Push-Forward of Product Measures on the Disk

3.2 L p -Extremal Polynomials on Analytic Regions

3.2.1 Push-Forward of Product Measures on the Disk

In this section, we will derive norm asymptotics for the extremal polynomials corresponding to measures of the form considered in Theorem3.2.2. We will use Faber polynomials in conjunction with the extremal property to eventually derive an upper bound in the proof of Theorem3.2.2and we will use subharmonicity of appropriate functions to derive a lower bound. For the remainder of

this section, we will let q > 0 be fixed but arbitrary, and we will denoteP_n(z;µ, q) by P_n(µ) and kP_n(µ)k_Lq(µ) by kP_n(µ)k_µ when there is no possibility for confusion. We begin with the following crude estimate, which applies even whenν is not a Szeg˝o measure:

Proposition 3.2.3. If µ is as in Theorem3.2.2then µis regular.

Proof. We will in fact show thatµsatisfies Widom’s criterion (see Section2.2) from which regularity immediately follows by Theorem 4.1.6 in [65].

For eachr∈(ρ,1], the equilibrium measure of the curve Γris absolutely continuous with respect to arc-length measure with continuous derivative bounded above and below by positive constants (see Theorem II.4.7 in [14]; the constants are allowed to depend onr). LetCbe a carrier ofµ. Since ν⁰(θ)>0 Lebesgue almost everywhere, we conclude that

λr(C∩Γr) =`(Γr)

forτalmost everyr∈(ρ,1] whereλris arc-length measure on Γrand`(Γr) is the length of the curve Γr. It follows that there is a sequence rn →1 such thatωΓ_rn(C) = 1 while clearly cap(Γr_n)→1.

This showsµsatisfies Widom’s criterion.

We will now begin developing the ideas necessary to prove the more refined estimate ofkP_n(µ)k^q_µ given in Theorem 3.2.2. Let {Fn}_n∈N be the sequence of Faber polynomials corresponding to the region G. Since we are assuming cap(G) = 1, we recover from our earlier discussion of Faber polynomials (see Section1.3) the following two facts:

1. Fn(z) is a monic polynomial of degreen, 2. forρ <|z| ≤1 we have

Fn(ψ(z)) =zⁿ+O(ρⁿ₀), (3.2.4)

where ρ₀ ∈ ( ˜ρ, ρ) and the implied constant is uniformly bounded from above in the annulus considered.

We will henceforth assume that some value ofρ0∈( ˜ρ, ρ) has been fixed so that (2) holds.

We begin with a lemma that immediately highlights the importance of these important polyno- mials to our results.

Lemma 3.2.4. LetN ⊆Nbe a subsequence such that w-lim_n→∞

n∈N

|Fn(z)|^qdµ(z) a_n =dγ

whereγis a measure on∂Gand{a_n}_n∈Nis a sequence of positive real numbers satisfyinglim_n→∞a_na⁻¹_n+1= 1. Then for any fixed k∈N, we have

w-lim_n→∞

n∈N

|Fn−k(z)|^qdµ an

=dγ.

Proof. Recall our notation G_ρ = {ψ(z) : ρ ≤ |z| ≤ 1}. It is clear from our observations above (specifically (3.2.4)) that all weak limits in question are measures on ∂Gand thatF_n has no zeros inG_ρ for all sufficiently largen. Now, letf be a continuous function onG_ρ. We have

Z

G_ρ

f(z)|Fn(z)|^q an

dµ(z)−

Z

G_ρ

f(z)|Fn−k(z)|^q an

dµ(z) =

= Z

G_ρ

f(z)

1−|Fn−k(z)|^q

|Fn(z)|^q

|Fn(z)|^q an

dµ(z)

= Z

G_ρ

f(z)

1−|φ(z)|^q(n−k)+O(ρⁿ₀)

|φ(z)|^qn+O(ρⁿ₀)

|Fn(z)|^q an

dµ(z)

→ Z

∂G

f(z) 1− |φ(z)|^−qk dγ(z)

= 0 since|φ(z)|= 1 whenz∈∂G.

Our next lemma will identify some ideal choices for the sequence {an}n∈Nof Lemma 3.2.4.

Lemma 3.2.5. Letγbe a probability measure on the unit interval[0,1]and letc_t=c_t(γ)be defined as in (3.2.1). The following are equivalent:

1. 1∈supp(γ), 2. lim_t→∞c^1/t_t = 1, 3. lim_n→∞c_q(n+1)c⁻¹_qn = 1.

Proof. It is obvious that (1)⇒(2) and (3)⇒(1) so we need only prove that (2)⇒(3). To this end, we have

c_qn+q cqn

= 1 + R1

0 r^qn(r^q−1)dγ(r) R1

0 r^qndγ(r) . If lim_t→∞c^1/t_t = 1, then the measures R^r1^qndγ(r)

0 r^qndγ(r) converge weakly to the point mass at 1 asn→ ∞, which implies the desired conclusion.

Now we can prove the following lemma, which will be of critical importance in our proof of Theorem3.2.2.

Lemma 3.2.6. Let κbe a measure onGand γ a measure on∂Dand let N ⊆Nbe a subsequence such that

w-lim

n→∞

n∈N

|Fn(z)|^q

a_n dκ=d(ψ_∗γ), where{an}_n∈Nis as in Lemma 3.2.4. Then

lim sup

n→∞

n∈N

kPn(κ)k^q_κ an

≤exp Z 2π

0

log(γ⁰(θ))dθ 2π

.

Proof. By the extremal property, we havekPn(κ)k^q_κ ≤ kFn−k(z)Pk(ψ∗γ)k^q_κ. By Lemma 3.2.4, we can write

Z

G

|P_k(z;ψ_∗γ)|^q|F_n−k(z)|^q an

dκ(z)→ Z

∂G

|P_k(z;ψ_∗γ)|^qd(ψ_∗γ) asn→ ∞throughN. Therefore

lim sup

n→∞

n∈N

a⁻¹_n kPn(κ)k^q_κ≤ kPk(ψ_∗γ)k^q_ψ

∗γ

for every k > 0. Since k here is arbitrary, we can take the infimum over allk, which is no larger than the limit asktends to infinity. The result now follows from Theorem2.3.1.

The following calculation will be useful also.

Proposition 3.2.7. If x6∈Gandr∈[ρ,1], then Z 2π

0

log|ψ(re^iθ)−x|^qdθ

2π = log|φ(x)|^q.

Proof. First, consider the case whenx6∈G. It is clear that cap(G_r) =r. Define ψ_r(z) =ψ(rz) on {z:|z|>ρr˜ ⁻¹}. Then we calculate

log|φ(x)|^q = Z 2π

0

log|e^iθ−φ(x)r⁻¹|^qdθ

2π +qlog(r)

=−qU^ωD φ(x)

r

+qlog(r)

=qg_C\D(φ(x)r⁻¹,∞) +qlog(r)

=qg

C\Gr(x,∞) +qlog(cap(Gr))

= Z

Γ_r

log|y−x|^qdω_G

r(y)

= Z 2π

0

log|ψr(e^iθ)−x|^qdθ 2π.

The first line follows from Example 0.5.7 in [50]. The second line is just the definition of the logarithmic potential. The third line then follows from (1.1.1) above and the fact that D has

logarithmic capacity 1. The fourth line then follows from the conformal invariance of the Green’s function (see Section1.1). The fifth line follows as the third did from the first. Finally, the last line follows from the definition of equilibrium measure as given in Theorem 3.1 in [71].

The case x∈∂Gfollows by dominated convergence as in Example 0.5.7 in [50].

Now we are ready to prove Theorem3.2.2.

Proof of Theorem 3.2.2. For now, let us assume that ` = m = 0 in our definition of µ. We will appeal to Lemma 3.2.6to prove our upper bound. As mentioned in the proof of Lemma 3.2.4, all weak limits of the measures considered there are supported on∂G, so it suffices to consider functions that are continuous in a neighborhood of∂G. For anyk∈N0, we have

n→∞lim Z

G_ρ

φ(z)^k|F_n(z)|^q

cqn(τ) dµ(z) = lim

n→∞

R1 ρ

R2π

0 r^k+qne^ikθh(re^iθ)dν(θ)dτ(r)

cqn(τ) + lim

n→∞

R

Dz^k|zⁿ|^qdσ₂ cqn(τ)

= Z 2π

0

e^ikθh(e^iθ)dν(θ) = Z

∂G

φ(z)^kd(ψ_∗(hν)).

It follows that the measures ^|F_c^n(z)|2

qn(τ) dµconverge weakly tod(ψ_∗(hν)) as measures onG. The upper bound in this case now follows from Lemma3.2.6.

If we add finitely many pure points outsideG, we get the desired upper bound by placing a single zero at eachzi andζi. More precisely, if we define the polynomialsy_∞(z) and Υ_∞(z) by

y_∞(z) =

m

Y

j=1

(z−zj) , Υ_∞(z) =

m

Y

j=1

(z−ζj), (3.2.5)

then we have

kPn(µ)k^q_µ≤ ky_∞Υ_∞P_{n−m−`</sub>(|y<sub>∞</sub>(z)Υ<sub>∞</sub>(z)|<sup>q</sup>µ)k<sup>q</sup><sub>µ</sub>=kP<sub>n−m−`}(|y_∞(z)Υ_∞(z)|^qµ)k^q_|y

∞(z)Υ∞(z)|^qµ

and then proceed as in the case when`=m= 0 and apply Proposition3.2.7.

For the lower bound we will use an argument inspired by the proof of Theorem 2.4.2. Recall that Theorem1.1.3(and the remark following it) implies that for eachzi, we can choose a sequence {wi,n}n∈Nso thatPn(wi,n;µ) = 0 and limn→∞wi,n=zi(we will establish later that such a sequence has a unique tail, but we do not need this now). Define

yn(z) =

m

Y

j=1

(z−wj,n) (3.2.6)

(so thaty_n(z)→y_∞(z) pointwise). We now can calculate

kPn(z;µ)k^q_µ≥ Z 1

ρ

Z 2π 0

Pn(ψ(re^iθ)) yn(ψ(re^iθ))

q m

Y

j=1

|ψ(re^iθ)−wj,n|^qh(re^iθ)dνac(θ)dτ(r) (3.2.7)

For|z|>1 andr∈[ρ,1], define the functions

S_r,n(z) = exp



− 1 2qπ

Z 2π 0

log





m

Y

j=1

|ψ(re^iθ)−w_j,n|^qh(re^iθ)ν⁰(θ)



 e^iθ+z e^iθ−zdθ



.

By our discussion in Section1.4, we can rewrite (3.2.7) as

kPn(z;µ)k^q_µ≥ Z 1

ρ

Z 2π 0

Pn(ψ(re^iθ)) e^i(n−m)θy_n(ψ(re^iθ))

q

|Sr,n(e^iθ)|^qdθ 2πdτ(r)

(notice that we arbitrarily added a factor ofe^{−i(n−m)θ}to the integrand, which is acceptable since it is inside the absolute value bars). For each fixedr, we invoke the subharmonicity of the integrand (or equation (2.3.2)) to obtain

kPn(z;µ)k^q_µ≥ Z 1

ρ

r^qn−qmS_r,n(∞)^qdτ(r). (3.2.8)

Sincewj,nconverges to zj asn→ ∞for eachj (by construction), we find that

lim inf

n→∞

kPn(z;µ)k^q_µ cqn(τ) ≥exp

Z 2π 0

log h(e^iθ)ν⁰(θ)) dθ 2π

^m Y

j=1

|φ(zj)|^q

by Proposition3.2.7. This is the desired lower bound.

2

The proof of Theorem3.2.2produces several interesting corollaries. The first of these shows that certain parts of the measure µ contribute only negligibly to the norm of the extremal polynomial.

The following corollary is reminiscent of Theorem 2.4.1(vii) in [56].

Corollary 3.2.8. If µis as in Theorem 3.2.2with ν a Szeg˝o measure on∂D, then

n→∞lim Z

D

|p_n(ψ(re^iθ);µ)|^qh(re^iθ)dν_sing(θ)dτ(r) + Z

G

|p_n(z;µ)|^qdσ₁(z) + +

Z

D

|pn(ψ(re^iθ);µ)|^qdσ2(re^iθ) +

m

X

j=1

bj|pn(zj;µ)|^q+

`

X

j=1

βj|pn(ζj;µ)|^q

= 0.

Proof. Let us writeµ=µ⁰+µ¹ whereµ⁰=ψ_∗(h(ν⊗τ)) +Pm

j=1d_jδ_zj. Then kPn(µ)k^q_µ

cqn(τ) =kPn(µ)k^q_µ0

cqn(τ) +kPn(µ)k^q_µ1

cqn(τ) . (3.2.9)

The proof of Theorem3.2.2shows that the left-hand side of (3.2.9) and the first term on the right- hand side of (3.2.9) both converge to the right-hand side of (3.2.2). This shows that everything exceptµ⁰contributes only negligibly to the norm ofp_n(z;µ). To show that the pure points outside Gcontribute only negligibly to the norm, we keep our definition ofw_1,nfrom the proof of Theorem 3.2.2and we writeµ⁰=µ⁰₁+w₁δ_z1. We can now calculate

1≥ R

C

Pn(z;µ) z−w1,n

q

|z−w1,n|^qdµ⁰₁ kPn(µ)k^qµ

+d₁|pn(z₁)|^q

≥ kPn−1(|z−w1,n|^qµ⁰₁)k^q_|z−w

1,n|^qµ⁰₁

kPn(µ)k^qµ

+d₁|p_n(z₁)|^q

=

cqn(τ) expR2π

0 log h(e^iθ)ν⁰(θ)|ψ(e^iθ)−w1,n|^q_dθ

2π

Qm

j=2|φ(zj)|^q cqn(τ) expR2π

0 log (h(e^iθ)ν⁰(θ))^dθ_2π Qm

j=1|φ(zj)|^q

+

+d1|pn(z1)|^q+o(1)

= 1 +o(1) +d1|pn(z1)|^q,

which implies the desired conclusion forz1. An identical proof works for eachzj forj= 2,3, . . . , m.

Remark. As a consequence of Corollary3.2.8, we see that ifK⊆Gis compact and µis of the form considered in Theorem3.2.2withν a Szeg˝o measure on ∂D, then

Z

K

|pn(z;µ, q)|^qdµ(z)→0

asn→ ∞.

An additional consequence of Theorem 3.2.2is the following corollary, which is a refinement of Theorem1.1.3.

Corollary 3.2.9. Letµbe as in Theorem3.2.2withν a Szeg˝o measure on∂D. There exists aδ >0 andN ∈Nsuch that for all n≥N, the polynomialPn(µ)has a single zero in{u:|u−zi|< δ}for each i≤m. If we denote this zero by w_i,n, then there is ana >0 so that |wi,n−z_i| ≤e^−an for all largen.

Proof. Theorem1.1.3(and the remark following it) establishes the existence of at least one zero of Pn(µ) in{u:|u−zi|< δ}for alliand all largen. Now, fix >0 (but small) and let{w1, . . . , w_t(n)}

denote the collection of zeros ofP_n(µ) outside Γ₁₊. Define for |z|>1 the functions

Sr,n(z) = exp



− 1 2qπ

Z 2π 0

log





t(n)

Y

j=1

|ψ(re^iθ)−wj|^qh(re^iθ)ν⁰(θ)



 e^iθ+z e^iθ−zdθ



.

As in the proof of Theorem3.2.2, we calculate kPn(z;µ)k^q_µ

c_qn(τ) ≥ R1

ρ r^qn−qt(n)Sr,n(∞)^qdτ(r)

c_qn(τ) ≥

R1

ρ r^qn−qt(n)Sr,n(∞)^qdτ(r) c_qn−qt(n)(τ)

= R1

ρ r^qn−qt(n)exp

1 2π

R2π

0 log h(re^iθ)ν⁰(θ) dθ

dτ(r) c_qn−qt(n)(τ)

t(n)

Y

j=1

|φ(wj)|^q, (3.2.10)

where we used Proposition3.2.7. From this expression, it follows thatn−t(n) tends to infinity as n→ ∞, for if it did not, then since|φ(wj)|>1 +for everyj≤t(n), we would havekPn(z;µ)k^1/nµ >

1 +for allnin some subsequenceN ⊆N, which violates the fact that cap(G) = 1 andµis regular (see Theorem III.3.1 in [50]).

Since n−t(n) → ∞, the first factor in (3.2.10) converges to exp

1 2π

R2π

0 log h(e^iθ)ν⁰(θ) dθ as n → ∞ while the left-hand side has limit given by the right-hand side of (3.2.2). If for each i∈ {1, . . . , m}we pick a sequence{w_i,n}_n∈Nas in the proof of Theorem3.2.2, then the corresponding factor in the product (3.2.10) converges to|φ(zi)|^q as n→ ∞. Therefore, it must be that

lim sup

n→∞

t(n)

Y

j=1,w_j6=wi,n

|φ(wj)|^q≤1.

However, each factor in this product is larger than (1 +)^q. We conclude that t(n) = m for all sufficiently large n. This implies P_n(µ) has a single zero near each z_j for j = 1, . . . , m when n is sufficiently large.

The proof of the exponential attraction now proceeds exactly as in the last portion of the proof of Theorem 8.1.11 in [56] and we provide it here for completeness. For eachi∈ {1, . . . , m}, let{w_i,n}_n∈N be as above. We have just shown that this sequence has a unique tail. Suppose for contradiction that there existsi∈ {1, . . . , m}so that for all a >0 it is true that|wi,n−zi|> e^−an for infinitely many values ofn∈N. Then we can find a subsequencen(j) so that|w_i,n(j)−zi|^1/n(j)→1 asj→ ∞.

The above proof shows that all accumulation points of the zeros of the extremal polynomials are in Pch(µ). Therefore, Corollary 1.1.5 in [65] (whose proof only depends on the extremal property and not orthogonality) implies that

lim inf

j→∞ |p_n(j)(z;µ, q)|^1/n(j)≥exp (−U^ωG(z)) (3.2.11)

for allz in some punctured neighborhood ofz_i, and in particular, uniformly on some small circle C_i centered at z_i. It is then clear that we retain the uniformity on C_i in (3.2.11) if we replace p_n(j)(z;µ, q) by p_n(j)(z;µ, q)/(z−w_i,n(j)). Our above analysis implies that for all sufficiently large n∈N, the polynomialp_n(j)(z;µ, q)/(z−w_i,n(j)) has no zeros in a neighborhood ofzi, and all other zeros tend toG∪ {z`}<sub>`6=i</sub>. Therefore,p_n(j)(z;µ, q)/(z−w_i,n(j)) is free of zeros insideCi, and so the minimum principle implies

lim inf

j→∞

p_n(j)(z;µ, q) z−w_i,n(j)

1/n(j)

≥exp (−U^ωG(z)) (3.2.12)

uniformly for allz in some open set containingz_i. By our definition of the subsequence{n(j)}j∈N, we conclude that

lim inf

j→∞ |p_n(j)(z_i;µ, q)|^1/n(j)≥exp (−U^ωG(z_i))>1,

which is clearly impossible by the normalization of pn. This contradiction gives us the desired conclusion.

Remark 1. We can actually quantify the parameters δ and a in the statement of Corollary3.2.9.

The proof of Corollary 3.2.9 shows that we may take δ to be any positive real number so that {w:|w−zi| ≤δ} ∩supp(µ) ={zi}for eachi≤m. Furthermore, equation (3.2.12) shows that

lim sup

n→∞

|zi−wi,n|^1/n≤exp (U^ωG(zi))

for alli≤m.

Remark 2. Corollary3.2.9tells us that the polynomialPn(µ, q) has a single zero extremely close to z_i for each i∈ {1, . . . , m} and the remaining n−m zeros are placed so as to minimize the L^q(µ) norm with respect to a varying – yet converging – weight. It would be interesting to look at the measureµonD∪ {z₁, . . . , z_m}given bydµ=dA(z) +Pm

j=1δ_zj (wheredA(z) refers to area measure on the unit disk) and see if the results from [36] continue to hold in this case, where the polynomial weight would bey_∞(z) (see (3.2.5) above).

The upper bound in the proof of Theorem 3.2.2 came from Lemma 3.2.6, which applies to arbitrary finite measures (not just product measures). We can also state the lower bound used in the proof of Theorem3.2.2in a more general form.

Proposition 3.2.10. Let µ˜ be a measure on Gso that µ˜≥µandµis the push-forward (via ψ) of the measure w(re^iθ)_2π^dθdτ(r) where1∈supp(τ)and w∈L¹(_2π^dθ ⊗dτ(r)). Then

kPn(˜µ)k^q_µ˜≥ Z 1

0

r^nqexp Z 2π

0

log(w(re^iθ))dθ 2π

dτ(r).

Remark. The statement here is very general because we do not insist on any continuity of w.

Proof. By the inequality of the measures and the extremal property, we have kP_n(˜µ)k^q_µ˜≥ kP_n(˜µ)k^q_µ≥ kP_n(µ)k^q_µ,

so it suffices to put the desired bound onkPn(µ, q)k^q_Lq(µ). Let X ⊆[0,1] be the collection of all r so thatw(re^iθ)_2π^dθ is a Szeg˝o measure on∂D. The proposition is trivial unlessτ(X)>0. Therefore, we assume this is the case, and forr∈X we define

Sr(z) = exp

− 1 2qπ

Z 2π 0

log w(re^iθ)e^iθ+z e^iθ−zdθ

, |z|>1 and write

kPn(µ)k^q_µ≥ Z

X

r^nq|Sr(∞)|^qdτ(r) as in (3.2.8). This is the desired lower bound.

We conclude this section with an example showing how one can apply Lemma3.2.6to a region without analytic boundary.

Example. LetG be the region

z:|z³−1|<1 and assume q > 1. Notice thatG has capacity 1 since φ(z)³ = z³−1 (see Example 3.8 in [35]). Therefore, when n is a multiple of 3 we have F3m(z) = (z³−1)^m.

Figure 3.1: The regionGof the example.

Let τ be a probability measure on (0,1) with 1 ∈supp(τ). The region G can be decomposed

into level sets Ξ_rwhere

Ξ_r=

z:|z³−1|=r

and r runs from 0 to 1. Letνr be arc-length measure on each component of Ξr and let h(z) be a function that is continuous onG and is invariant under rotations by ^2π₃ so that φ_∗(hν₁) has Z3

symmetry as a measure on∂Das in Example 1.6.14 in [56]. Let us defineµby Z

G

f(z)dµ(z) = Z 1

0

Z

Ξ_r

f(z)h(z)dν_r(z)dτ(r).

Consider the measurehν1 on∂G. Ifm∈Nis fixed, then by the extremal property we have that for any choice of complex numbersa0, . . . , am−1 andam= 1

kP3n(hν₁, q)k^q_Lq(hν1)≤

m

X

j=0

a_jF_3(j+n−m)(z)

q

L^q(hν₁)

=

m

X

j=0

a_jφ(z)^3(j+n−m)

q

L^q(hν₁)

.

Following the proof of the upper bound in Theorem 7.1 in [17] we get

kP3n(hν1, q)k^q_Lq(hν1)≤ Z 2π

0

1 +

m

X

k=1

γke^3kiθ

q

dφ∗(hν1) (3.2.13)

for anym≤nand any choice of constantsγ₁, . . . , γ_m. The assumed Z3 symmetry of the measure implies that P_3m(z;φ_∗(hν₁), q) =R_m(z³) for some monic polynomialR_m of degreem (this follows from the uniqueness of the extremal polynomial in the case q > 1; see Example 1.6.14 in [56]).

Therefore, we can chooseγ₁, . . . , γ_mappropriately so that the right-hand side of (3.2.13) is equal to kP3m(φ_∗(hν1), q)k^q_φ

∗(hν₁). The reasoning of Lemma3.2.6then implies lim sup

n→∞

kP3n(hν1, q)k^q_Lq(hν₁)≤exp Z 2π

0

log (φ_∗(hν1)⁰(θ))dθ 2π

.

Now, as in Lemma 3.2.6, we calculate (for f ∈C(G))

cqm(τ)⁻¹ Z

G

f(z)|F3m(z)|^qdµ(z) =cqm(τ)⁻¹ Z 1

0

Z

Ξr

f(z)h(z)dνr(z)

r^qmdτ(r)

→ Z

Ξ₁

f(z)h(z)dν₁(z)

asm→ ∞. Therefore, the measures ^|F_c ^3m|q

qm(τ)dµconverge weakly tohdν1and the reasoning of Lemma 3.2.6implies

lim sup

n→∞

kP3n(z;µ, q)k^q_Lq(µ)

c_qn(τ) ≤exp Z 2π

0

log(φ_∗(hν1)⁰(θ))dθ 2π

.

2

In the next section, we explore more detailed asymptotic properties of the polynomialsP_n(z;µ, q) andp_n(z;µ, q).