Asymptotics of Christoffel functions on arcs and curves

Asymptotics of Christoffel functions on arcs and curves

Vilmos Totik

November 11, 2013

1Supported by NSF DMS0968530 and by ERC No. 267055

Contents

1 Results 1

2 Upper estimate for Christoffel functions 4 2.1 Part I: Γ0is a Jordan curve . . . 4 2.2 Part II: Γ0 is a Jordan arc . . . 25 3 The lower estimate for the Christoffel functions in Theorem 1.1

for positive weights 28

4 Proof of Theorems 1.1 and 1.2 36

5 Appendix 38

References 40

Abstract

For a system of smooth Jordan curves and arcs asymptotics for Christoffel func- tions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.

1 Results

Letν be a finite Borel measure on the plane with compact support consisting of infinitely many points. The Christoffel functions associated withν are defined as

λn(z, ν) = inf

Pn(z)=1

Z

|Pn|²dν,

where the infimum is taken for all polynomials of degree at most n that take the value 1 atz.

Christoffel functions are closely related to orthogonal polynomials (for a survey see [13] by P. Nevai and [21] by B. Simon), to statistical physics (see e.g.

[15] by L. Pastur), to universality in random matrix theory (see e.g. the recent breakthrough [10] by D. Lubinsky, as well as [2],[22],[24]), to spectral theory (see e.g. [20], [21] by B. Simon and [1] by Breuer, Last and Simon) and to several other fields in mathematics. For the role and various use of Christoffel functions see [4], [6], [20], and particularly [13] by P. Nevai and [21] by B. Simon.

In this paper we consider asymptotics of Christoffel functions on smooth (C^1+α) Jordan curves and arcs. Recall that a Jordan curve is the homeomorphic image of the unit circle while a Jordan arc is the homeomorphic image of [−1,1].

Thus, a Jordan arc has two endpoints. The asymptotics of Christoffel functions onC²-Jordan curves was established in [25] with a systematic use of polynomial inverse images of the unit circle (lemniscates). The idea of that paper was that many things can be carried over to lemniscates from the unit circle, and a system ofC² Jordan curves can be well approximated (in a very specific sense) by lemniscates both from the inside and from the outside. This method does not work for arcs, and, in fact, except for the case when the set is a subset of the real line, no result has been known regarding Christoffel function asymptotics for arcs. In this paper we develop a method that handles both Jordan curves and arcs. We emphasize that we need a new method (actually very different ones) for both the upper and lower estimates, for previous methods do not work in either cases.

Thus, let Γ be the union of finitely manyC^1+α,α >0, smooth Jordan curves and arcs lying exterior to one another, and letsΓ=sbe the arc measure on Γ.

Let Γk,k= 0,1, . . . , k0 be the disjoint components of Γ: Γ =∪^kk=0⁰ Γk. We call those Γk that are Jordan arcs the arc-components of Γ. Since we need C^1+α smoothness just to have higher smoothness than C¹, we may and shall always assume 0< α <1.

Our main theorem is

Theorem 1.1 LetΓbe a system ofC^1+α-smooth Jordan arcs and curves lying exterior to one another, let z0 ∈ Γ be a point on Γ that is different from the endpoints of the arc components of Γ, and assume that Γ is C²-smooth in a neighborhood of z0. Assume that dν = wdsΓ is a measure on Γ with density w (with respect to the arc measure sΓ) which is continuous on Γ and positive

sΓ-almost everywhere. Then

nlim→∞nλn(z0, ν) = dν(z0) dµΓ

, (1.1)

where µΓ denotes the equilibrium measure of Γ, and on the right-hand side dν(z)/dµΓ is the Radon-Nikodym derivative of ν with respect toµΓ.

For the concepts from potential theory (like equilibrium measure, logarithmic capacity, Greens’ function etc.) see e.g. [5], [9], [17], or [19].

In the case that we are considering the equilibrium measureµΓis absolutely continuous with respect to the arc measuresΓ on Γ: dµΓ(t) =ωΓ(t)dsΓ(t) with a C^α-continuous density function ωΓ (see Proposition 2.2), and with it (1.1) takes the form

nlim→∞nλn(z0, ν) = w(z0)

ωΓ(z0). (1.2)

TheC^1+α-smoothness could be replaced by piecewiseC^1+α-smoothness with- out cusps, in which case Γ could have corners, and then the result is claimed for z0 which is not an endpoint or a corner (at endpoints and at corners the order of the Christoffel function is no longer 1/n, see [28]).

The global positivity and continuity ofwwas assumed only to have an easy formulation, the proof actually gives a much more general result. To this end we recall the class Reg from [23]: a measureν with support Γ is said to be in theRegclass if

nlim→∞

sup

Pn

kPnk^Γ kPnkL²(ν)

1/n

= 1, (1.3)

where the supremum is taken for all polynomials of degree at mostn, and where kPnk^Γstands for the supremum norm ofPnon Γ. This is not the standard defi- nition of theRegclass (which is in terms of the leading coefficients of orthogonal polynomials), but it is equivalent to it, see [23, Theorem 3.4.3,(v)]. See [23] for several other equivalent formulations and for general criteria implyingν ∈Reg.

We only mention here thatν ∈Regis a very weak global assumption onν, e.g.

it holds ifdν(t)/dsΓ(t)>0sΓ-almost everywhere. Therefore, Theorem 1.1 is a special case of

Theorem 1.2 LetΓbe a system ofC^1+α-smooth Jordan arcs and curves lying exterior to one another, let z0 ∈ Γ be different from the endpoints of the arc components of Γ and assume that Γ is C²-smooth in a neighborhood of z0. Assume that dν = wdsΓ+dνsing is a measure on Γ with density w and with singular part νsing (with respect to the arc measure sΓ) which is in the Reg class. Then, if w is continuous at z0 andz0 is a Lebesgue-point for νsing, we have

nlim→∞nλn(z0, ν) = w(z0)

ωΓ(z0), (1.4)

whereωΓ denotes the density of the equilibrium measureµΓ (with respect tosΓ).

The Lebesgue-point property ofνsing mentioned in the statement is

νsing({ζ |ζ−z0| ≤τ}) =o(τ) asτ→0. (1.5) Let us mention that some kind of global condition likeν ∈Reg is needed, e.g. ifν vanishes on a subarc of Γ, then (1.4) is necessarily false because then

lim inf

n→∞ nλn(z0, ν)> w(z0)

ωΓ(z0). (1.6)

We shall give a detailed proof for Theorem 1.1, the proof of Theorem 1.2 follows by simple changes. During the proof of the upper estimate in Theorem 1.1 we shall also verify (see Proposition 2.4)

Theorem 1.3 LetΓbe a finite union of disjointC^1+αJordan curves and arcs.

Then there is a constant C and for every n= 1,2, . . . there are monic polyno- mials Pn(z) =zⁿ+· · · of degreensuch that

kPnk^Γ≤Ccap(Γ)ⁿ, wherecap(Γ) denotes the logarithmic capacity ofΓ.

This should be compared to the fact (see e.g. [17, Theorem 5.5.4]) that for any nand monic polynomialPn(z) =zⁿ+· · · we have

kPnkΓ ≥cap(Γ)ⁿ.

Thus, the theorem says that on unions of smooth curves and arcs this theoretical lower bound can be achieved for every n disregarding a constant factor. For C^2+αcurves and arcs this follows from deep results of Widom [30]. Let us also mention that if there are at least two components, or Γ is a single smooth arc, then the better estimate

kPnk^Γ= (1 +o(1))cap(Γ)ⁿ

is impossible for alln([27], [26]). It is a delicate problem (connected with simul- taneous Diphantine approximation of the harmonic measures of the components of Γ) how close one can get by the norm of monic polynomials of degree nto the theoretical lower bound cap(Γ)ⁿ, see the papers [26] and [30].

First we shall deal with Theorem 1.1 in the special case whenwis continuous and positive on Γ. The general case of Theorem 1.1 will follow from this via a simple argument. The proof of the upper and lower estimates are distinctively different. The upper estimate will be obtained by a careful discretization of the equilibrium measure. That part of the proof holds at every Lebesgue-point ofν (Lebesgue-point with respect to arc-measure) and the localC² property is not needed there. The lower estimate will be reduced to the case when there are no arc-components of Γ.

2 Upper estimate for Christoffel functions

In this section we establish that lim sup

n→∞ nλn(z0, ν)≤dν(z0) dµΓ

. (2.1)

We need the concept of Lebesgue-point of a measure on Γ. Thus, letν be a Borel-measure on Γ and dν(t) =w(t)ds(t) +dνsing its decomposition into its absolutely continuous and singular parts with respect to arc measure s =sΓ. We say that z0 ∈ Γ, which is not an endpoint of an arc-component of Γ, is a Lebesgue-point forν (with respect to arc measure) if for everyε >0 there is a ρ >0 such that if 0≤τ ≤ρthen

Z

|ζ−z0|≤τ|w(ζ)−w(z0)|ds(ζ)≤ετ (2.2) and

νsing({ζ |ζ−z0| ≤τ})≤ετ . (2.3) Since the derivative ofνsingwith respect tosΓis 0sΓ-almost everywhere (see [18, Theorem 7.13]), standard proof shows thatsΓ-almost every point is a Lebesgue- point forν.

The main theorem of this part of the paper is

Theorem 2.1 Let Γ be a finite union of disjoint C^1+α Jordan curves or arcs lying exterior to one another, andν a Borel measure onΓ. Ifz0∈Γ is not an endpoint of an arc-component of Γ and z0 is a Lebesgue-point (with respect to arc measuresΓ)ofν, then

lim sup

n→∞ nλn(ν, z0)≤ dν(z0) dµΓ

.

For the proof of Theorem 2.1, let, as before, Γk be the disjoint components of Γ with Γ0 being the one containing z0. There is a change in the argument when Γ0is a Jordan arc as opposed to the case when it is a Jordan curve. First we consider the latter case, and return to the arc case after we have presented the proof for curves.

2.1 Part I: Γ

is a Jordan curve

Without loss of generality we may assume that z0 = 0 and that the real line is the tangent line to Γ at 0. Then in a neighborhood of 0 the curve Γ0 has a parametrization t+iγ(t) with γ^′ ∈ C^α and γ(0) = 0, γ^′(0) = 0; hence

|γ^′(t)| ≤C|t|^α,|γ(t)| ≤C|t|^1+α.

Letθk =µΓ(Γk), and for annconsider the integersnk = [θkn]. Divide each Γk intonk arcsI_j^k, (for eachkthe number of suchj’s isnk), each having equal weightθk/nk with respect toµΓ, i.e. µΓ(I_j^k) =θk/nk. Then

θk

nk − 1 n

=|µΓ(I_j^k)−1/n| ≤C/n². (2.4) Let

ξ_j^k= 1 µΓ(I_j^k)

Z

I_j^k

u dµΓ(u) (2.5)

be the center of mass with respect toµΓ. Simple argument shows that on Γ0we can choose theI_j⁰’s so that the real part of one of theξ⁰_j’s lying close to 0 is 0, sayℜξ₀⁰= 0. Indeed, since Γ0is a closed curve, the aforementioned subdivision can be started from any point on Γ0, i.e. if P ∈Γ0 is any point then there is a unique subdivision σP such that P is one of the division points. Take now any subdivisionσ, and in that subdivision let 0 lie in the subarc bc, with, say,b ℜb ≤ 0,ℜc ≥0 (recall that at 0 the x-axis is tangent to Γ), and let the two neighboring arcs of that subdivision beabb andcdb with ℜa <0,ℜc >0. Call a the left endpoint of ab. Now ifb P is moving on Γ0 froma toc in a continuous manner, then the subarcI(P) inσP for whichP is its left endpoint moves from abb to cd. Since the first one lies in the negative half-planeb ℜz ≤0, while the latter lies in the positive half-planeℜz≥0, in the first case the center of mass lies inℜz <0, while in the second case it lies in ℜz >0. Therefore, there will be a moment for which the center of mass of I(P) lies on the imaginary axis, and thenσP is the required subdivision, and we selectI(P) asI₀⁰. It then easily follows that ξ₀⁰ lies closest to 0 among theξ_j^k’s.

Consider now the polynomial

Rn(z) =Y

j,k

(z−ξ_j^k) (2.6)

of degree at most n. We claim that the polynomial

Pn(z) =Rn(z)/(z−ξ₀⁰) (2.7) verifies Theorem 2.1. We prove this via a series of propositions.

In what followsA∼B means that the ratioA/Bis bounded away from zero and infinity.

Proposition 2.2 dµΓ(t) =ωΓ(t)ds(t)with a positive density functionωΓwhich is C^α-smooth away from the endpoints of the arc-components ofΓ. If E is an endpoint of an arc-component ofΓ, thenωΓ(z)∼1/|z−E|^1/2 around E.

This is a standard result. When Γ consists of one component which is a Jordan curve it immediately follows from the Kellogg-Warschawski theorem (see [16,

E=a₁^k b =a₁^k ₂^k

b =a₂^k ₃^k

b =a₃^k ₄^k

I

I

I

x

x

x

Figure 1: The choice of the intervalsI_j^k and of the pointsξ_j^k

Theorem 3.6]). When Γ consists of several components, we could not find it in the appropriate form in the literature, hence we will present a proof in the Appendix to this paper. Actually, the proof gives that around an endpointE of an arc-component of Γ the functionωΓ(t)|t−E|^1/2 is a positive Lipαfunction.

Since away from endpoints of arc-components of Γ the densityωΓis bounded away from 0 and infinity, it follows that away from the endpoints we have s(I_j^k) ∼ 1/n, and if a^k_j, b^k_j are the endpoints of the arc I_j^k, then in this case

|ξ_j^k−a^k_j| ∼1/n, |ξ_j^k−b^k_j| ∼1/nand|ξ_j^k−ξ_i^k| ∼ |j−i|/n.

Proposition 2.3 If E is an endpoint of an arc-component of Γ, say E ∈ I₁^k andI₁^k, I₂^k, . . .follow one another in this order onΓ, then|ξ_j^k−E| ∼(j/n)²and s(I_j^k)∼j/n² in a neighborhood of E. Furthermore, if the endpoints of the arc I_j^k area^k_j, b^k_j then

|ξ_j^k−a^k_j| ∼ |ξ_j^k−b^k_j| ∼s(I_j^k)∼j/n², (2.8) and

|ξ_j^k−ξ^k_i| ∼ |j²−i²|

n² . (2.9)

See Figure 1.

Proof. LetI_j^k be the arcad^k_jb^k_j witha^k_j lying closer toE. Then, by Proposition 2.2,

jθk

nk = Z

c

Eb^k_j

ωΓ(t)ds(t)∼ Z

c

Eb^k_j |t−E|^−1/2ds(t),

and since|t−E| ∼s(cEt) we can continue this as Z

c

Eb^k_j

s(cEt)^−1/2ds(t)∼s(Ebd^k_j)^1/2∼ |E−b^k_j|^1/2.

Therefore, |E−b^k_j| ∼ (j/n)² and s(I₁^k) ∼ 1/n² follow because θk/nk ∼ 1/n.

Sincea^k_j =b^k_j−1, we also get forj ≥2 the relation|E−a^k_j| ∼(j/n)². Therefore, forj≥2

θk

nk

= Z

d

a^k_jb^k_j

ωΓ(t)ds(t)∼ Z

d

a^k_jb^k_j

((j/n)²)^−1/2ds(t)∼s(I_j^k)(n/j), which, in view again ofθk/nk∼1/n, givess(I_j^k)∼j/n².

Since ξ_j^k lies close to I_j^k, |ξ_j^k −E| ∼ (j/n)² is immediate for j ≥ 2. To prove it for j = 1 we may assume temporarily (i.e. just for the proof of this relation) that E = 0 and R₊ is the half-tangent to the arc Γk of Γ. Let the orthogonal projection of the arc I₁^k onto the real line be [0, d]. Then, as we have just seen, d ∼ 1/n², and ℜξ₁^k is the center of mass of a measure ρ(t)dt on [0, d] for whichρ(t)∼t^−1/2. Elementary estimate shows then thatℜξ₁^k/d is bounded away from 0 and infinity (no matter how smalldis), which combined with diam(I₁^k)∼1/n² yields the desired estimate|ξ₁^k| ∼(1/n)².

The same argument verifies (2.8), while (2.9) follows from the other state- ments in the proposition: for example ifi < j ≤2i,i6=j then

|ξ^k_j −ξ_i^k| ∼s(ab ^k_ib^k_j) = Xj τ=i

s(I_τ^k)∼ Xj τ=i

(τ /n²)∼(j²−i²)/n², while ifj >2ithen (use also the preceding relation withj= 2i)

|ξ^k_j −ξ_i^k| ∼ |E−ξ_j^k| ∼j²/n²∼(j²−i²)/n².

Proposition 2.4 For the polynomials (2.6) we have

kRnk^Γ ≤Ccap(Γ)ⁿ (2.10) with some C independent ofn.

This almost proves Theorem 1.3, the only problem is that the degree ofRn

is P

k[θkn], which may be smaller than n but at most by k0. To have exact degreen one should divide some of the Γk’s into not [θkn] but [θkn] + 1 parts so as to get totallynarcs, and proceed as below.

Proof. By Frostman’s theorem (see [17, Theorem 3.3.4]) Z

log|z−t|dµΓ(t) = log cap(Γ), z∈Γ. (2.11) Note that (with log⁺= max(0,log))

Z

log⁺|z−t|dµΓ(t)≤log⁺diam(Γ),

hence Z

|log|z−t||dµΓ(t)≤2 log⁺diam(Γ)−log cap(Γ). (2.12) Now we write in view of (2.11)

nlog cap(Γ) = X

j,k

n− 1 µΓ(I_j^k)

! Z

I_j^k

log|z−t|dµΓ(t)

+ X

j,k

1 µΓ(I_j^k)

Z

I_j^k

log|z−t|dµΓ(t) = Σ1+ Σ2. (2.13) Here, by (2.4) and (2.12),

|Σ1| ≤X

j,k

O(1) Z

I_j^k

log|z−t|µΓ(t)

=O(1). (2.14) Therefore, to prove the claim we have to show that on Γ

log|Rn(z)| −Σ2=X

j,k

1 µΓ(I_j^k)

Z

I_j^k

log

z−ξ_j^k z−t

ωΓ(t)ds(t)≤C. (2.15) The proof uses the idea of [19, Theorem VI.4.2]. It is more involved around endpoints of arc-components of Γ, so we give it only there. Thus, let z lie in an arc I_j^l₀ that lies around an endpoint E of an arc-component Γl of Γ, on which, say, the arcs I_j^l are following each other in the order I₁^l, . . . , I_j^l₀, ... with I₁^l containingE. zand (j0, l) will always have this meaning below. We consider the sum

X

(j,k)6=(j0,l)

1 µΓ(I_j^k)

Z

I_j^k

log

z−ξ_j^k z−t

ωΓ(t)ds(t) =: X

(j,k)6=(j0,l)

Lj,k(z), (2.16)

and prove that it is uniformly bounded (both from below and above). Note that this sum differs from the one on the right of (2.15) in one term (the term with

E=a₁^l

z a b

j +1₀

j +1₀ j₀

j₀

I

x

I

x

Figure 2: The position ofz, a, b

integral overI_j^l₀ is missing), and we shall actually show that not just the sum, but also the sum consisting of the absolute values|Lj,k|is uniformly bounded,

i.e. X

(j,k)6=(j0,l)

|Lj,k(z)|=O(1). (2.17) First we verify that the individual terms Lj,k(z) in (2.16) are uniformly bounded. This is clear for k6=l (i.e. whenI_j^k is on a different component of Γ thanz) or for k=lbutj 6=j0±1 (thej=j0term is not in the sum), for then in the integrand

|z−ξ^k_j| ∼dist{I_j^l₀, I_j^k} ∼ |z−t| for allt∈I_j^k.

So let j = j0±1, say j = j0+ 1. Then we know from Proposition 2.3 that

|z−ξ^l_j0+1| ∼s(I_j^l₀₊₁)∼j0/n², and from Propositions 2.2 and 2.3 thatωΓ(t)≤ Cn/j0 onI_j^l₀₊₁. LetI_j^l₀₊₁ be the arcab, see Figure 2. Clearlyb

Lj0+1,l(z) = 1 µΓ(I_j^l₀₊₁)

Z

I_j^l

0 +1

log

z−ξ_j^l₀₊₁ z−t

ωΓ(t)ds(t)

≤ Cnn j0

Z

I_j^l

0 +1

log|z−ξ_j^l₀₊₁|+ log 1

|a−t|

ds(t). (2.18) HereZ

I_j^l

0 +1

log 1

|a−t|ds(t)≤ Z

I_j^l

0 +1

log C0

s(at)b ds(t) =s(I_j^l₀₊₁)(logC0+1−logs(I_j^l₀₊₁)).

Therefore, the integral on the right of (2.18) equals s(I_j^l₀₊₁) log|z−ξ^l_j0+1|

s(I_j^l₀₊₁) +O s(I_j^l₀₊₁)

≤Cs(I_j^l₀₊₁)≤Cj0

n².

If we substitute this into (2.18) then we obtain the boundedness of Lj0+1,l(z) from above. Its boundedness from below is clear since for z∈I_j^l₀,t∈I_j^l₀₊₁ we

have

z−ξ_j^l₀₊₁ z−t

≥c >0 (2.19)

by (2.8).

The casej =j0−1 is similar providedj0−1>1, but for j0−1 = 1, we must proceed somewhat differently, for then ωΓ(t)≤ Cn/j0 is no longer true on I₁^l. In this case (i.e. when I_j^l₀₋₁ = I₁^l) we have µ(I₁^l) ∼ 1/n ∼ s(I₁^l)^1/2,

|z−t| ∼s(zt), sob

L1,l ≤ C s(ab)b ^1/α

Z b

ab

logCs(ab)b

s(zt)b s(at)b ^−1/2ds(t),

and the right-hand side will be shown to be bounded from above in the proof of (2.24)–(2.25) (the boundedness from below ofL1,l follows again from (2.19)).

These prove the uniform boundedness of the individual termsLj,k, (j, k)6= (j0, l).

It follows from Proposition 2.3 that there is anM such that if either k6=l ork=lbut |j−j0| ≥M then forz∈I_j^l₀ andt∈I_j^k we have

ξ_j^k−t z−ξ^k_j ≤1

2

(a closer look at the proof of Propositions 2.2 and 2.3 reveals thatM = 4 suffices for large n, but we do not need the best value ofM).

Thus, in this case for the integrands inLj,k(z) we get (use that log|1−u|= ℜlog(1−u) with any local branch of the log)

log

z−ξ_j^k z−t

=−log

1− ξ^k_j −t z−ξ_j^k

=ℜξ_j^k−t z−ξ_j^k +O





ξ_j^k−t z−ξ^k_j

2

.

Therefore, for such j andkwe have

|Lj,k(z)|= 1 µΓ(I_j^k)

Z

I_j^k

O





ξ^k_j −t z−ξ_j^k

2

dµΓ(t) =O s(I_j^k)²

|ξ_j^k−ξ^l_j0|²

!

, (2.20)

because the integral Z

I_j^kℜξ_j^k−t

z−ξ_j^kdµΓ(t) =ℜ Z

I^k_j

ξ^k_j −t z−ξ_j^kdµΓ(t)

vanishes by the choice ofξ_j^k.

The expression on the right of (2.20) is bounded by a constant timess(I_j^k)² when k 6= l or k = l but I_j^l is far from E (say farther than a fixed constant δ >0), and fork=l andI_j^l close toE (say for|ξ^k_j −E| ≤δ) it is at most (see Proposition 2.3) a constant times

s(Ij)²

|(j/n)²−(j0/n)²|² ∼ (j/n²)²

|(j/n)²−(j0/n)²|² = j²

|j²−j₀²|².

All in all, if we take into account the uniform boundedness of the termsLj,k

we obtain that the sum in (2.17) is at most X

|j−j0|≤M, j6=j0

|Lj,l| + X

|j−j0|>M

|Lj,l|+ X

j,k, k6=l

|Lj,k|

≤ (2M)C+C X

|j−j0|>M

j²

|j²−j₀²|²+CX

j,k

s(I_j^k)²≤C.

To complete the proof of the proposition we have to show that the additional term

1 µΓ(I_j^l₀)

Z

I_j^l

0

log

z−ξ_j^l₀ z−t

ωΓ(t)ds(t) (2.21)

in (2.15) is also bounded from above (from below we cannot claim boundedness forz can be very close toξ^l_j0). As before, we get from Proposition 2.3 that for j0>1 this term is at most

Cn Z

I_j0^l

logCs(I_j^l₀) s(zt)b

! j₀² n²

−1/2

ds(t),

which, withI_j^l₀=:ab, equalsb Cn²

j0

s(ab) log(Cs(b ab))b −s(zb) logb s(zb)b −s(caz) logs(caz) +s(ab)b

. (2.22) Now we use for 0≤x≤y≤1 the inequality

−2

e(x+y)≤xlogx+ylogy−(x+y) log(x+y)≤0, (2.23) which is immediate from the concavity of log and from the fact that on the interval (0,1) the minimum oftlogtis−1/e. Apply (2.23) withs(zb),b s(caz) in place ofx, y (in which casex+y=s(ab)) to continue (2.22) asb

≤Cn² j0

s(ab) log(Cs(b ab))b −s(ab) logb s(ab) +b O(s(ab))b

≤Cn² j0

s(ab)b ≤C,

E=a

b z

w

I

x

Figure 3: The choice ofw

where, in the last step we used that by Proposition 2.3,s(ab) =b s(I_j^l₀)∼j0/n². This gives the required estimate for (2.21) whenj0>1.

Whenj0= 1 thenEis an endpoint of the arcI_j^l₀, e.g. E=a. In that caseωΓ

is not bounded onI_j^l₀, so we have to proceed differently than before. Similarly as above, now we have withs(ab) =b s(I_j^l₀)∼1/n²,µΓ(I₁^l)∼1/n∼s(ab)b ^1/2the bound

C s(ab)b ^1/2

Z b

ab

logCs(ab)b

s(zt)b s(at)b ^−1/2ds(t) =:I (2.24) for the expression in (2.21). Recall that z lies on the arc abb = Eb, and letc w be the midpoint on the arcEzc in the sense thats(dEw) =s(wz), see Figure 3.c Now we split the integral in (2.24) over abb into three parts: the integrals over zb,b wzc andEw. For the first we have (use that the antiderivative ofd t^−1/2logt is 2t^1/2logt−4t^1/2)

Z b

zb

logCs(ab)b

s(zt)b s(cEt)^−1/2ds(t)≤ Z

b

zb

logCs(ab)b

s(zt)b s(zt)b ^−1/2ds(t)

= 2 log(Cs(ab))s(b zb)b ^1/2−2s(zb)b ^1/2logs(zb) + 4s(b zb)b ^1/2≤Cs(ab)b ^1/2 because, for anyC0> e² (by the monotonicity ofx^1/2log 1/xon (0, e⁻²)),

2s(zb)b ^1/2logC0s(ab)b

s(zb)b ≤2s(ab)b ^1/2logC0s(ab)b

s(ab)b = 2s(ab)b ^1/2logC0. The integral overwzc can be similarly handled. Finally, for the integral over Ewdwe have the bound

Z c

Ew

logCs(ab)b

s(dEw)s(cEt)^−1/2ds(t) ≤ logCs(ab)b

s(dEw)2s(dEw)^1/2≤logCs(ab)b

s(ab)b 2s(ab)b ^1/2

= 2(logC)s(ab)b ^1/2.

Substituting all these into (2.24) we get

I≤C, (2.25)

and with this the upper boundedness of (2.21) forj0= 1, as well.

Proposition 2.5 For the polynomials from (2.7) we have

|Pn(z)| ∼ncap(Γ)ⁿ (2.26)

uniformly in nandz∈I₀⁰, in particular

|Pn(0)| ∼ncap(Γ)ⁿ. (2.27) Forz∈Γ\I₀⁰

|Pn(z)| ≤Ccap(Γ)ⁿ 1

|z| (2.28)

with some C independent ofn.

Proof. Let z ∈ I₀⁰, i.e. with the notations of the preceding proof we have l=i0= 0. By the proof of Proposition 2.4 (see in particular (2.11)–(2.14)) and (2.17)) we have uniformly innandz∈I₀⁰

log|Pn(z)| −nlog cap(Γ) + 1 µΓ(I₀⁰)

Z

I₀⁰

log|z−t|ωΓ(t)ds(t) =O(1). (2.29) Now use that |z−t|= (1 +o(1))s(zt) (forb z−t ∼0) to get with I₀⁰=:abb for the last term in (2.29)

1 µΓ(I₀⁰)

Z

I₀⁰

log|z−t|ωΓ(t)ds(t) = 1 µΓ(I₀⁰)

Z

I₀⁰

log

(1 +o(1))s(zt)b

ωΓ(t)ds(t)

= o(1) + 1 µΓ(I₀⁰)

Z

I⁰₀

logs(zt)ωb Γ(t)ds(t).

Here we need that for t∈I₀⁰ Proposition 2.2 yields

|ωΓ(t)−ωΓ(0)| ≤C|t|^α≤Cn^−α to continue the preceding estimates as

= o(1) +ωΓ(0)(1 +O(n^−α)) µΓ(I₀⁰)

Z

I₀⁰

logs(zt)ds(t)b

= o(1) +ωΓ(0)(1 +O(n^−α))

µΓ(I₀⁰) (s(caz) logs(az) +c s(zb) logb s(zb)b −s(ab))b

= o(1) +ωΓ(0)(1 +O(n^−α)) µΓ(I₀⁰)

s(ab) logb s(ab) +b O(s(ab))b ,

where, in the last step we used again (2.23) with s(caz) ands(zb) playing theb role ofx, y (note that then x+y=s(ab)).b

Since

ωΓ(0)

µΓ(I₀⁰)s(ab) = 1 +b O(n^−α) and

logs(ab) =b O(1)−logn

are also true (the latter one follows from the first one in view of µΓ(I₀⁰) = (1 +o(1))/n), finally we can conclude

1 µΓ(I₀⁰)

Z

I⁰₀

log|z−t|ωΓ(t)ds(t) =O(1)−logn.

This and (2.29) prove (2.26).

The claim (2.28) follows immediately from Proposition 2.4, for|z−ξ₀⁰| ∼ |z| whenz∈Γ\I₀⁰.

Label the pointsξ_j⁰around 0 in such a way that, as their real part increases, they follow each other in the order

· · ·<ℜξ₋⁰₂<ℜξ₋⁰₁<0 =ℜξ₀⁰<ℜξ⁰₁<ℜξ₂⁰<· · ·.

We may also assume that this labeling is such that the range ofj includes all integers in [−τ n, τ n] for someτ >0.

Proposition 2.6 For all j we have

ξ_j⁰− j nωΓ(0)

≤C

|j| n

1+α

. (2.30)

Proof. Enough to prove this for |ξ_j⁰| ≤ δ with some small δ > 0 (otherwise the discussion below gives|j| ≥cδnand then the statement is obvious).

Recall that the real line is the tangent line to Γ at 0 and in a neighborhood of 0 the curve Γ has a parametrization t+iγ(t) withγ^′ ∈ C^α and γ(0) = 0, γ^′(0) = 0,|γ^′(t)| ≤C|t|^α,|γ(t)| ≤C|t|^1+α.

Letu=t+iγ(t)∈Γ. Then ds(u) =p

1 + (γ^′(t))²dt=dt+O(|u|^2α)dt. (2.31) Let a, bbe the endpoints of I₀⁰, ℜa <ℜb. We know that|a|,|b| ∼1/n (this is immediate from the facts thatℜξ₀⁰= 0, the equilibrium densityωΓis continuous and positive at 0, andds(u)∼dtby (2.31)). We can write

0 = ℜξ⁰₀= n0

θ0

Z

I₀⁰ℜu ωΓ(u)ds(u) =n0

θ0

Z

I⁰₀ℜu ωΓ(0)ds(u) +O

n1 nn^−α1

n

= n0

θ0

Z ℜb ℜa

tωΓ(0)dt+O(n^−1−α)

= n0ωΓ(0) θ0

1

2((ℜb)²−(ℜa)²) +O(n^−1−α),

from which it follows that (note n∼n0,ℜb− ℜa∼1/n)

|ℜb+ℜa|=O(n^−1−α).

Now lett0=ℜ(a+b)/2 +iγ(ℜ(a+b)/2). (2.31) implies

s(tc0b) =ℜb− ℜt0+O(n^−1−α); s(atc0) =ℜt0− ℜa+O(n^−1−α) and hence

s(tc0b)−s(atc0) =O(n^−1−α).

Therefore, ifξ⁰_j is the midpoint of the arc I_j⁰ with respect to arc length, then

|t0−ξ⁰₀| ≤Cn^−1−α. Since

|t0−ξ₀⁰| ≤ |t0|+|ξ₀⁰| ≤Cn^−1−α

is also true, finally we obtain |ξ₀⁰−ξ⁰₀| ≤Cn^−1−α. Note that by the definition ofξ⁰_j and theC^α-smoothness ofωΓ we also have

µΓ(aξd⁰₀) = 1

2µΓ(ab) +b O(n^−α) µΓ(ξc⁰₀b) = 1

2µΓ(ab) +b O(n^−α)

Sinceξ_j⁰,ξ⁰_j are geometric quantities defined in terms ofωΓ andsΓ, the same argument can be given for allj and we obtain

|ξ_j⁰−ξ⁰_j| ≤Cn^−1−α, |ξj| ≤δ, (2.32)

and (withaj, bj being the endpoints ofI_j⁰) µΓ(adjξ⁰_j) = 1

2µΓ(adjbj) +O(n^−α) µΓ(ξd⁰_jbj) = 1

2µΓ(adjbj) +O(n^−α).

These latter imply forj6= 0, say for j >0, jθ0

n0

= µ

j−1

[

l=0

I_l⁰

!

=µΓ(ξd⁰₀ξ⁰_j) +O(n^−1−α)

= Z ξ⁰_j

ξ⁰0

ωΓ(0)ds(u) +O(|ξ⁰_j|^1+α) +O(n^−1−α)

= Z ℜξ⁰_j

ℜξ⁰₀

ωΓ(0)dt+O(|ξ⁰_j|^1+α) = (ℜξ⁰_j− ℜξ⁰₀)ωΓ(0) +O(|ξ⁰_j|^1+α)

= (ξ⁰_j−ξ⁰₀)ωΓ(0) +O(|ξ⁰_j|^1+α),

which, in view of|ξ⁰₀| ≤Cn^−1−α,|ξ⁰_j| ≤Cj/nand (2.4) implies

ξ⁰_j− j nωΓ(0)

≤C

|j| n

^1+α .

The argument for negativej is just the same. Finally, this inequality combined with (2.32) gives (2.30).

Fix a large integer numberM and a smallρ >0, so small that evenρ^αM is small. Let

Qn(z) = Y

M³<|j|≤ρn

(z−ξ⁰_j). (2.33)

Proposition 2.7 . For z∈Γ,|z| ≤M/nwe have

Qn(z)

Qn(0) −1

≤

CM ρ^α+ 1 M

(2.34) with a C that depends only onΓ.

Proof.

log Qn(z)

Qn(0)

= X

M³<|j|≤ρn

log 1− z

ξ_j⁰ ,

and on applying (2.30) this can be written as X

M³<j≤ρn

log

1− z

j/nωΓ(0) +O((j/n)^1+α) 1− z

−j/nωΓ(0) +O((j/n)^1+α)

= X

M³<j≤ρn

log

1 +O(|z|(j/n)^1+α) +O(|z|²) (j/n)²

= X

M³<j≤ρn

O

|z|(j/n)^α−1+ |z|² (|j|/n)²

=O

(M/n)n^1−α(ρn)^α+(M/n)² M³/n²

,

from which the claim follows.

The key statement in the proof of Theorem 2.1 is

Proposition 2.8 Let Γδ be the part of Γ that lies of distance ≥ δ from the origin. Then

δlim→0ℜ Z

Γδ

dµΓ(u)

u = 0. (2.35)

The statement is that the real part of the principal value integral PV

Z

Γ

dµΓ(u)

u (2.36)

is zero at 0. This is due to the fact that the tangent line to Γ at 0 is horizontal.

Proof. Let Γδ be the complementary arc, i.e. the set of points on Γ which are closer to 0 thanδ. With some local branch of log we have to show that

δlim→0ℜ Z

Γδ

(log(z−u))^′

z= 0dµΓ(u) = 0.

Here, with z =x+iγ(x) ∈ Γ, u= t+iγ(t)∈ Γ (with some global t+iγ(t) parametrization of Γ that extends the local parametrizationt+iγ(t) around the origin, see the discussion before (2.4))

ℜ Z

Γδ

(log(z−u))^′

z= 0dµΓ(u) =

= lim

x→0

1 x+iγ(x)

Z

t+iγ(t)∈Γδ

log|(x+iγ(x))−(t+iγ(t))|

|t+iγ(t)| dµΓ(t+iγ(t)).

Since the whole integral Z

Γ

log|z−u|dµΓ(u) (2.37)

is constant on Γ (see (2.11)), what we need to show is that the previous expres- sion with Γδreplaced by Γδ tends to 0 asδ→0 (in this case the existence of the limit/derivative follows from what we have just done and from the constancy of (2.37)).

Sincex/(x+iγ(x))→1 asx→0, we need to show that 1

x Z

t+iγ(t)∈Γδ

log|(x+iγ(x))−(t+iγ(t))|

|t+iγ(t)| dµΓ(t+iγ(t)) =: 1

xI (2.38) is as small in absolute value as we wish for small|x|and small, but fixedδ >0.

Without loss of generality assume x > 0. Let the endpoints of Γδ be −δ1+ iγ(−δ1) andδ2+iγ(δ2),δ1, δ2>0. Thenδ²_j+γ(δj)²=δ², and hence, in view ofγ(δj) =O(δ_j^1+α), we have

δj=δ+O(δ^1+2α), j= 1,2. (2.39) With some largeN

I = Z δ2

−δ1

log|(x+iγ(x))−(t+iγ(t))|

|t+iγ(t)| ωΓ(t+iγ(t))p

1 + (γ^′(t))²dt

=

Z −N x

−δ1

+ Z N x

−N x

+ Z δ2

N x

=I1+I2+I3. First we deal withI2. It is the sum of

I21= Z N x

−N x

log|(x+iγ(x))−(t+iγ(t))|

|x−t| ωΓ(t+iγ(t))p

1 + (γ^′(t))²dt,

I22=− Z N x

−N x

log|t+iγ(t)|

|t| ωΓ(t+iγ(t))p

1 + (γ^′(t))²dt and

I23= Z N x

−N x

log|x−t|

|t| ωΓ(t+iγ(t))p

1 + (γ^′(t))²dt.

In I21 the log term is log(1 +O((N x)^α) = O((N x)^α)) because, with some ζ∈[−N x, N x],

|γ(x)−γ(t)|=|x−t||γ^′(ζ)| ≤ |x−t|C(N x)^α,

soI21=O(N^1+αx^1+α) =o(x) asx→0. Similarly,I22=o(x). Finally, I23 =

Z N x

−N x

log|x−t|

|t|

ωΓ(t+iγ(t))p

1 + (γ^′(t))²−ωΓ(0) dt

+ ωΓ(0) Z N x

−N x

log|x−t|

|t| dt=I231+I232.

The factor after the log term inI231is in absolute value≤C|t|^α≤C(N x)^αand Z N x

−N x

log|x−t|

|t| dt=x

Z N

−N

log|1−t|

|t|

dt≤CxlogN,

henceI231=O(N^α(logN)x^1+α) =o(x). ForI232, as simple calculation shows, we can write

|I232| ωΓ(0) =

Z N x (N−1)x

logx+t t dt≤

Z N x (N−1)x

x

tdt≤ x N−1. So|I2| ≤ωΓ(0)x/(N−1) +o(x).

ForI1+I3 we setJ = [−δ1,−N x]∪[N x, δ2] and note that the log term in the integrals in I1 andI3is

ℜlog

1−x+iγ(x) t+iγ(t)

= −ℜx+iγ(x)

t+iγ(t) +Ox t

2

= −xt+γ(x)γ(t)

t²+γ(t)² +Ox t

²

= −x t +xO

γ(t)² t³

+O

γ(x)γ(t) t²

+Ox t

2

Here on the rightγ(t)²/t³andγ(t)/t²are integrable, so the contribution to the integral over Γδ of the corresponding terms isxoδ(1) and γ(x)oδ(1) =xoδ(1), respectively, where oδ(1) means a quantity tending to 0 as δ→0. The contri- bution of the termO(x²/t²) is

Z

J

Ox t

2 dt=O

x² N x

=Ox N

.

Finally, the integral overJ of the term−x/t is equal to

− Z

J

x t

ωΓ(t+iγ(t))p

1 + (γ^′(t))²−ωΓ(0)

dt+ωΓ(0) Z

J

x

tdt=I4+I5.

InI4 we have

|ωΓ(t+iγ(t))p

1 + (γ^′(t))²−ωΓ(0)| ≤C|t|^α, so exactly as beforeI4=xoδ(1). Finally,

|I5|=ωΓ(0)x

Z max(δ1,δ2) min(δ1,δ2)

1

tdt≤Cxδ^1+2α

δ =xoδ(1)

where we used (2.39). All in all, we have|I| ≤xoδ(1) +o(x) +O(x/N) which shows that the term in (2.38) is as small as we wish if we select N large and thenδ >0 small (and also xsufficiently small after these selections).

Proposition 2.9 Let

Sn(z) = Y

|ξ^k_j|≥δ

(z−ξ^k_j). (2.40)

Then, for fixed M andz ∈Γ, |z| ≤ M/n, we have |Sn(z)/Sn(0)| = 1 +oδ(1) uniformly in n.

Proof. As always, we setz=x+iγ(x).

1 nlog

Sn(z)

Sn(0) = X

|ξ_j^k|≥δ

1 nlog

1− z

ξ_j^k

(2.41)

is easily seen to converge to Z

Γδ

log1− z u

dµΓ(u) (2.42)

(recall, that Γδis the part of Γ that lies of distance≥δfrom the origin). Indeed, the same sum on the right of (2.41) with 1/nreplaced byµΓ(I_j^k) andξ^k_j replaced byξ^k_j (that was the midpoint of I_j^k with respect to arc length) is essentially a Riemannian sum for the integral (2.42), and the sums with ξ_j^k, 1/n and with ξ^k_j,µΓ(I_j^k) are very close because of (2.4) and (2.32) and its analogue for other intervals. The integral in (2.42) is

Z

Γδ

ℜ

−z u

+O |z| u

2!!

dµΓ(u), and here

Z

Γδ

O |z| u

2!

dµΓ(u) =O |z|²

δ

,

while Z

Γδ

ℜ

−z u

dµΓ(u) =−xℜ Z

Γδ

dµΓ(u)

u +γ(x)ℑ Z

Γδ

dµΓ(u) u .

Here the second term isO(γ(x)/δ) =o(|x|) =o(|z|) asz→0, and for the first term Proposition 2.8 gives that it isxoδ(1).

Therefore, log

Sn(z)

Sn(0)

=n(1 +o(1))h

|z|oδ(1) +O(|z|²/δ) +o(|z|)i

≤2M oδ(1) for large nand|z| ≤M/n,z∈Γ. This proves the claim.

In the polynomialQnin (2.33) we put all factors (z−ξ_j⁰) withM³<|j| ≤ρn, whileSn in (2.40) contained the factors (z−ξ^k_j) with|ξ^k_j| ≥δ. For sufficiently small δ these latter include all ξ^k_j with k > 0 (i.e. which are created for the components Γk,k >0). Furthermore, for smallδ >0 if, with a sufficiently large fixedL we selectρ=ωΓ(0)δ−Lδ^1+α, then (2.30) shows thatQn and Sn have no common factors. On the other hand, if we selected ρ = ωΓ(0)δ+Lδ^1+α, then (2.30) shows that all the factors (z−ξ_j^k) except for (z−ξ_j⁰) with |j| ≤ M³ appear either in Qn or in Sn. We make the former selection, i.e. we set ρ=ωΓ(0)δ−Lδ^1+α, and let ˜Sn be the product of all factors (z−ξ_j⁰) for which

|j|> ρnbut |ξ⁰_j|< δ (these are the ones with|j|> M³ that appear neither in Qn nor inSn). According to what we have just said, their number is at most 4Lδ^1+αn.

Proposition 2.10 We have|S˜n(z)/S˜n(0)|= 1+oδ(1)uniformly innand|z| ≤ M/n,z∈Γ.

Proof. LetH be the set of j’s for which|j|> ρnbut |ξ⁰_j|< δ. Note that all suchξ_j⁰’s satisfy|ξ_j⁰| ≥δ/2 (see (2.30) and the definition ofρ). Now

log

S˜n(z) S˜n(0) =X

j∈H

log 1− z

ξ_j⁰ =X

j∈H

O |z|

|ξ_j⁰|

!

=O M

n

4Lδ^1+αn δ

,

from which the claim follows.

After these preparations we turn to the proof of Theorem 2.1.

From the definition of our polynomials it follows that Pn(z) =Qn(z)Sn(z) ˜Sn(z) Y

−M³≤j≤M³, j6=0

(z−ξ⁰_j) =:Qn(z)Sn(z) ˜Sn(z)Vn(z),