1>3.0.CO;2-M
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005. MR 2150803, DOI 10.1017/CBO9780511546617
David Jerison, Regularity of the Poisson kernel and free boundary problems, Colloq. Math. 60/61 (1990), no. 2, 547–568. MR 1096396, DOI 10.4064/cm-60-61-2-547-568
Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
Carlos E. Kenig and Tatiana Toro, Harmonic measure on locally flat domains, Duke Math. J. 87 (1997), no. 3, 509–551. MR 1446617, DOI 10.1215/S0012-7094-97-08717-2
Carlos E. Kenig and Tatiana Toro, Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math. (2) 150 (1999), no. 2, 369–454. MR 1726699, DOI 10.2307/121086
Carlos E. Kenig and Tatiana Toro, Poisson kernel characterization of Reifenberg flat chord arc domains, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 3, 323–401 (English, with English and French summaries). MR 1977823, DOI 10.1016/S0012-9593(03)00012-0
Carlos Kenig and Tatiana Toro, Free boundary regularity below the continuous threshold: 2-phase problems, J. Reine Angew. Math. 596 (2006), 1–44. MR 2254803, DOI 10.1515/CRELLE.2006.050
M. Lavrent′ev, Boundary problems in the theory of univalent functions, Amer. Math. Soc. Transl. (2) 32 (1963), 1–35. MR 0155970
Ch. Pommerenke, On univalent functions, Bloch functions and VMOA, Math. Ann. 236 (1978), no. 3, 199–208. MR 492206, DOI 10.1007/BF01351365
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. MR 114145, DOI 10.1007/BF02547186
Stephen Semmes, Chord-arc surfaces with small constant. I, Adv. Math. 85 (1991), no. 2, 198–223. MR 1093006, DOI 10.1016/0001-8708(91)90056-D
Stephen Semmes, Analysis vs. geometry on a class of rectifiable hypersurfaces in $\textbf {R}^n$, Indiana Univ. Math. J. 39 (1990), no. 4, 1005–1035. MR 1087183, DOI 10.1512/iumj.1990.39.39048
- [AC]
- Alt, H. W., and Caffarelli, L., Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144. MR 0618549
- [B]
- Capogna, Luca; Kenig, Carlos E.; and Lanzani, Loredana, Harmonic Measure: Geometric and Analytic Points of View, University Lecture Series 35, American Mathematical Society, 2005. MR 2139304
- [BCGJ]
- Bishop, C. J.; Carleson, L.; Garnett, J.; and Jones, P. W., Harmonic measures supported on curves, Pac. J. Math., 138 (1989), 233-236. MR 0996199
- [BJ]
- Bishop, C. J., and Jones, P. W., Harmonic measure, estimates and the Schwarzian derivative, J. Anal. Math., 62 (1994), 77-113. MR 1269200
- [DJ]
- David, G., and Jerison, D. S., Lipschitz approximation to hypersurfaces, harmonic measure and singular integrals, Indiana Math. J., 39 (1990), 831-846. MR 1078740
- [DKT]
- David, G.; Kenig, C. E.; and Toro, T., Asymptotically optimally doubling measures and Reifenberg flat sets with vanishing constant, Comm. Pure Appl. Math., 54 (2001), no. 4, 385-449. MR 1808649
- [EG]
- Evans, L. C., and Gariepy, R., Measure theory and fine properties of functions, CRC Press, Boca Raton, FL, 1992. MR 1158660
- [GM]
- Garnett, J. B., and Marshall, D. E., Harmonic measure, Cambridge U. Press, 2005. MR 2150803
- [J]
- Jerison, D., Regularity of the Poisson kernel and free boundary problems, Colloq. Math., 60-61 (1990), 547-567. MR 1096396
- [JK]
- Jerison, D., and Kenig, C. E., Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math., 46 (1982), 90-147. MR 0676987
- [KT1]
- Kenig, C. E., and Toro, T., Harmonic measure on locally flat domains, Duke Math. J., 87 (1997), 509-551. MR 1446617
- [KT2]
- Kenig, C. E., and Toro, T., Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math., 150 (1999), 369-454. MR 1726699
- [KT3]
- Kenig, C. E., and Toro, T., Poisson kernel characterization of Reifenberg flat chord arc domains, Ann. Sci. École Norm. Sup. (4), 36 (2003), no. 3., 323-401. MR 1977823
- [KT4]
- Kenig, C. E., and Toro, T., Free boundary regularity below the continuous threshold: 2-phase problems, J. Reine Angew. Mat., 596 (2006), 1-44. MR 2254803
- [L]
- Lavrentiev, M., Boundary problems in the theory of univalent functions, Mat. Sb. (N.S.), 1 (1936), 815-844. English transl. in Amer. Math. Soc. Transl. Ser. 2 32, AMS, 1963, 1-35. MR 0155970
- [P1]
- Pommerenke, C., On univalent functions, Bloch functions and VMOA, Math. Ann., 236 (1978), 199-208. MR 0492206
- [P2]
- Pommerenke, C., Boundary behavior of conformal maps, Springer-Verlag, 1992. MR 1217706
- [R]
- Reifenberg, E., Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92. MR 0114145
- [S1]
- Semmes, S., Chord arc surfaces with small constant, I, Adv. Math., 85 (1991), 198-223. MR 1093006
- [S2]
- Semmes, S., Analysis vs. geometry on a class of rectifiable hypersurfaces in , Indiana Univ. Math. J., 39 (1990), 1005-1035. MR 1087183
Review Information:
Reviewer:
John B. Garnett
Affiliation:
The University of California, Los Angeles
Email:
jbg@math.ucla.edu
Journal:
Bull. Amer. Math. Soc.
45 (2008), 321-327
DOI:
https://doi.org/10.1090/S0273-0979-07-01164-0
Published electronically:
July 11, 2007
Review copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.