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Book Review

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Book Information:

Authors: John B. Garnett and Donald E. Marshall
Title: Harmonic measure
Additional book information: Cambridge University Press, 2005, xv + 571 pp., ISBN 0-521-47018-8, US$110$

Christopher J. Bishop, Some questions concerning harmonic measure, Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990) IMA Vol. Math. Appl., vol. 42, Springer, New York, 1992, pp. 89–97. MR 1155854, DOI 10.1007/978-1-4612-2898-1_{7}

Christopher J. Bishop, Quasiconformal Lipschitz maps, Sullivan’s convex hull theorem and Brennan’s conjecture, Ark. Mat. 40 (2002), no. 1, 1–26. MR 1948883, DOI 10.1007/BF02384499

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 and arclength, Ann. of Math. (2) 132 (1990), no. 3, 511–547. MR 1078268, DOI 10.2307/1971428

J. Bourgain, On the Hausdorff dimension of harmonic measure in higher dimension, Invent. Math. 87 (1987), no. 3, 477–483. MR 874032, DOI 10.1007/BF01389238

Lennart Carleson, On the distortion of sets on a Jordan curve under conformal mapping, Duke Math. J. 40 (1973), 547–559. MR 330430

Lennart Carleson, On the support of harmonic measure for sets of Cantor type, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 113–123. MR 802473, DOI 10.5186/aasfm.1985.1014

Sunhi Choi, The lower density conjecture for harmonic measure, J. Anal. Math. 93 (2004), 237–269. MR 2110330, DOI 10.1007/BF02789309

José L. Fernández, Juha Heinonen, and Olli Martio, Quasilines and conformal mappings, J. Analyse Math. 52 (1989), 117–132. MR 981499, DOI 10.1007/BF02820475

Peter W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 24–68. MR 1013815, DOI 10.1007/BFb0086793

Peter W. Jones and Thomas H. Wolff, Hausdorff dimension of harmonic measures in the plane, Acta Math. 161 (1988), no. 1-2, 131–144. MR 962097, DOI 10.1007/BF02392296

Robert Kaufman and Jang Mei Wu, Distortion of the boundary under conformal mapping, Michigan Math. J. 29 (1982), no. 3, 267–280. MR 674280

M. Lavrent′ev, Boundary problems in the theory of univalent functions, Amer. Math. Soc. Transl. (2) 32 (1963), 1–35. MR 0155970

John L. Lewis, Gregory C. Verchota, and Andrew L. Vogel, Wolff snowflakes, Pacific J. Math. 218 (2005), no. 1, 139–166. MR 2224593, DOI 10.2140/pjm.2005.218.139

N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), no. 2, 369–384. MR 794117, DOI 10.1112/plms/s3-51.2.369

J. E. McMillan, Boundary behavior of a conformal mapping, Acta Math. 123 (1969), 43–67. MR 257330, DOI 10.1007/BF02392384

John E. McMillan and George Piranian, Compression and expansion of boundary sets, Duke Math. J. 40 (1973), 599–605. MR 318492

Ch. Pommerenke, On conformal mapping and linear measure, J. Analyse Math. 46 (1986), 231–238. MR 861701, DOI 10.1007/BF02796587

Feliks Przytycki, Mariusz Urbański, and Anna Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I, Ann. of Math. (2) 130 (1989), no. 1, 1–40. MR 1005606, DOI 10.2307/1971475

[Riesz and Riesz(1916)]

F. Riesz and M. Riesz.
Über die ranwerte einer analytischen Funcktionen.
Quartriéme Congress des Math. Scand., pages 27-44, 1916.

Byron L. Walden, $L^p$-integrability of derivatives of Riemann mappings on Ahlfors-David regular curves, J. Anal. Math. 63 (1994), 231–253. MR 1269221, DOI 10.1007/BF03008425

Thomas H. Wolff, Plane harmonic measures live on sets of $\sigma$-finite length, Ark. Mat. 31 (1993), no. 1, 137–172. MR 1230270, DOI 10.1007/BF02559503

Thomas H. Wolff, Counterexamples with harmonic gradients in $\textbf {R}^3$, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 321–384. MR 1315554

Review Information:

Reviewer: Christopher J. Bishop
Affiliation: SUNY Stony Brook
Email: bishop@math.sunysb.edu