| 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 |
| 98 | 6-8 | Delete the sentence ``By definitions of regular point and capacity...still diverges'' |
| 98 | 13, 18 | Replace $n\log 2/\gamma_n$ with $(n-1)\log 2/\gamma_n$ in (7.4) and (7.5) |
| 98 | 16 | Replace the sentence "By our smoothness assumption on..." with "Then $W_j$ is harmonic and positive on $\Omega$ and continuous at $0 \notin A_n$, $N_j \le n < N_{j+1}$. |
| 99 | 7-8 | Similar to the preceeding changes on page 98, we can remove the sentence ``Again we may replace....still converges.'' In this case, the function $F$ on line 13 should be the solution to the Dirichlet problem on $\Omega$ with the stated boundary conditions. Because $\zeta=0$ is a regular point, $F$ is still continuous at $0$. |
| 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, Fig.VII.5 | 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 |