Harmonic Measure

John B. Garnett, UCLA and
Donald E. Marshall, Univ. of Washington
Cambridge University Press, 2005.


Page Line Change
various various The printer (TechBooks) increased the thickness of some lines in some of the figures, but not uniformly, without informing us or Cambridge. Here is a list of the affected figures and a link to the original (correct) versions.
various various The changes to the lines described above appears to have also affected some fraction bars and overlines, which are now dashed or broken. Here is a list of the affected pages: 38, 118, 131, 190, 235, 295, 445, 475, 488.
xv 9b P. Poggi-Corradini (misspelled)
12 12 $\alpha>1$
14 10 $\gamma_{\delta}= \D \cap \partial B(\zeta,\delta)$
21 20 T_j should be S_j
24 3b (,) instead of [,]
25 10b proof of Lemma 2.3
28 4 divide the right side by 2\pi.
29 7b change line to:"follow the proof of Caratheodory's theorem."
31 5 "preceding" should be "following"
35 6 The upper limit in the integral should be $2\pi$
36 prob. 24(a) arctan should be arcsin
38 2b,4b,7b solid bar on $\Omega$
42 2 blackboard bold D (unit disc) instead of D
42 6 under a one-to-one and analytic map $\psi$ defined on a neighborhood of $(-1,1)$.
43 2b "(g" should be "g" (remove extra parenthesis)
44 5b \Gamma should be \gamma (lower case)
46 7b, 6b In a neighborhood $N$ of a point $\zeta_0\in \partial \Omega$, the function $g$ is the real part of a conformal map $\varphi=g+i \widetilde{g}$ and
46 5b, 3b 1/(2\pi) missing on two integrals
46 3b second integral is over N intersected with the boundary of Omega.
48 9 replace "open arc" with "proper open arc"
49 3 replace "Theorem I.1.3" with "Theorem I.2.1"
50 11b a closed parenthesis ) is missing
51 15 The second < should be \le, less than or equal to
52 1 add to the end of the sentence: "and $b_0=0$."
54 11 the lower limit of integration should be $-\pi$.
58 5 "us help" should be "help us"
60 1 $\varphi^'$ should be $\psi^'$
62 5 multiply the integral on the right by $|\varphi'(0)|^{\lambda}$
71 1 Worse yet, there exists
74 3 $a \notin \overline{\Omega}$
76 10-11 the interval [-2,2] should be the complement of the interval [-2,2].
82 7b Assume R=L instead.
83 2 delete second occurence of ``$E_n = \C \setminus \Omega_n$"
89 4b The first integral takes place on the boundary of $\Omega_n$.
95 8 on the behavior
104 4 $\Om \cap E$ should be $\Om\setminus E$
116 5 2^k should be 2^n.
116 16-18 commas at end, not periods.
117 6,7 $\Omega_n$ should be $\Omega$
117 6,10,11,13 s_n should be replaced by r_{n-2} (7 times)
117 9 B_n should be B_{n-2}
117 13 s_n should be replaced by r{n-2} (two places); r_{n} in the numerator should be replaced by r_{n+1} and |z|\le r_n should be |z|\le r_{n+1}.
117 17 s_0 should be r_0
118 4 s_{2n} should be r_{2n-2}
118 Figure III.7 The broken overline on E should be solid.
118 4b The broken overline on E should be solid (three places).
118 1b,3b fraction lines should be solid (6 places).
124 6,7 F^+=F\cap {Im z \ge 0}, F^-=F\cap {Im z <0}
124 15 z should be replaced by w
124 2b z^*=-|Re z|+i Im z
125 14 Extra | before .Then
129 1b III.22 should be III.24
131 (1.4),(1.6) fraction lines should be solid.
133 3b Fuglede, B., Acta Math. 98(1957), 171-219 proved that for any curve family $\Gamma$ we can remove a subset $\Gamma_0$ without affecting the extremal length and find an extremal metric for the family $\Gamma\setminus\Gamma_0$. The removed family $\Gamma_0$ has infinite extremal length (zero modulus). An example is a rectangle with vertical ends E and F. If we add a point z_0 lying on the top edge of the rectangle to E then the extremal distance from E to F is unchanged but the constant metric is no longer extremal since there are short curves from z_0 to F. But if we discard the curves from z_0 to F from the curve family of curves from E to F then the constant metric is extremal, and the length is unchanged. The discarded family is so small that it has infinite extremal length. See Exercise IV.9.
137 9 (2.3) should be (3.3)
150 4b $\subset \Gamma$ should be $\in \Gamma$
152 17 Take $z_0\in A$ such that $dist(z_0,B)=dist(A,B)$.
155 2 n-1 should be n-2
155 4 1/$\pi$ should be 2/$\pi$ in the formula for $\omega$
155 7 n=3 should be n=4
171 16-17 line break too big
172 6b-7b line break too big
195 2b Exercise III.22 should be Exercise III.24(b)
206 13b Lindel\"of's theorem refers to Exercise II.3(d)
206 6b Replace Borel with $\omega$-measurable.
207 2 We should have said there is a subset $A_0$ of $A$ which is an $F_{\sigma}$, hence Borel, with $\omega(A_0)=1$. Thus $A$ is $\omega$-measurable.
Proof: By Egoroff's theorem and Lemma 3.1, there are closed $K_n$ such that $|\b D\setminus K_n|<1/n$ and such that $\varphi$ is continuous on the compact set $\cup_{K_n}\overline{\Gamma_{\pi/2}(e^{i\theta})}.$ Hence $\varphi(K_n)$ is compact and $\omega(\cup_n \varphi (K_n)) =1.$ Thus A is $\omega$-measurable and (3.2) holds. Continue as in text, except delete the word "also" in the last line of the proof.
A proof that A is Borel can be found in S. Mazurkiewicz, "Uber erreichbare Punkte", Fund. Math 26 (1936), 150-155.
209 Figure VI.3 The region U is outlined by a thick curve. T_n(w) should have thinner boundary (it is not part of the boundary of U).
210 Figure VI.4 There are dashed and dotted lines, which are difficult to discern because the line thicknesses were changed.
212 12-14 Replace the sentence: "The exterior domain ... countable subset of \Gamma." with "If \Gamma' is a copy of \Gamma rotated by 90 degrees and scaled by 1/\sqrt{3} then six copies of \Gamma' in a hexagonal pattern fit exactly along the outer boundary of \Gamma."
212 3b $E \subset \partial \D$
212 11b "a of measure 0" should be "a set of harmonic measure 0"
220 10b curve, then at
221 1,2,4,9,11,13 $\varphi(G_j)$ should be $\varphi_j(G_j)$ (16 occurances).
221 table, line 1 $\varphi_1(G_1)\cap\varphi_2(G_2) \subset T_n(\Gamma)$.
235 several overline and fraction bars should be solid (6 places).
239 5-11,
for a new version of this see new page 239
242 10b orientation preserving ACL homeomorphism
244 13b the "only if" statement is correct. The "if" statement is correct for quasicircles, but false in general. Here is a counter-example due to N. Karamanlis (11/2014).
244 9b Kahane's measure from Example 2.6 is *not* a doubling measure, as asserted, because it assigns zero mass to some non-empty open intervals. However, if the construction in Example 2.6 is changed so that the densities satisfy d_{n+1,k} = d_{n,j}/2 when k =4j or 4j+3 and d_{n+1,k} = 3d_{n,j}/2 when k = 4j+1 or 4j + 2 then the limit measure is doubling and singular.
244 2b Here is a corrected version, due to N. Karamanlis, of the first half of the proof of the Jerison-Kenig Theorem (Theorem 3.5). The second half of the proof in the text is correct because the "iff" statement on line 13b is correct for quasicircles. For another treatment of the Jerison-Kenig Theorem, which depends on a result of Kryzyz, see Broch, Hag, and Junge, [2011] "A note on the harmonic measure doubling condition", Conf. Geo. Dyn. (15), 1-6.
252 5b,7b remove 2\pi.
269 8-9 such that if $g(z)$ is a Bloch function on $\D$, then for a.e. $\zeta \in \b\D$
274 5 (C) should be (c)
276 5b & 4b B(\zeta_1,diam(\gamma)) should be B(w_1,diam(\gamma))
277 Figure VIII.1 No subscript on $D$ or on $\widetilde D$.
282 15 remove black box at end of line
285 11 $\chi_j(\zeta)$ should be $\chi_j(z)$
285 12 and all $z\in A_1(\zeta)\cap\partial\Om$
295 2b fraction bars should be solid.
300 3b $\Omega_J$ should be $\Omega_n^J$
310 1 "bounded universal integral means spectrum" should be boldface.
311 10b delete the incorrect sentence: "It is clear that..."
316 8 liminf not limsup
323 12 change "By the central limit...distribution," to "By the elementary theory of large deviations
(R. Durrett, {\it Probability: Theory and Examples,} 2nd Edition (1996) pp. 70 -76),"
337 13 and 19 replace "maximal" with "largest"
338 18 replace (3.7) with (3.7a)
338 20 number this displayed formula (3.7b)
338 7b "Q^0" should be "Q" in two places.
339 5 replace (3.7) with (3.7b)
339 14 "=" is missing between \alpha(B) and Max. (the first = sign emphasizes that $\alpha$ depends on B)
340 4 replace (3.7) with (3.7a)
340 6 replace $C^{1+\epsilon}$ with $C R^{1+\epsilon}$
331-340 See additional corrections to Section IX.3. For a more complete exposition, see Cuf\'i, Tolsa, Verdera, About the Jones-Wolff Theorem on the Hausdorff dimension of harmonic measure, Catalan Institution for Research and Advanced Studies, 2018, preprint. See: https://arxiv.org/abs/1809.08026
343 3&7 Batakas should be Batakis
367 22 For more details see page 367 correction.
373 8b ||F'|| should be ||F||.
393 6b should be =
393 4b-3b The integrals take place on the unit disk
395 7 (g')^2/2 should be (g')^2 by (1.8) and (6.1).
396 2 (1-|z|^2)^2
396 9b (1-|z|^2)^3 should be (1-|z|^2)
397-411 all for a new version of sections 7 and 8, see new sectX7-X8
399 5b (r.e.) double dots - should be just one.
400 1b (r.e.) 1 should be 1/2
429 11 in the def of l_k(E), inf should be sup.
435 8b Theorem A.1 (mislabelled)
436 9 Corollary A.2 (mislabelled)
436-439 various P_z should not be multiplied by 1/(2\pi) since it is included in the definition.
440 14 "almost everywhere" should be "everywhere".
446 19b-16b Replace the last sentence "On $\Omega_d$... solved on $\Omega_d$." with:
Set $\sigma=\partial \Omega_2\setminus \partial \Omega$ and $\sigma^\prime=\tau(\sigma)$ (as subsets of $\Omega_d$). Then $\Omega_2$ and $\Omega_2^\prime=\tau(\Omega_2)=\Omega_d\setminus \overline{\Omega_1}$ are conformally equivalent to finitely connected Jordan domains, so that by Theorem II.1.1, we can solve the Dirichlet problem on $\Omega_2$ and on $\Omega_2^\prime$. Moreover $\Omega_d=\Omega_2 \cup \Omega_2^\prime$ and $\sigma\cap \sigma^\prime$ is a finite subset of the end points $E$ of $J$. So by the Schwarz Alternating Method from the proof of Theorem II.1.1, given a continuous function $f$ on $\partial \Omega_d$, we can find $u$ harmonic on $\Omega_d$, continuous on $\Omega_d\cup(\partial \Omega_d\setminus E)$ and equal to $f$ on $\partial \Omega_d\setminus E$. Note that by Corollary I.2.5 and Theorem I.1.3 applied to the conformal image of a small neighborhood in $\Omega_d$ of $p\in E$, $u$ also extends to be continuous at each $p\in E$.
446 6b See also Tsuji[1959], p. 31
450 17b $\Omega$ should be $\D$
465 5b contradicts (D.16)
466 3 Corollary D.2 implies
508 1 and 2 should be for a.a. $\theta$
534 15-26 Betsakos, Betsakos and Solynin references should be before Beurling.
541 23 Iberoamericana
541 10b add page numbers 1-8.
544 18 On V.I. Smirnov domains
549 1 The journal name should be in italics
555-558 various add pages: Batakis 343, Durrett 323, Tsuji 446, Lindel\"of 206, Ostrowski 239.
560 21b Tn(\Gamma) page number reference should be 211 (first occurance).
571 20-21;col2 should be a space between Zygmund's theorem and Oksendal conjecture

Thanks to all who have pointed out errors, including:
Enrique Alvarado, Florian Bauer, Dimitrios Betsakos, Ryan Card, Bob Carlson, Fausto Di Biase, Maria Jose Gonzalez, Kari Hag, Jon Handy, Sa'ar Hersonsky, Juha Heinonen, Steffen Junge, Nikolaos Karamanlis, William Meyerson, Olena Ostapyuk, Jordi Pau, Yuval Peres, Fernando Perez-Gonzalez, Pietro Poggi-Corradini, Istvan Prause, Jouni Rättya, Bill Ross, Pablo Shmerkin, Carl Sundberg, Jason Tedor, Huy Tran, Ignacio Uriarte-tuero, and Joan Verdera

Please send corrections to
Don Marshall
Subject: Harmonic Measure