Appendix A. The Dirichlet Problem and Harmonic Measures

In this appendix we prove Wiener's theorem on regular boundary points for Green functions and Dirichlet problems. In particular, it will follow that these two con• cepts are identical.

A.I Regularity with Respect to Green Functions

Let Gee be a domain such that aG is of positive capacity and, for a E G, let gG(z, a) be the Green function of G with pole at a. Recall from Sections 1.4 and 11.4 that gG(z, a) is defined as the unique function on G satisfying the following properties:

(i) ga(z, a) is nonnegative and harmonic in G \ {a} and bounded as z stays away from a, 1 (ii) gG(z, a) - log -- is bounded in a neighborhood of a, Iz -al (iii) lim gG(z, a) = 0 for quasi-every x E aG z~x,ZEG with (ii) replaced by

(ii)' ga(z, a) - log Izl is bounded in a neighborhood of 00 when a = 00. We call a point x E aG on the boundary of G a regular point (with respect to the Green function gG(z, a» if

lim gG(z, a) = O. z~x, ZEG

Soon we shall see that this notion is independent of the choice of a; therefore, we shall just speak of regular boundary points. Note also that, by definition (which, however, depends on the existence theorem for Green functions), quasi-every point on the boundary of G is a regular point.

Theorem 1.1 (Wiener's Theorem). Let 0 < A < 1 and set 450 Appendix A. The Dirichlet Problem and Harmonic Measures

Then x E aG, x i= 00, is a regular boundary point of G if and only if

00 n (1.1) ?; 10g(l/cap(An(x» = 00. In particular, regularity is a local property.

Proof. First we consider the case when G is an unbounded domain and a = 00. Set E := C \ G and 1 V(E):= log--. cap(E) It is immediate that the condition (1.1) does not change if An (x) is defined as

(note that equality is allowed at both places on the right); hence in what follows we can work with this definition of An (x), which is more convenient than the original one, for then An (x) is compact. We may assume without loss of generality that x = O. We start with the proof of the sufficiency of condition (1.1). In view of the representation

(1.2)

(see (1.4.8», which we use to extend gG to the whole plane, the inequality

UILE (z) ::::; V (E), (1.3) and the lower semi-continuity of UILE, it suffices to show that (1.1) implies

UILE(O) = V(E). (1.4)

Assume to the contrary that (1.4) is not true, i.e.

fJ := V(E) - UILE(O) > O.

Then, in view of Theorem 1.4.1, we will have

V (E*) - UILE' (0) ~ fJ (1.5) for every compact subset E* of E of positive capacity. One can easily verify from the discussion at the end of the present proof that A can be replaced by Ak with any fixed k, so we can assume A as small as we like. In particular, we can assume that A < 1/4 is so small that log(l/(l - A» < fJ/2 is satisfied. If 00 n ?; log(l/cap(A2n(0» = 00, then we set En = A2n(0); otherwise, we choose En = A2n-l(0). In any case the sets En are disjoint and satisfy A.l Regularity with Respect to Green Functions 451

00 n " -- - 00 (1.6) ~ V(En) - .

Finally, we choose E* = {OJ U (Un~noEn) with an no so large that we have E* C {z I Izl :s 1/2}. Then E* is compact and E* C E. Now let z E Ei and t E Ej with j =1= i and cap(Ei) > O. If j < i then Iz - tl ::: (1 - AWl, while for j > i we have Iz - tl ::: Itl. Thus, in any case 1 1 1 log - + log -- > log --. Itl 1 - A - Iz - tl

Let us integrate this inequality with respect to d/-LE*(t) on E* \Ei . Then, observing that both z and t lie in the disk DI/2(0), the left-hand side will be at most

UIlE'(O) + log 1 ~ A < V(E*) - f3 + i, while the right-hand side is

UIlE' (z) - [ log _1-d/-LE* (t). lEi Iz-tl Since UIlE' (z) = V (E*)

for quasi-every z E Ei , we obtain

[ log _1-d /-LE.(t) ::: ~, for q.e. z E E i • lEi Iz-tl 2

Let Vi be the restriction of /-LE' to Ei . Then the preceding inequality takes the form UVi(Z) > ~ for q.e. z E E . - 2' i Thus, UllEi 1 f f3 /-LE·(E i ) = vi(Ei) = f V(Ei) dVi = V(Ei ) UVid/-LEi::: 2V(Ei )· But this implies

UIlE' (0) = log ~ d/-LE* (t) f It I

l"i-l ::: 2,8 log - ~ -- = 00, A i V(Ei )

which is a contradiction by (1.5) and (l.6). This contradiction was the result of the assumption that (Ll) holds but (1.4) does not; hence (Ll) implies regularity. 452 Appendix A. The Dirichlet Problem and Harmonic Measures

Now we turn to the necessity, and let us assume that (1.1) does not hold. Then there are arbitrary small r's such that the circle {z I Izl = r} does not intersect E := C \ G (see Lemma 1.2.1). For such an r, let K\ be the intersection of E with the disk Dr(O) := {z Ilzl ::: r}, and set K2 := E \ K\. Then both K\ and K2 are compact, and we can choose a bounded closed neighborhood K; of K2 such that also K; is disjoint from K \ . On applying Theorem 1.4.1 as before, it is enough to prove the irregularity of x = °with respect to any domain G\ ~ G containing x on its boundary; thus we can assume without loss of generality that infinitely many (or all if we like) An (0) are of positive capacity. This implies, in particular, that K\ has positive capacity. Let ko be the smallest integer with the property that for k > ko the sets Ak := Ak(O) are disjoint from K2. In what follows we shall assume that Ako is entirely contained in the disk Dr(O); if this is not the case, then in the following discussion we have to split Ako into Ako n Dr (0) and Ako \ Dr (0) and make the necessary changes. Finally, without loss of generality we may assume r < 1/2 is so small that 00 k 1

~ V(Ak ) < 210g(1/).) is satisfied. Since for cap(Ak) > 0, k ::: ko, we have with v := ILK) IAk

( UiLAk 1 f V(K\) V ILK) (Ak) = JAk V (Ak) dILK) = V (Ak) U dILAk ::: V (Ak) ,

it follows that

00 ( 1 ) look 1 UiLK) (0) ::: L log k ILK) (Ak) ::: V(K\) log - L -- < - V(Kd, k=ko ). ). k=ko V (Ak) 2

and this means that gC\K/O, 00) > 0. Using the mean value inequality for the subharmonic function gC\K) (z, 00) on circles {z I Izl = r} that do not intersect E (as we have seen, there are such circles with arbitrary small r > 0) we can conclude

lim sup gC\K) (z, 00) > 0. (1.7) Z--->O,ZEG Now let M be larger than the maximum of gC\K) on the boundary of K; and m the minimum of gG on the same set. Since K; is a neighborhood of K2 disjoint from K\, we have the relation aK; c G; therefore m > ° by the minimum principle. Replacing M by a larger number if necessary, we can assume that M 1m::: 1. Let us now apply the generalized minimum principle (Theorem 1.2.4) to the function (Mlm)gG - gC\K) on the set G* := C \ (K\ U KD. This is a superharmonic function in G* which is bounded from below (around infinity it behaves like «M 1m) - 1) log Izl) and which has nonnegative boundary limits A.I Regularity with Respect to Green Functions 453

quasi-everywhere on 3G* = 3K1 U3Ki (recall also that on 3Ki both gC\K[ and gG are continuous). Therefore we can conclude that (M jm)gG - gC\K[ is nonnegative on G*, and we have in view of (l. 7)

limsup gc(z,oo) 2:: lim sup m gC\K[(Z, oo) > 0, Z-+O.ZEG Z-+O.ZEG M which verifies that 0 is not a regular boundary point of G. Thus, the proof of Wiener's theorem is complete in the case when a = 00.

Now let us tum to the case of an arbitrary domain G with cap(3G) > O. The mapping z -+ z' := 1/(z - a) maps G into an unbounded domain G' and gG(z, a) is transformed to gG'(z', (0). Thus, x E 3G is a regular boundary point with respect to G if and only if x' = I j (x - a) is a regular boundary point with respect to G'. Therefore, we only have to show that the Wiener condition (l.I) is also preserved under this mapping. Let D be a disk around x not containing a on its boundary, and let D' be its image. Then on D the mapping z -+ z' is a constant times a nonexpansive mapping (i.e. which can only shrink distances), and on D' the same is true for its inverse. We have mentioned after the proof of Lemma I.2.1 that a nonexpansive mapping does not increase the capacity; therefore there is a constant M such that for every subset K of D we have 1 Mcap(K) ::: cap(K') ::: Mcap(K), where K' denotes the image of K. In particular, for every sufficiently large n we have (l.8)

The sets An (x)' are not the sets Am (x') for the point x', but it immediately follows from what we have just said about the mapping z -+ z' that there is an L such that for every large n

n+L An(x') S; U An(x)', j=n-L and conversely n+L An(x)' S; U An(x'). (1.9) j=n-L The first containment immediately implies

< log(l jcap(An (x'))) 10g(ljcap(Uj:~~z An(x)'»

j=n+L 1 < (1.1 0) j~L 10g(1jcap(An (x)'»' 454 Appendix A. The Dirichlet Problem and Harmonic Measures where in the last step we also applied the inequality Theorem 1.6.2(e) (for large n all of the above sets are contained in a disk of radius 1/2, so we can choose M = 1 in that inequality). On multiplying (1.10) by n, summing over n and making use of (1.8) we can see that

implies

~ 10g(1/ca;(An (X))) = 00. The converse can be proved in the same way from (1.9), and so the invariance of the Wiener condition under the mapping z -+ z' has been verified. 0

A.2 Regularity with Respect to Dirichlet Problems

Next we shall discuss regular boundary points with respect to the Dirichlet prob• lem. First we recall the definition of the Perron-Wiener-Brelot solution of the Dirichlet problem from Section 1.2. Consider a domain Gee such that C \ G has positive capacity, and suppose that I is a bounded Borel measurable function defined on aGo The upper and lower classes of functions corresponding to I and G are defined as

1ij,G := {g I g superharmonic and bounded below on G,

liminf g(z) ::: I(x) for x E aG} z-+x, ZEG and 1iiG := {g I g subharmonic and bounded above on G, limsupg(z) ~ I(x) for x E aG}, z-+x, ZEG and the upper and lower solutions of the Dirichlet problem for the boundary function I are given by -G . H f (z):= mf g(z), Z E G, gE'H;.,G

and H7 (z):= sup g(z), Z E G. gE'HjG

Always H~ ~ H7 and if H~ == H7, then this function H1 is called the Perron• Wiener-Brelot solution of the Dirichlet problem on G for the boundary function I. In what follows we shall assume that aG is a compact subset of C. This can always be achieved by a fractional linear transformation. A.2 Regularity with Respect to Dirichlet Problems 455

First we show that the upper and lower solutions are harmonic functions, and if I is a continuous function, then the Perron-Wiener-Brelot solution exists, is harmonic in G, and has boundary limit I(x) for quasi-every x E aGo We shall do this in several steps. I. Let II I :5 M on aG. Then in the upper and lower classes we can restrict ourselves to functions that have values in the interval [-M, M] (i.e. the lower and upper limits do not change if we only take them for such functions). In fact, if, for example, g is an upper function, then min(M, g) is again an upper function, and by the minimum principle it is at least as large as min I ~ - M in G. Thus, in what follows we shall restrict our attention to functions with values in the interval [-M, M].

II. Let ..1 C G be a closed finite subdisk of G. For every upper function g E 1ij.G we can construct another g* E 1ij,G in such a way that g* :5 g, g* is harmonic in ..1, and the mapping g -+ g* is monotone. In fact, the restriction of g to the boundary of ..1 is a lower semi-continuous function, so there is an increasing sequence {hml of continuous functions on aL1 that tends to g on aL1. We can extend each hm into ..1 harmonically (Corollary 0.4.4) and we denote this extension also by hm. Now in ..1 the sequence {hml is a bounded and increasing sequence of harmonic functions; hence its limit h is also harmonic in ..1 by Harnack's principle (Theorem 0.4.10). Now let g* coincide with g outside ..1, and with h inside ..1. Since g is superharmonic, we have hm(z) :5 g(z) for all m and z E ..1, so the inequality g* :5 g is clear. In a similar fashion it easily follows from the minimum principle that if gl :5 g2, then gj :5 g~. Thus it remains to prove that each g* is in the upper class 1ij,G. Since g* coincides with g outside ..1, only the superharmonicity of g* has to be proved. The lower semi-continuity of g* on ..1 follows from the fact that there g* = h is the limit of an increasing sequence of continuous functions. Since g* = g outside ..1, and g is lower semi-continuous, the lower semi-continuity of g* on the whole domain G follows. Thus, to be able to conclude the superharmonicity of g*, it remains to show that for every Zo E G there is an ro > 0 such that for all 0 < r < ro we have

-21 17C g*(zo + reit)dt :5 g*(zo) Jr -7C

(see Remark 0.5.3). For Zo E ..1 obviously there is such an ro because of the mean value property of g* in ..1, while for Zo fj. ..1 any ro will be suitable for which the disk Dro (zo) lies in G because the corresponding inequality holds for g and g* :5 g but g*(zo) = g(Zo). III. We claim that both the upper and lower solutions to the Dirichlet problem are harmonic functions in G. Consider, for example, the upper solution. It is enough to prove the harmonicity on every finite disk ..1 C G with ..1 c G, for then H~ (z) will also be harmonic at infinity in case G is an unbounded domain (indeed, the upper solution is bounded by M and so Corollary 0.3.5 applies). Let 456 Appendix A. The Dirichlet Problem and Harmonic Measures us fix a Zo E ..1, and let gn E 1ij.G be functions such that gn (zo) -+ H~ (zo) as n -+ 00. Since the minimum of two upper functions is again an upper function, we can assume that gn+1 ::::; gn' Consider the corresponding gZ constructed in the preceding paragraph. We also have g~+1 ::::; g~, so these g~'s converge to a g* that is harmonic in ..1 by Harnack's principle. Since g* ::::; gn for all n, we clearly have g*(zo) = H~ (zo). We claim that this equality holds at every other point of ..1, which yields the harmonicity in ..1 of the upper solution H~ (zo). In fact, let Zl be another point of ..1, and choose upper functions hn such that {hn(zd} converges -G to H f (Zl). As before, we can assume hn+1 ::::; hn, and even that hn ::::; gn' Then {h~} converges to a function h* ::::; g* which is harmonic in ..1. But we must have h*(zo) = H~ (zo) = g*(zo); therefore, the two functions h* and g* coincide on L1 by the minimum principle. Thus, g*(zl) = h*(zl) = H~ (Zl) as we claimed. IV. Next we show that -G lim sup Hf (z) ::::; f(x) (2.1) z..... x. ZEG and liminf Hl(z) :::: f(x) (2.2) z ..... x. ZEG at every point x E aG that is a regular point for the Green functions of G and at which f is continuous. In particular, these relations hold quasi-everywhere when f is continuous. Let us consider the first relation at a regular point x E BG. Since G is a domain, there is a broken line in G that intersects any neighborhood of x. Thus, if r > 0 is sufficiently small, and K = (C \ G) n Dr(x), then x will lie on the outer boundary of the compact set K. Since regularity is a local property, we get that x is a regular boundary point with respect to Green functions in C \ K, i.e. the Green function gC\K (z, 00) is continuous and vanishes at x. Now let e2r be the maximum of the differences If(x) - f(y)1 for all y E aG n D2r(X), Y the minimum of gC\K(Z, 00) for Iz -xl:::: 2r, and M := SUPYEJG If(y)1 the maximum of If I. Then y > 0 and it immediately follows that M g(z) := f(x) + e2r + - gC\K(Z, 00), Z E G, y is an upper function for f. Hence

lim sup H~ (z) ::::; lim sup g(z) ::::; f(x) + e2r, z..... x. ZEG z..... x. ZEG and since r > 0 was arbitrary, and e2r -+ 0 as r -+ 0 by the assumed continuity f at x, the relation (2.1) follows. V. Finally, we show that if f is a continuous function, then the upper and lower solutions coincide, and their boundary limits agree with f quasi-everywhere. In fact, by what we have just proved, Hl (z) - H~ (z) is a nonpositive harmonic A.2 Regularity with Respect to Dirichlet Problems 457

function in G which has boundary limit zero at quasi-every x E aGo Thus, this function is zero by the generalized minimum principle, i.e. the upper and lower so• lutions coincide. This proves the existence of the solution of the Dirichlet problem with boundary function f. Inequalities (2.1) and (2.2) verify the claim concerning the boundary limits of this solution. Note also that the preceding proof of

lim HJ(z) = f(x) (3.3) Z-+X.ZEO used the continuity of f only at the point x, so if the Dirichlet problem is solvable for an f in G and x E aG is regular with respect to Green functions, then we have (2.3) provided f is continuous at x E aGo Now we are in position to prove the equivalence of the regularity of a point with respect to Dirichlet's problem and Green functions. Recall that x E aG is called a regular boundary point with respect to the Dirichlet problem in G if (2.3) holds for every continuous f.

Theorem 2.1. Let G be a domain with cap(aG) > 0, and x E aGo Then the following properties are pairwise equivalent. (i) x is regular with respect to the Dirichlet problem, i.e. (2.3) is true for every continuous f. (ii) If the Dirichlet problem for the boundary function f is solvable in G and f is continuous at x, then (2.3) is true. (iii) x is regular with respect to Greenfunctions in G, i.e. if a E G and go(z, a) is the Green function with pole at a, then

lim go(z, a) = 0. (2.4) Z-+X, ZEO (iv) Wiener's condition holds, i.e.

00 n ?; 10g(1/cap(An(x))) = 00,

where the sets An (x) were defined in Theorem 1.1. (v) x is afine limit point o/e \ G. Proof. The equivalence of (iii) and (iv) was the content of Theorem 1.1; (ii) obvi• ously implies (i), while it was proved above that (iii) implies (ii). That (iv) implies (v) follows from Lemma 1.5.5 (the proof of this lemma used Wiener's theorem Theorem 1.1 which we have verified above, so we can use Lemma I.5.5 here). The implication (v) ~ (iii) follows from the definition of fine topology (which means via the Riesz decomposition theorem that every super or subharmonic function is continuous in that topology) as follows: the Green function is zero at every z ¢ G except for an F,,-set E of zero capacity. By Lemma I.5.3, x is also the fine limit point of the set (C \ G) \ E; hence 458 Appendix A. The Dirichlet Problem and Harmonic Measures

gG(x, a) = lim gG(z, a) = 0, z~x,z¥GUE and this is exactly (iii), for gG(z, a) is upper semi-continuous (in this proof we used the standard extension of gG(z, a) to a subharmonic function to C \ {a}). Thus, it is left to prove that (i) implies (iii). As before, we can assume that aG is a compact subset of C. For z E aG let us define f(z) := Iz - xl, and let H? be the solution of the corresponding Dirichlet problem (the existence of which has already been proved). Then H? is a positive function in G, so if r > 0 is some small fixed number, then there is a constant c > 0 such that H?(z) ~ c gG(z, a) for Iz - al = r (here and in what follows replace Iz - al = r with Izl = l/r when a = 00 ). But then in the domain G* := G \ Dr(a) the function H?(z) - c gG(z, a) is bounded from below, is harmonic there and has nonnegative boundary limits at quasi-every point of aG*, so by the generalized minimum principle it is nonnegative on the whole G*. Since H? (z) tends to f (x) = 0 as z --+ x, Z E G, it follows that the Green function ga(z, a) also has zero boundary limit as z --+ x, and this is exactly property (iii). o

A.3 Harmonic Measures and the Generalized Poisson Formula

Let G be a d~main with cap(aG) > 0, and let a be ayoint of G. Let us form the balayage oa of the Dirac mass oa onto aGo Then oa is called the harmonic measure of the point a with respect to G. We have discussed some properties of the harmonic measures in Section 11.4, where we showed that the Green function of G with pole at a coincides with

1 ~ gG(z, a) = log -- - UOa(z) + Ca, (3.1) Iz -al

where Ca = 0 if G is bounded, and Ca = gG(a, 00) if G contains the point infinity (see formula (11.4.31». With the help of harmonic measures we can define the generalized Poisson integral (3.2) PIG(f, z) = JaG ( f(t) d8;(t) for functions f defined on the boundary aG of G. In this section we shall show that the solution of the Dirichlet problem is given by this generalized Poisson integral whenever this integral exists. We shall also show that if a, bEG, then there is a positive constant Ca,b such that 8: ~ ca,b8;. Therefore, any two harmonic measures are comparable, and hence the integrability of f with respect to any of them is equivalent to the integrability with respect to any other one, so there will be no ambiguity in the expression "f is integrable with respect to harmonic measures". A.3 Harmonic Measures and the Generalized Poisson Formula 459

Theorem 3.1 (Brelot's Theorem). Let G be a domain such that aG is compact and ofpositive capacity, and let f be a finite, Borel measurable jUnction defined on aGo Then the Dirichlet problem in G is solvable for f ifand only iff is integrable with respect to harmonic measures, and then the solution is given by the generalized Poisson integral PIG(f, z) = { fd8;. JaG In particular, this is true for every bounded Borel measurable function.

The proof of the theorem yields that the conclusions are true when the existence of the generalized Poisson integral is assumed in the weaker sense that it can be finite or infinite (i.e. the only noncovered case is when the integrals of both the positive and the negative parts of f are infinite). The theorem allows us to give another meaning to harmonic measures. Let E C aG, and let us consider the Dirichlet problem in G with boundary function equal to 1 on E and equal to 0 on aG \ E. This is solvable, and the solution WE,G is (also) called the harmonic measure associated with E and G, though it is not a measure but a harmonic function. WE,G can be used to estimate harmonic functions if some information is known on them on the boundary; therefore these harmonic measures WE,G play an important role in harmonic analysis. In view of the preceding theorem WE,G is given by a generalized Poisson integral, so we deduce the formula WE,a(Z) = 8;(E) (3.3) connecting the two notions of harmonic measures. Proof. The proof of the theorem will be given in several steps. First we assume the integrability of f with respect to harmonic measures.

I. Here we show that ifx E aG is a regular boundary point, then

8;. -+ Ox as a -+ x (3.4) in the weak* topology. In fact, suppose this is not true. Then there is a p < 1/2 and an c > 0 such that for a sequence of points a = aI, a2, ... converging to x we have 8.; (Dp (x» :s I-c. First let G be bounded. Let K = D p/ 2 (x) naG, and M = 10g(l/cap(K». Then Wiener's criterion yields that x is also a regular boundary point for the domain C\ K, so UI-'K(X) = M.

On the other hand, the equilibrium potential is strictly less than M outside D p (x), i.e. there is an CI > 0 such that

From Lemma 1.6.10 we know that there exists an increasing sequence {Kn} of compact subsets of K such that ILK(K ) -+ I, and with ILn = ILKI all the n K. 460 Appendix A. The Dirichlet Problem and Harmonic Measures

potentials U I1n are continuous. By the monotone convergence theorem it follows that for sufficiently large n we have

while Ul1n(z) ~ M - 2E], Z ¢ Dp(x).

This latter inequality also implies that Ul1n (z) ~ M for every z (note that the potential of Itn is not larger than that of itKin D p (x), which is at most M everywhere). Now Ul1n is a continuous function that is harmonic in G, so we can apply property (c) of Theorem 11.4.1 to write for a = aj, j = 1,2, ... ,

Ul1n(a) f Ul1nd~ ~ ~(C \ Dp(x))(M - 2E]) + 8;, (Dp(x))M

< E(M - 2Ed + (1 - E)M = M - 2EE] ~ Ul1n(x) - EE] which, for j -+ 00, contradicts the continuity of Ul1n. This contradiction verifies (3.4) for bounded domains. When G is unbounded, the proof is similar if we use Riesz' formula (11.4.25). In fact, by this formula,

Ul1n(a) = f Ul1nd~ - IIltnllgG(a, 00), and we can reason as before:

Ul1n(a) f Ul1nd~ - IIltnllgcCa, 00)

< M - 2EE] - IlltnllgcCa, 00)

< Ul1n(x) - EE] -lIltnllgcCa, 00), and since here gG(aj, 00) -+ 0 as j -+ 00 by the regularity of x E aG, we arrive again at a contradiction with the assumed continuity of U I1n . II. Suppose that I is continuous. We are going to verify below that PIG (f, z) is harmonic in G. What we have just proven gives

lim PIG(f, z) = I(x) z-x,ZEG for every regular point x E aG; hence this is true quasi-everywhere. Now we can apply the simple Lemma 1.2.6 to conclude that PIG(f, z) is indeed the solution of the Dirichlet problem, i.e. for continuous I the theorem is verified. III. Next we show the validity of the theorem for semi-continuous boundary functions f. Let us suppose for example, that I is lower semi-continuous. A.3 Harmonic Measures and the Generalized Poisson Formula 461

Let g be an upper function for f, and let us consider the function

g*(x) := liminf g(z), x E aGo (3.5) Z-->X,ZEG

Then g* is lower semi-continuous, and g* ::: f. Thus, there is a sequence {g~} of continuous functions converging monotone increasingly to g* (on aG, of course). Now PIG (g~ ,z) solves the Dirichlet problem for g~, and g is an upper func• tion for the latter function, so PIG(g~, z) ~ g(z). For n -+ 00 we obtain from the monotone convergence theorem the inequality PIG(g*, z) ~ g(z). But PIG(f, z) ~ PIaCg*, z), so by taking the infinium for all upper functions g we -G can deduce that PIG(f, z) ~ H f (z). Note that this argument did not use the semi-continuity of f and can be repeated for lower functions, as well. Thus, we have proved that

Hy (z) ~ PIG (f, z) ~ H~ (z) (3.6) for every z and f for which the generalized Poisson integral converges. Now we use the lower semi-continuity of f. It implies via (3.4) that

liminf PIG(f, z) ::: f(x) Z-->X,ZEG for every x E G that is a regular boundary point of aG; hence this is true quasi• everywhere. We claim that the set E where this inequality does not hold is an Fa-set. In fact, f is lower semi-continuous, so there are continuous functions fn(x) < f(x) converging monotonically to f(x) at every x E aGo Now if

E := {x E aG I liminf PIG(f, z) < fez)}' Z-->X,ZEG and En := {x E aG I liminf PIG(f, z) < fn(z)}' Z-->X,ZEG then E = U~l En and each En is compact. Thus, E is an Fa set of zero capacity, so for every z E G there is a finite measure v = Vz such that UV(z) < 00 but UV(x) = 00 for every x E E (see Lemma 1.2.3). Now if m is the infimum of U V on aG, then for every t: > 0 the sum PIG(f, x) + t: (UV(x) - m),

is an upper function by the choice of V. Hence -G PIG(f, x) + t: (UV(x) - m) ::: H g (z),

and for t: -+ 0 we obtain the converse of the right estimate in (3.6), by which we have verified (3.7) Finally, since f is lower semi-continuous, there is an increasing sequence of continuous functions Un} converging to f. By part II of this proof, then 462 Appendix A. The Dirichlet Problem and Hannonic Measures

HY(z) ~ HX(z) = PIG(fn,z), and here the right-hand side tends to PIG(f, z) as n -+ 00, by the monotone convergence theorem. Thus, HY(z) ~ PIG(f, z) holds. This and equality (3.7) prove the theorem for f. IV. Let f be any function integrable with respect to any of the 8;, z E G (and then with respect to any other one, see V below). Then, by the Vitali-Caratheodory theorem, there is a sequence {gn} of lower semi-continuous, and another sequence {hn} of upper semi-continuous functions such that hn :s f :s gn, and

f (gn - hn)d8;; -+ 0, (3.8) where Zo EGis some fixed point (see [195, Theorem 2.25]). Then, in view of the comparability of the harmonic measures, the same relation is true if Zo is replaced by any z E G. For hn and gn we can apply part III to conclude (see (3.6))

PIG (hn, z) = H~ (z) :s HY (z) :s PIG (f, z)

-G -G < H f (z) :s H gn (z) = PIa(gn, z). Now the proof is completed by the observation that here the difference of the left and right-hand sides tends to zero in view of (3.8). V. In this part of the proof we show that any two harmonic measures are com• parable. More precisely, if a and b both belong to a compact subset S of G, then [, :s CS,G~' (3.9) where the positive constant CS,G depends only on Sand G. It is enough to prove this for domains with C2 boundary. In fact, then for other G' s we can select an increasing sequence of domains {G n} with C2 boundary exhausting G. We have shown in Section 11.4 that then [,oG n -+ [,oG in the weak* topology, where the upper index indicates onto what set we take the balayage. It is also immediate from the proof (and from the rest of the proof below) that this

convergence is uniform in a E S; furthermore for large n the constants CS,G n are bounded. Then, letting n -+ 00 we can conclude from

the inequality f hd[, :s C~,G f hd~ for any nonnegative continuous h with compact support in C, which is enough to conclude (3.9). A.3 Harmonic Measures and the Generalized Poisson Formula 463

Thus, let us assume that G is of C2 boundary. Choose a closed set SI C G such that S is contained in its interior. There is a number C* such that on OSI we have gG(z, a) :s C*gc(z, b) for every z E OSI and a, b E S. Since both of these functions vanish on the boundary oG, an application of the maximum modulus theorem yields that the same inequality continues to hold in all of G \ SI. Thus, for the normal derivatives on oG in the direction of the inner normal, we also have ogc(s, a) * ogG(s, b) ----

~ 1 ogc(s, a) d8a (s) = - ds. 2rr on

VI. Finally, we verify that every generalized Poisson integral is harmonic in G. In the same way as before, we can assume that G has C 2 boundary (imagine / to be continuously extended to a neighborhood of oG). For r > 0, let Yr be the contour {s + rn(s)1 E oG} (see Fig. 3.1). It follows from the smoothness of the boundary that ogc(s + rn(s), a) ogG(s, a) ---+ ones) ones) uniformly in s E oG as r ---+ O. By the symmetry of the Green function (Theorem 11.4.8), the functions gc(s + rn(s), a) are harmonic in a, so the same is true of ogc(s + rn(s), a) ones) and then of course every integral Ogc(s + rn(s),a) 1 /(s) ds 3G ones) inherits this property. Now the harmonicity of

Fig. 3.1 464 Appendix A. The Dirichlet Problem and Harmonic Measures

~ I agG(S,a) 1 f(s)doa(s) = 1 f(s)- ds aG aG 2n an in a follows by letting r tend to O. This completes the proof of the sufficiency of the integrability condition. We still have to prove its necessity. Thus, suppose that the Dirichlet problem is solvable for f, and for an upper function g consider the associated function g* defined in (3.5), and analogously, for a lower function h set

h*(x):= liminf h(z), X E aGo z ..... x, ZEG

Fix z E G. We have verified in the proof above (see part III) that

PIdg*, z) ::: g(z), and similar reasoning shows that

h(z) ::: PIG(h*, z), i.e. h(z) ::: PIG(h*, z) ::: PIG(g*, z) ::: g(z). Observing that here, by assumption of the existence of the Dirichlet solution, the supremum of the left-hand side for all h coincides with the infimum of the right-hand side for all g, and furthermore that

h*(t) ::: f(t) ::: g*(t) for all t E aG, the integrability of f with respect to 8; immediately follows. 0 Appendix B. Weighted Approximation in eN by Thomas Bloom

In this appendix we will present multidimensional versions of some of the results on weighted approximation given in Chapters I, II and III. Those chapters rely on potential theory in one complex variable and its exten• sion to the weighted case. For the multidimensional generalizations we will use pluripotential theory. This theory has been developed over the last 30 years or so and in particular gives the "correct" version of capacity of sets in eN. We will use it to construct a weighted pluricomplex Green function V;,Q' given a closed set E C eN and an admissible weight w on E. The support of the associated Monge-Ampere measure is the set on which the "sup" norm of a weighted polynomial lives (see Theorems 2.6 and 2.11). In the one variable case, this is equivalent to the following: One uses Theorem 1.4.1 (restated for subharmonic functions and upper envelopes) to characterize Fw - UILw and then applies the Riesz decomposition theorem to obtain 1 II - -L1(F - UILw) f"'W - 2rr w •

In one variable this approach loses the "electrostatic" interpretation and it is not the most straightforward way to embark on explicit computation. However in several variables it is the only option. In eN, the Monge-Ampere measure (ddCV;,Q)N arises from the complex Monge-Ampere operator (ddC)N, which is the natural operator associated to pluripotential theory, and it is non-linear (if N > 1). Explicit methods for computing the support of the Monge-Ampere measure are not available in the several variable case and it would be interesting to develop some. In Section B.I we review some basic facts from pluripotential theory. In Section B.2 we prove the results on where the "sup" norm and sup norm of a weighted polynomial live. Our approach does not use results on Fekete points. In Section B.3 we define Fekete points in the weighted several variable case. Theorem 3.2 gives a result on the distribution of Fekete points but a generalization of Theorem III,1.3 is not as yet known in several variables (see Problem 3.3). 466 Appendix B. Weighted Approximation in eN

B.1 Pluripotential Theory

In this section we give a brief summary of various concepts from pluripotential theory. eN denotes complex N-space. We use z = (ZI, ... , ZN) with Zi E e as coordinates for eN. We may identifY eN with R2N (Euclidean 2N -space) with coordinates (XI, YI, ... , XN, YN) where Xk = Re(zd and Yk = Im(zd for k = 1, ... ,N. Under this identification of eN with R2N all the usual concepts from real analysis in Euclidean space (e.g. Lebesgue 2N-dimensional measure) apply to eN. The Euclidean norm of a point Z E eN is given by Izi := (lzll2 + ... + IZNI2)1/2. The open ball of center ZO and radius r, r > 0, is

An N multi-index a = (al, ... , aN) is an N-tuple of non-negative integers. The monomial (Z~I) ... (z';;') is denoted za. It is a monomial of degree la I = al + ... +aN· A polynomial p(z) = L caZa is of degree n if at least one of the coefficients lal::::n Ca with la I = n is non-zero. Let G be an open subset of eN.

Definition 1.1. A function u : G --+ [-00,00) is upper semi-continuous (u.s.c.) on G if for every ZO E G, limsupu(z) S u(zo). z~zo The function u is called lower semi-continuous (l.s.c.) if -u is upper semi• continuous. Given a function f : G --+ [-00,00), its upper semi-continuous regularization 1* is defined by

1* (zo) = lim sup f (z) for all ZO E G. (1.1) z-+zo Then f* ::: f and 1* is the smallest u.s.c. function with this property. That is, if u is u.s.c. on G and u ::: f, then u ::: 1* (see also notes to Section 11.2). The concept of upper semi-continuity given here is the same as that given in Chapter 0. In pluripotential theory, the standard convention is to work with the multivari• able version of subharmonic functions (rather than the multivariable version of superharmonic functions). Thus, we have

Definition 1.2. A function u : G :--+ [-00,00) is plurisubharmonic (p.s.h.) if it is u.s.c. on G, u ¢. -00 on any component of G, and, for every a E G, bEeN the function of the single complex variable A --+ u(a + Ab) is subharmonic or identically -00 on every component of the set {A Eel a + Ab E G}. B.l Pluripotential Theory 467

We will use the notation U E PSH(G). For example, let f be an analytic function on G with f ¥= o. Then log If(z)1 E PSH(G) (compare with Example 0.5.4). The set of plurisubharmonic functions on G that are locally bounded is denoted by PSH(G) n L~c(G). The p.s.h. functions on CN of at most logarithmic growth at 00 are

C := (u E PSH(CN ) I u ::s log+ Izl + C}. (1.2)

Also those p.s.h. functions on CN of logarithmic growth at 00 are

C+ := (u E PSH(CN ) Ilog+ Izl + D\ ::s U ::s log+ Izl + D 2 }. (1.3)

Here log+ Izl = max {log Izl, O} and in (1.2) and (1.3) the constants C, D\, D2 may depend on u. Clearly C+ c C. For example, let p(z) be a polynomial of degree d ::: I on C N . Then (ljd) log Ip(z)1 E C but only for N = I is (ljd) log Ip(z)1 E C+. This is be• cause, for N > I, the zero set of a non-constant polynomial is not compact. The starting point for potential theory in one complex variable is Laplace's equation and its solutions (harmonic functions). In pluripotential theory, the corre• sponding role is played by the (homogeneous) complex Monge-Ampere equation and its solutions (rather than Laplace's equation and harmonic functions on R 2N ). This is essentially because a "free upper envelope" of plurisubharmonic func• tions must satisfy the complex Monge-Ampere equation. A specific result in this direction is Theorem 1.3 below. We will first describe the complex Monge-Ampere equation. We consider the operators defined in terms of the real coordinates of R2N by, for k = I, ... , N,

(1.4) a:k := ~ (a~k - n a:J ' and the differential forms

Let u be real-valued and twice continuously differentiable on the open set G (u E C2(G». Then u satisfies the complex Monge-Ampere equation if

(1.6)

where i, j = I, ... , N, and det denotes determinant. For N = 1,

and (1.6) reduces to Laplace's equation. In contrast, however, to the one variable case, for N > 1 solutions of (1.6) are not necessarily real analytic. Indeed there exist solutions of (1.6) "in the sense of distributions" which are not C 2 . 468 Appendix B. Weighted Approximation in eN

For u E C 2 (G) we consider the 2N-fonn

(ddCu)N := ddcu 1\ ... 1\ ddcu, (1.7)

where (1.8)

Then,

and (1.9)

where d V is the standard 2N -dimensional volume fonn on R2N. The operator u -+ (ddCu)N has an extension "in the sense of distribu• tions" to locally bounded plurisubhannonic functions on G. For u E PSH(G) n L~c(G), (ddCu)N is a locally finite positive Borel measure on G. The proof of this fact is not a standard application of the theory of distributions as (1.9) shows that one must consider a product of distributions. The next theorem gives an important specific result showing that solutions of (1.6) play, in the several variable case, the role that hannonic functions do in the one variable case. Given a locally bounded plurisubhannonic function u, if u does not satisfy the complex Monge-Ampere equation in a neighborhood N of a point zo, then there exists another plurisubhannonic function u which equals u outside N, is strictly larger than u at some points of N and u satisfies (1.6) in a neighborhood of zo. The theorem implies that a "free upper envelope" of plurisubhannonic func• tions satisfies the complex Monge-Ampere equation.

Theorem 1.3. Let u E P S H (G) n L~c (G). Let Zo be a point of G and B(zO, R) c G, R > O. Then there exists a unique u E PSH(G) n L~c(G) such that (i) (ddCu)N = 0 on B(zo, R); (ii) u = u on G \ B(zo, R); (iii) u 2: u on B(zo, R). Indeed, on B(zo, R) we can take the Perron-Bremmerman envelope

u(z) = sup{v(z) I v E PSH(B(zo, R», lim sup v(~) :s u(f) ~->~' for all ~' E aB(zo, R)}.

This is the analogue of the Perron-Wiener-Brelot lower solution of the classical Dirichlet problem (see Section I.2). The next definition provides the generalization to several variables of sets of capacity zero. B.l Piuripotentiai Theory 469

Definition 1.4. A set F C C N is pluripolar if, for all a E F, there is a neighbor• hood B of a and a function U E PSH(B) such that F nBc {u = -oo}.

By a theorem of Josephson ([101], Theorem 4.7.4), given F pluripolar, there is a function U E PSH(CN ) such that F C (u = -oo). In fact, we may take U E.c ([101], Theorem 5.2.4). For example, if F C {z E CN I! (z) = O} and! is analytic on C N , f =1= 0, then F is pluripolar since logl!1 E PSH(CN ). As in the one variable case, we say that a property holds quasi-everywhere (q.e.) on a set S if it holds on S \ F, where F is pluripolar. An important property of capacity of sets in one variable which holds in several variables is

Theorem 1.5. ([ 10 1], Theorem 4.7.7) A countable union ofpluripolar sets is again pluripolar.

In one variable, sets which satisfy Definition 1.4 are known as polar sets (rather than pluripolar sets). A Borel set in the plane is of capacity zero if and only if it is polar. Given a compact set of capacity zero in the plane, Evans' Theorem (IlL I. 11 ) gives a potential whose value on the set is +00, thus exhibiting it as a polar set. In fact, in the several variable case, there is a theory of capacity of sets in which the pluripolar sets are precisely those of outer capacity zero but we will not need this aspect of the theory (see [101]). In pluripotential theory, specific functions are often constructed as an upper envelope of plurisubharmonic functions. The next theorem gives convenient con• ditions for the (regularized) upper envelope of a family of plurisubharmonic func• tions of at most logarithmic growth at 00 to again be a plurisubharmonic function of at most logarithmic growth at 00.

Theorem 1.6. ([101], Proposition 5.2.1) Let U = (UdiEI be afamily offunctions in .c. Let u(z) = SUPiEI Ui(Z). Suppose that (z E C N I u(z) < +oo} is not pluripolar. Then u* E .c.

In fact u as defined in Theorem 1.6 is not, in general, u.s.c .. The function u*, its upper semicontinuous regularization, is, and this property is needed for a function to be plurisubharmonic. Now it is important that, in the situation of Theorem 1.6, the function u* does not differ from u on a large set. Specifically, we have the following result of Bedford and Taylor.

Theorem 1.7. ([101], Theorem 4.7.6) In the situation of Theorem 1.6 the set {z I u*(z) > u(z)} is pluripolar.

We will now give the generalization to several variables of the Green function with pole at 00. Let E C CN be compact. 470 Appendix B. Weighted Approximation in eN

Definition 1.8. The pluricomplex Green function of E, denoted by VE, is defined for z E eN by

VE(Z) := sup{u(z) I U E £ and U SOon E}. (1.10)

The function Vl; (z) denotes its upper semi-continuous regularization.

Then E is pluripolar if and only if Vl; == +00 ([101], Corollary 5.2.2). If E is not pluripolar, the function Vl; (z) has the following properties (compare with those of the Green function, Section 1.4 or 11.4):

(i) Vl; (z) is non-negative and (by Theorem 1.3) satisfies the complex Monge• Ampere equation on eN \ E; (ii) Vl; E £+ since VE E £ (by Theorem l.6) and for C large, log+ Izl - C SO on E so Vl;(z) ~ log+ Izl - C; (iii) Vl;(z) = 0 q.e. on E (by Theorem l.7). The Monge-Ampere measure (ddCVl;)N has total mass (2n)N ([101], Corollary 5.5.3) and supp(ddCVl;)N C E. This measure is referred to as the equilibrium measure of E. We will give some examples of compact sets, pluricomplex Green functions and equilibrium measures without details of calculations. Further examples and a sample of the calculations involved can be found in [101].

Example 1.9. (i) Let E = {z E eNllzl S l}. E is the unit Euclidean ball in eN. Then VE = Vl; = log+ Izl and (ddCVnN is (up to normalization) the surface area on the sphere {z E eNllzl = I}. (ii) Let E = {(Zl, Z2) E e 21 IZII S 1, IZ21 S I}. The set E is the unit polydisc in e 2. Then

VE = Vl; = max{log+ Izd, log+ IZ2!}

and (ddCVE)2 is (up to normalization) the measure d8Id(}z on

where 81,82 are the angular parts of polar coordinates for Zl, Z2. VE(ZI, Z2) is not of class C2 on e 2 \ E but nevertheless satisfies the complex Monge-Ampere equation "in the sense of distributions" on C2 \ E. (iii) Let E = {(Zl, Z2) E e 211z1 S I} U {(Zl, Z2) E e 21Z2 = 0, Izd S 2}. The set E is the unit Euclidean ball in e 2 union a pluripolar set. We have

IOg+ Izl for zz i= 0 VE(z) = I log+ IZi/21 for zz = 0 B.2 Weighted Polynomials in eN 471 and log+ Izl. Note that

0, Izll > I}, and this is a pluripolar set. o

Two important theorems follow:

Theorem 1.10. Let U E PSH(G) n Lt':;'c(G). Then the measure (ddCu)N places zero mass on any pluripolar set.

Theorem 1.11 (Principle of Domination). Let u E 'c, v E ,C+ and suppose u ::: v almost everywhere with respect to the measure (ddCv)N on supp(ddCv)N. Then u ::: v on eN.

In particular, in view of Theorem 1.10 the hypotheses of Theorem 1.11 are satisfied if u ::: v q.e. on supp(ddCv)N. In the one variable case Theorem 1.11 includes the principle of domination of Section 1.3 which there is stated for (superharmonic) potentials.

B.2 Weighted Polynomials in eN

Let E C eN be a closed set and w a real-valued function on E such that w 2: O. The function w is called a weight function. As in the one variable case, we have

Definition 2.1. A weight function is admissible if it satisfies the following prop• erties:

(i) w is upper semi-continuous; (ii) the set {z EEl w(z) > O} (2.1) is not pluripolar; (iii) if E is unbounded then Izlw(z) -+ 0 as Izl -+ 00, z E E.

We define Q = Q w via Q = - log w. Then Q is l.s.c. on E, the set {z EEl Q(z) < +oo} is not pluripolar and, if E is unbounded,

lim (Q(z) - log Izl) = +00. (2.2) Izl ....HXl.ZEE

The weighted pluricomplex Green function of E with respect to Q is defined by VE.Q(z) := sup{u(z) I u E'c, u ::: Q on E}. (2.3) 472 Appendix B. Weighted Approximation in eN

We let V~.Q denote its upper semi-continuous regularization. Of course, in the unweighted case, that is w == 1, we have Q == 0 and v.~'. Q is, for E compact, the pluricomplex Green function of E (see (1.10». Using condition (ii) and Theorem 1.6 we conclude that V;.Q E C. Now, for p > 0 we let Ep := {z E E Ilzl s pl. We will now show that even if E is unbounded there is a compact subset of E that determines VE •Q .

Lemma 2.2. For p sufficiently large, VE,Q = VEp,Q' Proof. By condition (ii) E is not pluripolar. Thus, by Theorem 1.5, for some a > 0, Err is not pluripolar. Then, since V;.,Q E C we have, for some constant c > 0, V;.,Q s log+ Izl + c. (2.4) Using condition (2.2), we may choose p > a so large that

Q(z) - log Izl ~ c + 1 for z E E \ Ep. (2.5)

Now suppose u E C and u S Q on Ep. Then since u S V~.,Q we obtain from (2.4) and (2.5) that u(z) S Q(z) for z E E \ Ep. We conclude that u S Q on E if u s Q on Ep. This means that VEp,Q S VE,Q' The reverse inequality being obvious, the proof is concluded. 0

The Borel measure (ddcV;.Q)N has compact support since, by Lemma 2.2, it is equal to (ddCV; Q)N whose support is contained in Ep. p. We will use the notation

(2.6)

(2.7)

S~ := {z EEl V;,Q(z) ~ Q(z)}. (2.8)

This is the same notation as used in the one variable case (Sw and /-Lw are introduced in Theorem 1.1.3 and S~ in Theorem 111.1.2). We will justify the use of the same notation as the one-dimensional case in Lemma 2.4. That is, for E C e and w an admissible weight on E we may use Theorem 1.1.3, minimizing an energy integral to construct /-Lw or using the procedure of this section we construct the measure 2~ddcV;,Q' Then, in fact, both methods give the same Borel measure. First, however, we prove (compare with Theorem I.l.3(e» the following lemma.

Lemma 2.3. Let E C eN be a closed set and w an admissible weight function. Then Sw C S~.

Proof. Suppose ZO E E \ S~. That is V;,Q(zo) < Q(zo). We will show that (ddCV;,Q)N has zero mass on a ball centered at ZO i.e. V;,Q satisfies the complex Monge-Ampere equation in a neighborhood of ZO so ZO rt Sw. B.2 Weighted Polynomials in eN 473

Now, since V~.Q is u.s.c. and Q is l.s.c., there is a ball B(zo, r), r > 0, centered at ZO such that

sup V~,Q(z) < inf Q(z). zEB(zO,r) ZEB(zO ,r)nE

Applying Theorem 1.3 to u = V';;,Q we obtain u :::: u on B(zo, r) and u = u on aB(zo, r), u E C and (ddCu)N == 0 on B(zo, r). This implies, by the maximum principle, that for z E B(zo, r),

u(z):s sup V';; Q(z) < inf Q(z). ZEB(zO.r)' zEB(zO,r)nE (Although plurisubharmonic functions satisfy a maximum principle, in the above case it suffices to use the maximum principle for the subharmonic functions of one complex variable A -+ u(zo +AV) for all unit vectors v in eN.) Hence u(z) :s Q(z) for all z E E and so u = V';;,Q' Thus V';;,Q satisfies the Monge-Ampere equation inB(zo,r). 0

Lemma 2.4. Let E C e be closed and w an admissible weight function on E. Then V';;,Q = Fw - UiLw (2.9) and tLw = (l/2n)ddCV,;;, Q' (2.10) where Fw and UiLw are as in Theorem 1.1.3. Proof. Rephrasing Theorem 1.4.1 in terms of subharmonic functions and upper envelopes (rather than superharmonic functions and lower envelopes) we have

Fw - UiLw (z) = sup{u(z) I u E C, u is harmonic for Izllarge and u :s Q q.e. on E}. (2.11)

Now V,;;, Q is a member of the family of functions on the right side of (2.11) since VE,Q :s Q on E so, by Theorem 1.7, V';;,Q :s Q q.e. on E. Thus Fw-UiLw :::: V';;,Q and we must now prove the reverse inequality. The function Fw - UiLw is in £ and by Theorem I. 1.3 (d), Fw - UiLw :s Q q.e. on E. Since supp(ddCV';;,Q) is contained in E, using the principle of domination (Theorem 1.11), we have

Fw - UiLw :s V';;,Q for all z E e. This proves (2.11). By the Riesz decomposition theorem

tLw = (l/2nM(Fw - UiLw). But for u subharmonic, ddcu = .1u dm, where dm is Lebesgue measure on R2 and this proves (2.10). 0

Now, we have in eN (compare with Theorem 1.1.3) the following result. 474 Appendix B. Weighted Approximation in eN

Theorem 2.S. Let E C eN be a closed set and w an admissible weight fUnction. Then the following properties hold: (i) Sw is not pluripolar; (ii) Sw C S~; (iii) V;,Q .::: Q q.e. on E; (iv) VE,Q = Q for q.e. Z E Sw (or S~). Proof. Property (i) follows from Theorem 1.10. Property (ii) is the same as Lemma 2.3. Property (iii) follows from Theorem 1.7, since VE,Q .::: Q, and property (iv) follows from (ii) and (iii). 0

A weighted polynomial is, as in the one variable case, a function of the form wn Pn where Pn is a polynomial of degree.::: n. We have the following estimates.

Theorem 2.6. Suppose Pn is a polynomial ofdegree at most nand Iw n Pn (z) I .::: M for q.e. Z E Sw. Then

(i) IPn(z)l.::: Mexp(nV;,Q(Z)) forall Z E eN; (ii) Iw n Pn(z)1 .::: M exp[n(V;,Q(Z) - Q(z»] for all Z E E; (iii) Iw n Pn(z)1 .::: M for q.e. Z E E. Proof. The hypothesis implies that

-1 Iog I--Pn(z) I < Q(Z) for q.e. Z E Sw' n M Hence, by Theorem 2.5(iv)

-1 Iog I--Pn(z) I .::: V;,Q(z) for q.e. Z E Sw' n M Then, by the principle of domination,

1 Pn(z) N ;; log I~ I .::: VE,Q(z)* for all Z E e ,

which proves (i) and (ii). Using Theorem 2.5(iii) and (ii) above we obtain (iii). o

In particular, for Z E E \ S~, we have

(2.12)

We will use the notation II f II ~ for the "sup" of a function on a set K. That is,

IIfII~ = inf{IIfIIK\F I F is pluripolar, Fe K}. (2.13) B.2 Weighted Polynomials in eN 475

Remark 2.7. If K is not pluripolar in a neighborhood of any of its points (i.e. {z E K liz - ZO I < 8} is not pluripolar for any ZO E K, 8 > 0) and f is continuous on K, then IIfIIK = IIfll~. Theorem 2.6(iii) may be reformulated as follows:

(2.14) for all polynomials Pn of degree at most n (n = 1,2, .. .). We will show that Sw is the smallest compact set such that (2.14) is satisfied for all weighted polynomials. That is, the "sup" norm of a weighted polynomial lives on Sw (the precise statement is Theorem 2.11). Furthermore, if E is not pluripolar at any of its points and w is continuous, then the actual sup norm of any weighted polynomial lives on Sw (the precise statement is Theorem 2.12). These results generalize to eN Theorem II1.2.3 and Corollary III.2.6. First, we will need to represent the weighted pluricomplex Green function as an upper envelope of functions of the form [deg(p)]-Ilog IPI, rather than, as it is defined, an upper envelope of functions in .c. Let E be closed in eN and w an admissible weight function on E. For n = I, 2, ... consider

ct>E,n(Z) = ct>n(Z) := sup {lPn(z)1 I deg Pn :::: nand Ilwn PnllE :::: I}. (2.15)

Then, for each z E eN we have

(2.16)

We let (2.17) since from (2.16) it follows that the above limit exists and equals the supremum (see notes to Section III.3). The function ct> E is called the Siciak extremal function. Similarly, we define

o/E,n(Z) = o/n(Z) := sup {lPn(z)1 I deg Pn :::: nand IIwn Pnll'i; :::: l} (2.18) and sup ( o/n (z)) l/n. (2.19) n::::l Theorem 2.8. We have (i) logct>(z) = VE,Q(z); (ii) (log 0/ (z))* = V;, Q (z).

Proof. We may assume E is compact, since replacing E by E p all the quantities in the statement of Theorem 2.8 remain unchanged. 476 Appendix B. Weighted Approximation in eN

By definition, for n = 1,2, ... ,

1 -log l«Pn(z)1 :::: Q(z) for all z E E, n so by definition of the weighted pluricomplex Green function,

Using (2.17), we have

log «P(z) :::: VE.Q(z) for all Z E eN.

Similarly, if Pn is a polynomial in the family defined by the right side of (2.18)

1 -log IPn(z)1 :::: Q(z) for q.e. z E E. n Thus, by Theorem 2.5(iv),

1 -log IPn(z)1 :::: VE,Q(z) for q.e. Z E Sw n and by the principle of domination,

1 -log IPn(z)1 :::: Q(z) for all z E eN, n V; ' and so 10g1fr(z) :::: V;,Q(z) for all z E eN. Now to prove (i) it suffices to show that

(2.20)

But this will also prove (ii) since if (i) holds, (log «P)* = V;, Q and since log 1fr :::: log «P we get (log 1fr)* :::: V;,Q' We will use the following theorem of Siciak which shows that a general func• tion U E C may be approximated by functions of the form [deg(p)]-llogIP(z)l.

Theorem 2.9. Let U E C. Then there exists a sequence offunctions {ud, k = 1, 2, ..., satisfying, for all z E eN, (i) Uk+l(Z) :::: Uk(Z), k = 1,2, ... ; (ii) lim Uk(Z) = u(z); k-+oo (iii) for each k, there exist finitely many polynomials {Pj,dl:::j:9k each of degree :::: nj,k such that I Uk(Z) = sup -log IPj,k(z) I· l:::j:::tk nj,k B.2 Weighted Polynomials in eN 477

Now, we continue the proof of Theorem 2.8. Suppose U E £. and U S Q on E. To prove (2.20) we must show that U Slog <1>. Consider a sequence {Uk} as in Theorem 2.9. We may suppose U is bounded below on E (since Q is l.s.c. it is bounded below by, say, C on E and we may replace u by Max(u, C) if necessary). The sequence Uk decreases monotonically to u which is less than or equal to Q on E. By Dini' s theorem, given 8 > 0 there exists an integer ko such that u(Z) s Uk(Z) s Q(z) + 8 for all z E E and all k::: ko. Now, possibly adding a term cz~j·k to Pj,b with c small, we may assume nj,k deg(Pj,k) and

u(z) - 8 s Uk(Z) s Q(z) + 28 for all z E E, all k::: ko.

Thus, for d = d(k) = sup deg(Pj,d, l:oj :9k

Hence U - 38 Slog <1>. Since 8 is arbitrary, (2.20) follows. o

Now we have (compare with Theorem III.2.9) the following corollary.

Corollary 2.10. The function V,i;,Q = log 1/1 q.e. on E and if V,i;,Q is continuous at zo, then V,i;,Q(zo) = VE,Q(ZO) = 10g<1>(zo) = log 1/1 (zo). Proof. The first statement follows from Theorem 1.7 and the second statement from Theorem 2.8. 0

Theorem 2.11. Let S be any closed subset of E such that II w n Pn II S = II wn Pn 1I'i; for all polynomials Pn of degree at most n (n = 1,2, ... ). Then S :J Sw. Proof. By Theorem 2.8(ii), we have V;,Q = V,i;,Q' Hence

Sw = supp(ddcV,i;,Q)N = supp(ddCV;,Q)N c S. o

Theorem 2.12. Let E be a closed subset ofCN that is not pluripolar in a neigh• borhood of any of its points. Let w be a continuous admissible weight func• tion on E. Then II wn Pn II E = II w n Pn II SO' for all polynomials Pn of degree at most n (n = 1,2, ... ). Furthermore if S is any closed subset of E such that IIwnPnllE = IIwnPnlisforallpolynomials Pn ofdegreeatmostn(n = 1,2, ... ), then S :J Sw. Proof. Use Remark 2.7 and Theorem 2.11. o 478 Appendix B. Weighted Approximation in eN

In the case that the hypotheses of Theorem 2.12 are not satisfied, it is conve• nient to introduce a new weight function w defined by

w(z) = lim IIwll~(z 8)nE' (2.21) 8-->0 • The proofs of Theorems 2.13 and 2.14 are (replacing "capacity zero" by "pluripolar") identical to the proofs in the one variable case (see proof of Theorem IIL2.3) and so we merely state the results.

Theorem 2.13. The jUnction w is admissible, w ~ w, and w = w q.e. on E. Theorem 2.14.

IIwnPnlisw = IIwnPnlisw = IIwnPnllE = IIwnPnll'i;.

We also have the following relation between wand w (here Q = -log w).

Theorem 2.15. V;.Q = V;.Q'

Proof. Since Q ~ Q, V;.Q ~ V;.Q and we need only prove the reverse inequality.

Suppose U E £ and U ~ Q on E. Then U ~ Q q.e. on E so U ~ Q q.e. on Sw. It follows that u ~ V;,Q q.e. on Sw and by the principle of domination u ~ V;,Q on eN. This proves Theorem 2.15. 0

Remark 2.16. We note that the immediate consequences of Theorem 2.15 (see also Theorem IIL2.3) are

J.Lw J.Lw and Sw = Sw.

B.3 Fekete Points

We will consider Fekete points for subsets of eN. First we consider the monomials to be ordered lexicographically. That is Za > zf3 if lal > 1,81 or if lal = 1,81 and ai = ,8i for i = 1, ... , j but aj+\ > ,8j+\ (with an obvious meaning if j = 0). We use the notation ek(Z) for the k-th monomial under this ordering. For ek(Z) = za we write a = a(k). For example, in e2, the first six monomials under this ordering are

and

The Vandermonde determinant in several variables is defined as follows. Let T be a positive integer, T 2: 2, and ~\, ... , ~T points in eN. The Vandermonde determinant of order T is the TxT determinant B.3 Fekete Points 479

(3.l)

eT-) (l;T) It may be considered as a polynomial in NT variables (the coordinates of l;), ... , l;T). Let E be a closed subset of eN and w an admissible weight function on E. We also consider the functions W(l;) , ... , l;T) := V(l;), ... , l;T)W(l;))la(T)I ... W(l;T)la(T)I. (3.2) Note that fixing T - 1 of the entries on W we have a weighted polynomial of degree la(T)1 in the remaining entry.

Definition 3.1. A T -th Fekete set for E (associated to w) consists of points ~), ... , hE E such that IW(~), ... , h)1 = I sup W(l;) , ... , l;T)i. ~;EE For N = 1, a(T) = T - 1 and (3.l) gives n Il;i - l;jlw(l;i)W(l;j) )~i

Then L~(z, Fn) is a weighted polynomial of degree at most n for i = 1, ... , mn, which satisfies Li (c(n) 'L") = {I i = j (3.4) w 5J ,.rn 0 i =1= j and (3.5) By Lagrange's formula we have mn L w n Pn(~j)L~(z, Fn) (3.6) j=) so that (see (111.1.12)) IIwnPnllE :s mn II wnP n IIFn. (3.7) From (3.4), (3.5), and (2.12) we may conclude that ~t) E S: (see (111.1.2)) and Fn C S~ for n = 1, 2, .... We will derive further information on the distribution of Fekete points. Namely with F = U:) Fn we have 480 Appendix B. Weighted Approximation in eN

Theorem 3.2. Sw C F.

Proof. We will show VF,Q = V!:',Q; hence

Sw = supp(ddCV;,Q)N = supp(ddCV;,Q)N c F.

Now, suppose IIwn PnllF :s 1. Then, by (3.7), IIwn PnllE :s mn. Thus (see (2.15)) oo that VF,Q :s VE,Q' The reverse inequality being obvious, we are done. o

Problem 3.3. Consider the sequence of normalized counting measures

where ( ) denotes the Dirac a-measure at the indicated point. For N = 1, Theorem 111.1.3 shows that the weak* limit of the sequence {vn } is f-Lw = (l/2TC)ddCV;,Q' For N > 1 it is not known what measures are weak* limits of the sequence Vn although a reasonable conjecture is that the sequence {vn } converges weakly to (l/2TC)N (ddCV;,Q)N. Even in the unweighted case (w == 1, N > 1) the problem is unsolved.

Problem 3.4. Define Fekete polynomials, for each integer T ~ 1 by

V(~l""'~T'Z) PT ()Z = . V(~l'''.'~T) PT(Z) is a polynomial of degree la(T + 1)1. Is

-I' I I IPT(Z)I 1m og - V;,Q(Z) on eN \ E ? T--->oo la(T + 1)1 IIw 1a (T+1)I pTilE- For N = I, using Lemma 2.4 this is Corollary 1II.l.lO (slightly modified).

B.4 Notes and Historical References

Section E.1

The definition of (ddCu)N for u, a locally bounded plurisubharmonic function, is due to Bedford and Taylor ([8] also [101], Section 3.4). It is based on an earlier estimate due to Chern, Levine, and Nirenberg. BA Notes and Historical References 481

For N > 1 a "natural" domain for the operator u -+ (ddCu)N is not known, see [101], Section 3.8. Theorem 1.3 is due to Bedford and Taylor ([8], Theorem 9.1]). Theorem 1.6 is due to Siciak [206]. Theorem 1.11 is due to Bedford and Taylor ([10], Theorem 6.5]).

Section B.2 Lemma 2.3 is essentially the same as Proposition 9.3 in [10]. Theorem 2.8 (part (i) and w continuous) appears in [206]. The fact that the pluricomplex Green function can be represented as in Theorem 2.8(i), due, in this general case, to Siciak, is a crucial step in using pluripotential theory to obtain results on approximation by polynomials in several variables (see e.g. [18]). Theorem 2.9 is in [208]. Similar (though not identical) results are in [206] and [207]. Theorem 2.12, in the unweighted case, is a consequence of Theorem 7.1 in [9].

Section B.3 Problem 3.3 in the unweighted case is stated in [207]. Fekete points, in the mul• tivariable weighted setting, were defined in [206]. Interesting results relating Fekete points and Chebyshev constants in the mul• tivariable case had previously been proven by Zaharjuta [237]. Leja points in the (unweighted) multivariable setting were defined in [85] and [19]. Basic Results of Potential Theory

Balayage of measures Section 11.4, p. 110 Bernstein-Walsh lemma (generalized) Theorem III.2.1, p. 153 Boundedness of equilibrium potential Theorem 1.4.3, p. 51 Brelot's theorem Theorem A.3.1, p. 459 Capacity, properties of Theorem 1.6.2, p. 64 Chebyshev constants Section 111.3, p. 163 Continuity of equilibrium potential Theorem 1.4.4, p. 51 Continuity theorem Theorem 11.3.5, p. 107 Dirichlet's problem Section 1.2, Appendix A, pp. 41,454 Discretization of potential Section VI.4, p. 326 Evans' theorem Theorem III.l.ll, p. 152 Existence and uniqueness of equilibrium measures Theorem 1.1.3, p. 27 Existence and uniqueness of equilibrium measures for a condenser Theorem VIII.I.4, p. 383 Fekete points Sections 111.1, VIII.3, B.3, pp. 142,396,478 Fine topology Section 1.5, p. 58 Gauss' theorem Theorem 11.1.1, p. 83 Gauss-Frostman Theorem Theorem 11.5.16, p. 135 Generalized minimum principle for superharmonic functions Theorem 1.2.4, p. 39 Green energy Section 11.5, p. 127 Green functions Section 1.4, Appendix A, pp. 53,449 Harmonic measure Section A.3, p. 458 484 Basic Results of Potential Theory

Harnack's Principle Theorem 0.4.10, p. 17 Helly's theorem Section 0.1.3, p. 3 Isolated singularities of harmonic functions Theorem 0.3.4, p. II Kellogg's theorem Section IV.2, p. 210 Leja-Siciak extremal function Section III.5, p. 177 Lower envelope theorem Theorem L6.9, p. 73 Maximum principle for harmonic functions Theorem 0.2.6, p. 8 Maximum principle for logarithmic potentials Corollary 11.3.3, p. 104 Maximum principle for Green potentials Corollary 11.5.9, p. 131 Mean-value of potential Theorem 11.1.2, p. 84 Mean-value property for harmonic functions Theorem 0.2.4, p. 7 Poisson's formula Theorems 0.4.1, A.3.1, pp. 12,459 Principle of descent Theorem L6.8, p. 70 Principle of domination Theorem 11.3.2, p. 104 Principle of domination for Green potentials Theorem IL5.8, p. 130 Privaloff's theorem Section IV.2, p. 210 Recovering a measure from its potential Section 11.1, p. 83 Regular points Section 1.4, Appendix A, pp. 54,449 Riesz decomposition theorem Theorem 11.3.1, p. 100 Schwarz's theorem Theorem 0.4.2, p. 13 Transfinite diameter Section III. I , p. 142 Unicity theorem Theorem IL2.1, p. 97 Unicity for Green potentials Theorem 11.5.3, p. 126 Uniqueness of measures Lemma L 1.8, p. 29 Uniqueness for Green potentials Theorem IL5.6, p. 129 Wiener's theorem Theorem L4.6, Appendix A, pp. 54,449 Bibliography

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Mhaskar-Rakhmanov-Saff number, p.204 weighted Leja points, p.258

B(x; fl, ex) beta function, p. 344 e complex plane, p. 1 e Riemann sphere, p. 1 C(E, F) condenser capacity, pp. 132, 393

CoeD) continuous functions with compact support in D, p. 101

C5(D) two-times continuously differentiable functions in Co(D), p.101

CoCO) continuous functions vanishing outside 0, p. 281 cap(E) logarithmic capacity, p.25 cap(w, E) weighted capacity for w restricted to E, p. 64

characteristic function of E, p.64

characteristic function of disk Izl :s 8, p.98 c" minimal carrier capacity, p.373 eN complex N -space, p. 466 ConCA) convex hull of A, p.165

C(zo) circlelz-zol=r, p.35 weighted capacity, p.63

weighted "signed" capacity for condenser (17" 172 ), p.396 Laplacian, p. 20

n(n - 1)/2-th root of maximum Vandermonde, p.142 8(17) transfinite diameter of 17, p.142 unit mass at t, p.112 496 List of Symbols

8:;-' n (n - 1) /2-th root of maximum weighted Vandennonde, p. 143

8w weighted transfinite diameter, p.144 dist (A, B) Hausdorff distance, p. 196 dn(Y, X) Kolmogorov n-width, p.349 D(r\, r2) ring {zl r\ S Izl s r2}, p.421

D~(zo) punctured disk 0 < Iz - zol < R, p.11

Dr(z\) open disk Iz - z\1 < r, p.8 £ signed measures with finite Green energy, p. 128 positive measures with finite Green energy, p. 128 k-th monomial in several variables, p.478

inf {IIWPn IlL" I Pn E lln}' p.372

sign of charge on E j , p.382

infllWn(x - rn)ll.~;" p.418 rn F(K) F-functional, p.194 IIfII';; supremum on K ignoring zero capacity sets, p.154 n point Fekete set, p.143

(rational) Fekete set on E j , p.395 modified Robin constant (= Vw - f QdJ-Lw), p.27

Robin constant (= F\ + F2) for condenser, p.418

SUPzEE If(z)w(z)l, p.349 constant r(A/2)r(l/2)/2r(()... + 1)/2), p.239 leading coefficient of orthogonal polynomials, p. 360 Green function of G with pole at a, pp.53, 109 . . ... log(l/w(x)) H(a, fJ) contmuous weIghts on R satIsfymg hm f! = a, Ix 1---+00 x p.344 h * g(z) convolution, p.98 1-i? set of lower functions on R, p.41

1-iu ,R set of upper functions on R, p.41 f HI Perron-Wiener-Brelot solution of Dirichlet problem, p.41 List of Symbols 497

upper and lower solutions of Dirichlet problem on R, p.41 hn(CX1, fh, CX2, fh) sup {IIPnWllI/IlPnW211 I deg Pn :s n}, p.343 Hn(x) Hermite polynomials, p.361

I(f.L) energy integral, p. 24

Iw(f.L) weighted energy integral, p. 26 indy(Z) winding number of Z with respect to y, p.422 "inf" infimum ignoring zero capacity sets, p. 43 ZEH

min (log~, log ~), p.98 Izi S L(UIL; Zo, r) mean value of UIL over circle Cr(zo), p.84 m two-dimensional Lebesgue measure, p. 83

M signed measures of the form Lf=l SjUj, Uj 2: 0, p.382 M(L') unit Borel measures on L', p. 24

MAXUV set of maximum points of U V , p.374

MIL set of weak* limit measures of v(Pn(f.L)), p.374 f.L+ L f.Lj, p.391 ej=l f.L- L f.Lj, p.391 ej=-l measure f.L restricted to D, p.97 equilibrium distribution for L', p. 24 Green equilibrium distribution for E, p.132 equilibrium distribution for weight w, p.27 Green equilibrium distribution for w, p.132

total mass of measure f.L, p.38

Green energy ff gGCz, n df.L(z) df.L(n, p.l27

mutual Green energy ff gG(z, n df.L(z) dv({), p.127

f.L(w, E) equilibrium distribution for w restricted to E, p.64 498 List of Symbols

(v, J..) mutual logarithmic energy log _1_dv(z)dJ..(t), p.387 if Iz - tl balayage of measure v, p.lIO normalized counting measure on H, p. 145 normalized zero counting measure, p.373 equilibrium distribution for K, p. 194 aD boundary of D, p. 17 outer boundary of E, p. 68 ajan normal derivative, p. 83 Pc(E) polynomial convex hull of 17, p.53 Fekete polynomial associated with w, p.150 Leja-Siciak function, p. 177 Siciak extremal function for 17 C eN, p.475 set of monic polynomials of degree n, p.364

orthonormal polynomials with respect to d{t, p.360 Poisson kernel, p. 13 q.e. quasi-everywhere, p. 25

Qw, Q external field log(ljw) for weight w, p.26 R closure of R, p.83

Rw {z EEl (UJLw + Q) (z) < Fw}, p.157

17+ Uoj=IEj , p.391

17_ Uoj=-l E j , p.391 17* {z EEl 17 has positive capacity in every neighborhood of z}, p.269 Eo {z EEl w(z) > OJ, p.26 LCiai, p.388 i#j Ullman distribution, p.238 "c" inclusion except for a set of zero capacity, p.196 "sup" supremum ignoring zero capacity sets, p.43 zEH

supp({t) support of {t, p. 3 List of Symbols 499

Sw support of ILw, p.27

S*w {Z EEl (UILw + Q) (Z) ~ Fw }, p.144 SW restricted support of ILw, p.281 Tn weighted Zolotarjov numbers, p.398 tw Chebyshev constants, p. 163 tnW Chebyshev numbers, p. 163 tw restricted Chebyshev constant, p. 163 -w tn restricted Chebyshev numbers, p. 163 Tnw, Tn Chebyshev polynomials, p. 163 UIL logarithmic potential, p. 21

v uG Green potential, p.124 V minimal logarithmic energy, p.24 Vw minimal weighted energy, p.27

V(x\, X2,.'" xn) Vandermonde determinant, p.142 Vandermonde determinant in several variables, p. 479

W weight function, p. 26 w(z) lim8 ..... o+ IIwlI~,(z), p.154 XE(W) function space, p. 349 Index

absolute continuity boundary point of balayage, 122, 216 regular, 54,449 of extremal measure, 216 irregular, 54,449 admissible weight, 26,471 Brelot's theorem, 459 analytic completion, 7 Brelot-Cartan theorem, 138 approximation by weighted polynomials, 301,304, C-absolutely continuous measure, 115 307,354 capacity by incomplete polynomials, 283 of a Borel set, 25 by weighted rationals, 447 minimal carrier -, 373 problem of type II, 307 signed -, 396 property, 281 of a circle/disk, 25,45 of signum function, 317, 409, 417 of a condenser, 132,444 asymptotically extremal polynomials, of a line segment, 25,45 169 outer -, 76 zeros of -, 174 weighted -, 63 asymptotically minimal rational functions, weighted signed -, 396 398 Carleson's theorem, 123 asymptotic zero distribution, 374 carrier of /1-, 373 asymptotics Cauchy-Riemann equations, 6 for n-widths, 350 Chebyshev for Chebyshev polynomials, 163,169 constant, 163 for Fekete polynomials, 151 weighted -, 163 for orthogonal polynomials, 362 weighted rational -, 398 strong, 364 numbers, 163 restricted -, 163 Bagby points, 445 polynomials, 149,162,163 Baire category theorem, 2 asymptotic behavior of -, 169 balayage, 11 0 zeros of -, 165,168 absolute continuity of -, 122,216 Choquet's theorem, 76 measure, 110 Christoffel function, 251

of 800 , 118 circle onto a compact set, 110 capacity of -, 25 out of general open sets, 116 equilibrium distribution of -, 25 barrier, 79 circular symmetric weights, 245 Bernstein's circularly connected set, 348 formula, 365,372 closure, 18 problem, 307 condenser, 393,444 Bernstein-Walsh lemma, 153 capacity of, 132, 393 Besicovic covering theorem, 231 modulus of, 398, 443 best rational approximation, 409 potential, 393 502 Index conformal mapping of a segment, 45 extremal point method for -, 275 essential onto ring domain, 421,448 maximum modulus, 193 constrained energy problem, 76 supremum norm, 193 contact problem, 246 Evans' theorem, 37, 152, 187 continuity theorem, 107 exponential weights, 240, 284, 306, 308, contracted zero distribution, 361 321,344,360 convolution, 98 extremal measure, 27 counting measure, 145 absolute continuity of -, 216 covering in the Vitali narrow sense, 231 numerical determination of -, 258, 264 restricted support of -, 281 de La Vallee Poussin, 293 support of -, 192,281 density of states, 249 extremal point method Deny's theorem, 187 for conformal mapping, 275 Dini's theorem, 72 for Dirichlet problem, 269 Dirichlet problem, 15,40,54 lower function of -, 41,454 Fekete lower solution of -, 41,454 points/sets, 142,479 Perron-Wiener-Brelot solution of -, weighted -, 143,187 40,454 rational -, 396,446 regularity of polynomials, 150 point with respect to -, 54,449 F-functional, 194,320 domain with respect to -, 54 Fu-set, 37 upper function of -, 41,454 fine topology, 58 upper solution of -, 41,454 finite logarithmic energy, 27 discrepancy theorem, 427,448 Fourier method, 209 disk Freud weights, 204,250, 306, 308, 321, capacity of -, 25,45 360 equilibrium measure of -, 27 function distribution of eigenvalues, 249 boundary -, 18,458 domain, 6 lower semi-continuous -, domination lemma, 391 upper semi-continuous -, superharmonic, 18, 100 eigenvalues, 249 weight -, 26 elasticity, 246 energy Gauss' theorem, 83 Green, 127 Gauss-Frostman theorem, 135 integral, 24, 127 generalized minimum principle, 39 weighted -, 26 generalized Poisson integral, 458 semi-continuity of -, 70 geometric mean, 365 problem, 382 Green energy, 127,131 for signed measures, 382 Green equilibrium measure, 132 weighted -, 26, 131 Green's formula, 83 equilibrium Green function, 53 distribution/measure, 24,27, 132,383, and conformal maps, 109 470 extremal point method for -, 274 of a circle/disk, 25 for the complement of [ -I, I], 109 of a segment, 25 for the disk, 109 of several segments, 412 for the right half plane, 110 potential, 49,388 numerical calculation of -, 274 boundedness of -, 51 pluricomplex, 470, 471 continuity of -, 51,59 symmetry of, 119 of a circle/disk, 45 with pole at a, 109 Index 503

with pole at 00, 108 logarithmic Green potential, 123 energy, 24 maximum principle for -, 131 potential, 21 principle of domination for -, 130 lower envelope theorem, 73 unicity theorem for -, 126 lower function, 41,454 lower regularization, 138 harmonic lower semi-continuity, 1 conjugate, 7,9 lower solution, 41, 454 function, 6 l.s.c., 1 measure, 458 Harnack's Markoff-Bernstein-type inequality, 313 inequality, 16 Markoff's inequality, 267,317 principle, 17 maximum principle, 8 Hausdorff distance, 196 for Green potentials, 131 Helly's theorem, 3,30 for logarithmic potentials, 28, 104 Hermite polynomials, 361 mean value Hilbert's lemniscate theorem, 79 of potential, 84, 137 Hirschman's multiplier theorem, 220 property, 7 measure incomplete polynomials, 205,243,283, balayage -, 110 353 C-absolutely continuous -, 115 inequality carrier of -, 373 Bernstein-Markoff-type -, 313 counting -, 145 Bernstein-Walsh -, 153 equilibrium -, 27 Harnack's -, 16 extremal-, 27 Markoff's -, 267 normalized counting -, 145 infinite wire restricted support of, 281 charges on, 252 signed -, 382 infinite-finite range support of -, 3 inequality, 334,357 Mhaskar-Rakhmanov-Saff number, 203 inner normal, 83 minimal carrier capacity, 373 integrated density of states, 249 minimum principle, 19 irregular boundary point, 54 generalized, 39 modified Robin constant, 27 Jacobi polynomials, 187 modulus Jacobi weights, 206,241,285,354 of condenser, 398,443 Joukowski transformation, 45 of ring domain, 421 Monge-Ampere equation, 467 Kellogg's theorem, 210 monic orthogonal polynomials, 360 Kolmogoroff n-width, 349 monic polynomial, 149,360

Laguerre weights, 207,285,243,354 Nikolskii-type inequality, 342 Laplace's equation, 6 nontangential limit, 89 Laplacian operator, 83 normalized zero distribution, 361 Leja-Gorski points, 258 numerical calculation rational -, 399 of extremal measure, 258,264,402 weighted -, 258 of Green function, 274 Leja-Siciak function, 177 n-widths, 349 lemniscate sets, 164 line segment(s) oriented curve, 89 capacity of -, 25,45 orthogonal polynomials, 250 equilibrium measure of -, 25,412 asymptotics for -, 362 Lip ex condition, 86, 209 distribution of zeros of -, 374 504 Index

leading coefficient of -, 360 q.e., 25 monic -, 360 quasi-everywhere, 25 zeros of -, 360,374 outer random matrices,' 249 boundary, 48,53 rational capacity, 76 approximation, of signum function, domain, 53 409,417 Fekete points/sets, 396 Perron-Wiener-Brelot solution, 40,454 functions, 394 Phragmen-LindelOf principle, 78 asymptotically minimal -, 398 pluripolar, 469 fast decreasing, 356 plurisubharmonic, 466 weighted, 447 Poisson Leja-G6rski points, 399 formula, 12 Zolotarjov constant, 398 integral, 13,459 regular kernel, 13 boundary point, 54, 449 Poisson-Jensen formula, 121 domain, 54 polar sets, 76 restricted support, 281 polynomial convex hull, 53,79 Riesz decomposition theorem, 100 polynomials Riesz formula, 116 approximation by -, 301,304,307 Riesz representation theorem, 3 of asymptotically minimal norm, 169 Robin constant, 27 asymptotic behavior of -, 362 modified for w, 27 Chebyshev -, 149,162,163 fast decreasing -, 313 Schwarz's theorem, 13 Fekete -, 150 Siciak extremal function, 475 Hermite -, 361 semi -continuity, 1 incomplete -, 205,243,283,353 signed monic -, 149,360 capacity, 396 orthonormal -, 360 energy, 382 weighted -, 153 measure, 382 zero distribution of -, 169,176 signum function zeros of -, 427 approximation of -, 317 potential rational approximation of -, 409,417 approximation of - by continuous ones, Stone-Weierstrass theorem, 279 73 strong asymptotics condenser -, 393 for leading coefficient, 365 discrete approximation of -, 326 for LP norms, 372 equilibrium -, 49 subharmonic functions, 18 logarithmic -, 21 superharmonic functions, 18, 100 mean-value of -, 84 support, 3 on arcs, 89 of extremal measure, 192 vector -, 442 restricted -, 281 principal value (PV), 221 principle Taylor series, 9 of descent, 70 thin sets, 59 of domination, 43,104,471 Three lines theorem, 77 for Green potentials, 130 transfinite diameter, 142 Harnack's, 17 weighted -, 144 Privaloff s theorem, 210 Ullman distribution, 238,322,367 quantum systems, 249 unicity theorem, 97, 116 quasi-admissible weights, 63 for Green potentials, 126 Index 505 unitary matrix ensemble, 249 Leja-G6rski points, 258 upper polynomials, 153 function, 41,454 rational functions, 447 solutions, 41,454 rational Zolotarjov constants, 398 signed capacity, 396 transfinite diameter, 144 Vandermonde, 142,478 Wiener's theorem, 54,449 vector potential, 442 Vitali narrow sense, 231 zero distribution of asymptotically minimal polynomials, weak* 174 convergence, 30 of Chebyshev polynomials, 165 topology, 2,64 of orthogonal polynomials, 361,374 weight function, 26 contracted -, 361 admissible -, 26 zeros quasi-admissible -, 63 of asymptotically minimal polynomials, radially symmetric, 245 174 weight point, 287 of best approximating polynomials, weighted 188 capacity, 63,396 of Chebyshev polynomials, 165, 168 Chebyshev constant, 163 of orthogonal polynomials, 360,373 energy integral, 26 discrepancy of -, 427, 448 Fekete points/sets, 143,396,479 Zolotarjov constants, 398 LP-norm, 181 Zolotarjov problem, 394,398,445 Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

A Selection

210. GihmanlSkorohod: The Theory of Stochastic Processes I 211. ComfortJNegrepontis: The Theory of Ultrafilters 212. Switzer: Algebraic Topology - Homotopy and Homology 215. Schaefer: Banach Lattices and Positive Operators 217. Stenstrom: Rings of Quotients 218. GihmanlSkorohod: The Theory of Stochastic Processes II 219. DuvautlLions: Inequalities in Mechanics and Physics 220. Kirillov: Elements of the Theory of Representations 221. Mumford: Algebraic Geometry I: Complex Projective Varieties 222. Lang: Introduction to Modular Forms 223. BerghlLOfstrom: Interpolation Spaces. An Introduction 224. Gilbargffrudinger: Elliptic Partial Differential Equations of Second Order 225. Schutte: Proof Theory 226. Karoubi: K-Theory. An Introduction 227. GrauertlRemmert: Theorie der Steinschen Riiume 228. SegallKunze: Integrals and Operators 229. Hasse: Number Theory 230. Klingenberg: Lectures on Closed Geodesics 231. Lang: Elliptic Curves. Diophantine Analysis 232. GihmanlSkorohod: The Theory of Stochastic Processes III 233. StroocklVaradhan: Multidimensional Diffusion Processes 234. Aigner: Combinatorial Theory 235. DynkinlYusbkevich: Controlled Markov Processes 236. GrauertlRemmert: Theory of Stein Spaces 237. Kothe: Topological Vector Spaces II 238. GrahamlMcGehee: Essays in Commutative Harmonic Analysis 239. Elliott: Probabilistic Number Theory I 240. Elliott: Probabilistic Number Theory II 241. Rudin: Function Theory in the Unit Ball of en 242. Huppert/Blackbum: Finite Groups II 243. Huppert/Blackbum: Finite Groups III 244. KuberULang: Modular Units 245. ComfeldIForninlSinai: Ergodic Theory 246. NaimarklStem: Theory of Group Representations 247. Suzuki: Group Theory I 248. Suzuki: Group Theory II 249. Chung: Lectures from Markov Processes to Brownian Motion 250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations 251. ChowlHale: Methods of Bifurcation Theory 252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampere Equations 253. Dwork: Lectures on p-adic Differential Equations 254. Freitag: Siegelsche Modulfunktionen 255. Lang: Complex Multiplication 256. Horrnander: The Analysis of Linear Partial Differential Operators I 257. Horrnander: The Analysis of Linear Partial Differential Operators II 258. Smoller: Shock Waves and Reaction-Diffusion Equations 259. Duren: Univalent Functions 260. FreidlinlWentzell: Random Perturbations of Dynamical Systems 261. BoschlGuntzerlRemmert: Non Archimedian Analysis - A System Approach to Rigid Analytic Geometry 262. Doob: Classical Potential Theory and Its Probabilistic Counterpart 263. Krasnosel'skil'lZabreiko: Geometrical Methods of Nonlinear Analysis 264. Aubin/Cellina: Differential Inclusions 265. GrauertlRemmert: Coherent Analytic Sheaves 266. de Rham: Differentiable Manifolds 267. Arbarello/CornalbalGriffithslHarris: Geometry of Algebraic Curves, Vol. I 268. Arbarello/CornalbalGriffithslHarris: Geometry of Algebraic Curves, Vol. II 269. Schapira: Microdifferential Systems in the Complex Domain 270. Scharlau: Quadratic and Hermitian Forms 271. Ellis: Entropy, Large Deviations, and Statistical Mechanics 272. Elliott: Arithmetic Functions and Integer Products 273. Nikol'skiI: Treatise on the Shift Operator 274. Hormander: The Analysis of Linear Partial Differential Operators III 275. Hormander: The Analysis of Linear Partial Differential Operators IV 276. Ligget: Interacting Particle Systems 277. FultonlLang: Riemann-Roch Algebra 278. BarrIWells: Toposes, Triples and Theories 279. Bishop/Bridges: Constructive Analysis 280. Neukirch: Class Field Theory 281. Chandrasekharan: Elliptic Functions 282. LelonglGruman: Entire Functions of Several Complex Variables 283. Kodaira: Complex Manifolds and Deformation of Complex Structures 284. Finn: Equilibrium Capillary Surfaces 285. Burago/Zalgaller: Geometric Inequalities 286. Andrianaov: Quadratic Forms and Hecke Operators 287. Maskit: Kleinian Groups 288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes 289. Manin: Gauge Field Theory and Complex Geometry 290. Conway/Sloane: Sphere Packings, Lattices and Groups 291. Hahn/O'Meara: The Classical Groups and K-Theory 292. KashiwaralSchapira: Sheaves on Manifolds 293. RevuzIYor: Continuous Martingales and Brownian Motion 294. Knus: Quadratic and Hermitian Forms over Rings 295. DierkeslHildebrandtIKiisterlWohlrab: Minimal Surfaces I 296. DierkeslHiidebrandtlKiisterIWohlrab: Minimal Surfaces II 297. PasturlFigotin: Spectra of Random and Almost-Periodic Operators 298. Berline/GetzlerNergne: Heat Kernels and Dirac Operators 299. Pommerenke: Boundary Behaviour of Conformal Maps 300. OrliklTerao: Arrangements of Hyperplanes 301. Loday: Cyclic Homology 302. Lange/Birkenbake: Complex Abelian Varieties 303. DeVorelLorentz: Constructive Approximation 304. Lorentz/v. GolitschekIMakovoz: Construcitve Approximation. Advanced Problems 305. Hiriart-UrrutyILemarechal: Convex Analysis and Minimization Algorithms I. Fundamentals 306. Hiriart-Urruty!Lemarechal: Convex Analysis and Minimization Algorithms II. Advanced Theory and Bundle Methods 307. Schwarz: Quantum Field Theory and Topology 308. Schwarz: Topology for Physicists 309. AdemIMilgram: Cohomology of Finite Groups 310. GiaquintalHiidebrandt: Calculus of Variations I: The Lagrangian Formalism 311. GiaquintaIHiidebrandt: Calculus of Variations II: The Hamiltonian Formalism 312. Chung/Zhao: From Brownian Motion to Schrodinger's Equation 313. Malliavin: Stochastic Analysis 314. Adams/Hedberg: Function Spaces and Potential Theory 315. Biirgisser/Clausen/Shokrollahi: Algebraic Complexity Theory 316. SaffITotik: Logarithmic Potentials with External Fields 317. RockafellarlWets: Variational Analysis 318. Kobayashi: Hyperbolic Complex Spaces Springer and the environment

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