# Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

Authors: Ana Portilla; Yamilet Quintana; José M. Rodríguez; Eva Tourís
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Let ℙ be the space of polynomials with complex coefficients endowed with a nondiagonal
Sobolev norm∥·∥W1,p(Vμ), where the matrix V and the measure μ constitute a p-admissible pair for
1≤p≤∞. In this paper we establish the zero location and asymptotic behavior of extremal polynomials
associated to ∥·∥W1,p(Vμ), stating hypothesis on the matrix V rather than on the diagonal matrix appearing in its unitary factorization.


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## Body

## 1. Introduction

In the last decades the asymptotic behavior of Sobolev orthogonal polynomials has been one of the main topics of interest to investigators in the field. In [1] the authors obtain the nth root asymptotic of Sobolev orthogonal polynomials when the zeros of these polynomials are contained in a compact set of the complex plane; however, the boundedness of the zeros of Sobolev orthogonal polynomials is an open problem, but as was stated in [2], it could be obtained as a consequence of the boundedness of the multiplication operator Mf(z)=zf(z). Thus, finding conditions to ensure the boundedness of M would provide important information about the crucial issue of determining the asymptotic behavior of Sobolev orthogonal polynomials (see, e.g., [3–13]). The more general result on this topic is [3, Theorem 8.1] which characterizes in terms of equivalent norms in Sobolev spaces the boundedness of M for the classical diagonal norm
(1)∥q∥WN,p(μ0,μ1,…,μN)∶=(∑k=0N‍∥q(k)∥Lp(μk))1/p
(see Theorem 3 below, which is [3, Theorem 8.1] in the case N=1). The rest of the above mentioned papers provides conditions that ensure the equivalence of norms in Sobolev spaces, and consequently, the boundedness of M.Results related to nondiagonal Sobolev norms may be found in [5, 6, 14–19]. Particularly, in [5, 6, 15, 18, 19] the authors establish the asymptotic behavior of orthogonal polynomials with respect to nondiagonal Sobolev inner products and the authors in [5] deal with the asymptotic behavior of extremal polynomials with respect to the following nondiagonal Sobolev norms.Letℙ be the space of polynomials with complex coefficients and let μ be a finite Borel positive measure with compact support S(μ) consisting of infinitely many points in the complex plane; let us consider the diagonal matrix Λ∶=diag(λj),j=0,…,N, with λj being positive μ-almost everywhere measurable functions, and U∶=(ujk),0≤j,k≤N, a matrix of measurable functions such that the matrix  U(x)=(ujk(x)), 0≤j,k≤N is unitary μ-almost everywhere. If  V∶=UΛU*, where U* denotes the transpose conjugate of U (note that then V is a positive definite matrix μ-almost everywhere), and 1≤p<∞ we define the Sobolev norm on the space of polynomials ℙ(2)∥q∥WN,p(Vμ)∶=(∫[(q,q′,…,q(N))*]p/2(q,q′,…,q(N))V2/p×(q,q′,…,q(N))*]p/2dμ)1/p∶=(∫[(q,q′,…,q(N))*]p/2(q,q′,…,q(N))UΛ2/pU*×(q,q′,…,q(N))*]p/2dμ)1/p.In [20, Chapter XIII] certain general conditions imposed on the matrix V are requested in order to guarantee the existence of an unitary representation with measurable entries.IfU is not the identity matrix μ-almost everywhere, then (2) defines a nondiagonal Sobolev norm in which the product of derivatives of different order appears. We say that qn(z)=zn+an-1zn-1+⋯+a1z+a0 is an nth monic extremal polynomial with respect to the norm (2) if
(3)∥qn∥WN,p(Vμ)=inf{∥q∥WN,p(Vμ):q(z)=zn+bn-1zn-1+⋯+b1z+b0,bj∈ℂ}.It is clear that there exists at least annth monic extremal polynomial. Furthermore, it is unique if 1<p<∞. If p=2, then the nth monic extremal polynomial is precisely the nth monic Sobolev orthogonal polynomial with respect to the inner product corresponding to (2).In [5, Theorem 1] the authors showed that the zeros of the polynomials in {qn}n≥0 are uniformly bounded in the complex plane, whenever there exists a constant C such that λj≤Cλk, μ-almost everywhere for 0≤j,k≤N. This property made possible to obtain the nth root asymptotic behavior of extremal polynomials (see [5, Theorems 2 and 6]). Although it is required compact support for μ, this is, certainly, a natural hypothesis: if S(μ) is not bounded, then we cannot expect to have zeros uniformly bounded, not even in the classical case (orthogonal polynomials in L2); see [21].TakingN=1, 1≤p≤2 and setting up hypothesis on the matrix V (see (4)) rather than on the diagonal matrix λ, the authors of [22] the following equivalent result to [5, Theorem  1].Theorem 1 (see [22, Theorem 4.3]).
Letγ be a finite union of rectifiable compact curves in the complex plane, μ a finite Borel measure with compact support S(μ)=γ, V a positive definite matrix μ-almost everywhere and
(4)V2/p=(apbpbp¯cp).
Assume that 1≤p≤2, (cpp/2dμ/ds)-1∈L1/(p-1)(γ), and the norms in W1,p((app/2+cpp/2)μ,cpp/2μ) and W1,p(app/2μ,cpp/2μ) are equivalent on ℙ. Let {qn}n≥0 be a sequence of extremal polynomials with respect to (2). Then the multiplication operator is bounded with the norm W1,p(Vμ) and the zeros of {qn}n≥0 lie in the bounded disk {z:|z|≤2∥M∥}.In this paper we improve Theorem1 in two directions: on the one hand, we enlarge the class of measures μ considered and, on the other hand, we prove our result for 1≤p<∞ (see Theorem 19). In order to describe the measures we will deal with, we introduce the definition of p-admissible pairs as follows: given 1≤p<∞, we say that the pair (V,μ) is p-admissible if μ is a finite Borel measure which can be written as μ=μ1+μ2, its support S(μ) is a compact subset of the complex plane which contains infinitely many points, and V is a positive definite matrix  μ-almost everywhere with |bp|2≤(1-ε0)apcp,   μ1-almost everywhere for some fixed 0<ε0≤1; the support S(μ2) is contained in a finite union of rectifiable compact curves γ with (cpp/2dμ2/ds)-1∈L1/(p-1)(γ) if γ≠∅, V2/p:=(apbpbp¯cp) and dμ2/ds is the Radon-Nykodim derivative of μ2 with respect to the Euclidean length in γ.We want to make three remarks about this definition. First of all, sinceV=(a2b2b2¯c2) is a positive definite matrix μ-almost everywhere, V2/p also has this property and hence |bp|2<apcp, μ-almost everywhere.In order to obtain(cpp/2dμ2/ds)-1∈L1/(p-1)(γ) the best choice for μ2 is the restriction of μ to γ.Note that the support ofμ is an arbitrary compact set: we just require that S(μ2) (the part of S(μ) in which V2/p is about to be a degenerated quadratic form, when |bp|2 is very close to apcp) is a union of curves.Therefore, with the results onp-admissible pairs we complement and improve the study started in [22], where the case μ=μ2 with 1≤p≤2 was considered.Another interesting property which could be studied is the asymptotic estimate for the behavior of extremal polynomials because, in this setting, there does not exist the usual three-term recurrence relation for orthogonal polynomials inL2 and this makes it really difficult to find an explicit expression for the extremal polynomial of degree n. In this regard, Theorems 22 and 23 deduce the asymptotic behavior of extremal polynomials as an application of Theorems 18 and 19. More precisely, we obtain the nth root and the zero counting measure asymptotic both of those polynomials and their derivatives to any order. The study of the nth root asymptotic is a classical problem in the theory of orthogonal polynomials; see for instance, [1, 2, 5, 23, 24].Furthermore, in Theorem23 we find the following asymptotic relation:
(5)limn→∞qn(j+1)(z)nqn(j)(z)=∫dωS(μ)(x)z-x
for any j≥0.The main idea of [5, 6, 22] and this paper is to compare nondiagonal and diagonal norms.When it comes to compare nondiagonal and diagonal norms, [25] is remarkable, since the authors show that symmetric Sobolev bilinear forms, like symmetric matrices, can be rewritten with a diagonal representation; unfortunately, the entries of these diagonal matrices are real measures, and we cannot use this representation since we need positive measures for the Sobolev norms.Finally, we would like to note that the central obstacle in order to generalize the results given in this paper and [22] to the case of more derivatives is that there are too many entries in the matrix V and just a few relations to control them (see Lemma 8 and notice that some limits appearing in that Lemma do not provide any new information). In that case we have just three entries (ap,bp,cp), but in the simple case of two derivatives (N=2) we have
(6)V∶=(a11a12a13a12¯a22a23a13¯a23¯a33),
and we would need to control six functions (a11,a12,a13,a22,a23,a33); in the general case with N derivatives, we would need to control (N+1)(N+2)/2 functions.The outline of the paper is as follows. In Section2 we provide some background and previous results on the multiplication operator and the location of zeros of extremal polynomials. We have devoted Section 3 to some technical lemmas in order to simplify the proof of Theorem 17 about the equivalence of norms; in fact, in these lemmas the hardest part of this proof is collected. In Section 4 we give the proof of that Theorem and in Section 5 we deduce some results on asymptotic of extremal polynomials.


## 2. Background and Previous Results

In what follows, given1≤p<∞ we define
(7)∥f∥W1,p(app/2μ,cpp/2μ)∶=(∫(app/2|f|p+cpp/2|f′|p)dμ)1/p,∥f∥W1,p(Dμ)∶=(∫(ap|f|2+cp|f′|2)p/2dμ)1/p,∥f∥W1,p(Vμ)∶=(∫(ap|f|2+cp|f′|22ℜ(bpff′¯))p/2dμ)1/p,
for every polynomial f.It is obviously much easier to deal with the norms∥·∥W1,p(app/2μ,cpp/2μ) and ∥·∥W1,p(Dμ) than with the one ∥·∥W1,p(Vμ). Therefore, one of our main goals is to provide weak hypotheses to guarantee the equivalence of these norms on the linear space of polynomials ℙ (see Section 4).In order to bound the zeros of polynomials, one of the most successful strategies has certainly been to bound the multiplication operator by the independent variableMf(z)=zf(z), where
(8)∥M∥:=sup{∥zq(z)∥W1,p(Vμ):∥q∥W1,p(Vμ)=1}.
Regarding this issue, the following result is known.Theorem 2 (see [5, Theorem 3]).
Letμ be a finite Borel measure in ℂ with compact support let and 1≤p<∞. Let {qn}n≥0 be a sequence of extremal polynomials with respect to (2). Then the zeros of {qn}n≥0 lie in the disk {z:|z|≤2∥M∥}.It is also known the following simple characterization of the boundedness ofM.Theorem 3 (see [3, Theorem 8.1]).
Letμ be a finite Borel measure in ℂ with compact support; α,β nonnegative measurable functions; and 1≤p<∞. Then the multiplication operator is bounded in W1,p(αμ,βμ) if and only if the following condition holds:
(9)thenormsinW1,p((α+β)μ,βμ)andW1,p(αμ,βμ)areequivalentonℙ.It is clear that if there exists a constantC such that β≤Cαμ-almost everywhere, then (9) holds. In [8, 13] some other very simple conditions implying (9) are shown.In what follows, we will fix ap-admissible pair (V,μ) with 1≤p<∞; then S(μ2) is contained in a finite union of rectifiable compact curves γ in the complex plane; each of these connected components of γ is not required to be either simple or closed.


## 3. Technical Lemmas

For the sake of clarity and readability, we have opted for proving all the technical lemmas in this section. This makes the proof of Theorem17 much more understandable.The following result is well known.Lemma 4.
Let us consider1≤α<∞. Then
(10)(x+y)α≤2α-1(xα+yα)foreveryx,y≥0.Lemma 5 (see [22, Lemma 3.1]).
Let us consider0<α≤1. Then  (1)
|y|α-|x|α≤|y-x|α for every x,y∈ℝ;(2)
2α-1(yα+xα)≤(y+x)α≤yα+xα for every x,y≥0.Lemma 6 (see [22, Lemma 3.2]).
Let{sn}n and {tn}n be two sequences of positive numbers. Then
(11)limn→∞2sntnsn2+tn2=1iiflimn→∞sntn=1.In what followsap,bp, and cp refer to the coefficients of the fixed matrix V2/p.Definition 7.
We say that{fn}n⊂ℙ is an extremal sequence for p if, for every n, ∥fn∥L∞(μ2)=1 and
(12)limn→∞∫‍|2bpfnfn′|p/2dμ2∫‍(ap|fn|2+cp|fn′|2)p/2dμ2=1.Lemma 8.
If1≤p<∞ and {fn}n is an extremal sequence for p, then
(13)limn→∞∫|bpfnfn′|p/2dμ2∫(apcp|fnfn′|)p/2dμ2=1,limn→∞∫(apcp|fnfn′|)p/2dμ2(∫(ap|fn|2)p/2dμ2)1/2(∫(cp|fn′|2)p/2dμ2)1/2=1,limn→∞2(∫(ap|fn|2)p/2dμ2)1/2(∫(cp|fn′|2)p/2dμ2)1/2∫((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2=1,limn→∞2p/2-1∫((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2∫(ap|fn|2+cp|fn′|2)p/2dμ2=1,limn→∞∫(ap|fn|2)p/2dμ2∫(cp|fn′|2)p/2dμ2=1.Proof.
The case1≤p≤2 is a consequence of [22, Lemmas 3.5 and 3.6]. We deal now with the case p>2. First note that we can rewrite limit (12) in Definition 7 as the limit of the following product:
(14)limn→∞∫|bpfnfn′|p/2dμ2∫(apcp|fnfn′|)p/2dμ2·∫(apcp|fnfn′|)p/2dμ2(∫(ap|fn|2)p/2dμ2)1/2(∫(cp|fn′|2)p/2dμ2)1/2·2(∫(ap|fn|2)p/2dμ2)1/2(∫(cp|fn′|2)p/2dμ2)1/2∫((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2·2p/2-1∫((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2∫(ap|fn|2+cp|fn′|2)p/2dμ2=1.
Since the limit of the product is 1, if we prove that the first, third, and fourth factors tend to 1 asn tends to infinity, then the limit of the second factor must also be 1.
So, our problem is reduced to show(15)limn→∞∫‍|bpfnfn′|p/2dμ2∫‍(apcp|fnfn′|)p/2dμ2=1,(16)limn→∞2(∫‍(ap|fn|2)p/2dμ2)1/2(∫‍(cp|fn′|2)p/2dμ2)1/2∫‍((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2=1,(17)limn→∞2p/2-1∫‍((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2∫‍(ap|fn|2+cp|fn′|2)p/2dμ2=1,(18)limn→∞∫‍(ap|fn|2)p/2dμ2∫‍(cp|fn′|2)p/2dμ2=1.
Again, we can rewrite the limit in the definition of extremal sequence as the limit of the following product:(19)limn→∞∫|bpfnfn′|p/2dμ2∫‍(apcp|fnfn′|)p/2dμ2·∫‍(2apcp|fnfn′|)p/2dμ2∫‍(ap|fn|2+cp|fn′|2)p/2dμ2=1.
The two factors above are nonnegative and less than or equal to 1 using, respectively, that|bp|2<apcpμ2-almost everywhere and 2xy≤x2+y2. Thus,
(20)limn→∞∫|bpfnfn′|p/2dμ2∫(apcp|fnfn′|)p/2dμ2=1=limn→∞∫(2apcp|fnfn′|)p/2dμ2∫(ap|fn|2+cp|fn′|2)p/2dμ2,
and (15) holds.
Givenε>0, for each n let us define the following sets:
(21)Fn,ε∶={z∈S(μ2):11+ε≤ap|fn|cp|fn′|≤1+ε},Fn,εc∶=S(μ2)∖Fn,ε.
Let us consider the strictly decreasing functionA(t):=2t/(t2+1) on [1,∞). If t≥1+ε, then A(t)≤A(1+ε)=:Cε<A(1)=1. Consequently, if x/y≥1+ε, then 2xy/(x2+y2)=A(x/y)≤Cε, and if x/y≤1/(1+ε), then y/x≥1+ε and 2xy/(x2+y2)=A(y/x)≤Cε. Therefore,
(22)∫Fn,εc‍(2apcp|fnfn′|)p/2dμ2∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2≤Cεp/2<1.
Using this fact and (20), we have
(23)1=limn→∞(((×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫Fn,ε‍(2apcp|fnfn′|)p/2dμ2)×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1)+(×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫Fn,εc‍(2apcp|fnfn′|)p/2dμ2)×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1))×(1+(×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ)-1))-1)≤(Cεp/2+liminfn→∞(×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1))×(1+liminfn→∞(×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)×(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1))-1.
If we assume thatliminfn→∞((∫Fn,ε(ap|fn|2+cp|fn′|2)p/2dμ2)(∫Fn,εc(ap|fn|2+cp|fn′|2)p/2dμ2)-1)=l<∞, then from the previous inequality we have 1≤((Cεp/2+l)/(1+l))<1, and this is a contradiction. Hence, limn→∞((∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1)=∞ and consequently,
(24)limn→∞∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2=0.
Since for eachn, we have
(25)∫Fn,εc‍(2apcp|fnfn′|)p/2dμ2∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2≤∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2,
then (24) implies that
(26)limn→∞∫Fn,εc‍(2apcp|fnfn′|)p/2dμ2∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2=0.
On the other hand, using (20) it is easy to deduce that
(27)limn→∞((((∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫Fn,ε‍(2apcp|fnfn′|)p/2dμ2)×(∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1)+((∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫Fn,εc‍(2apcp|fnfn′|)p/2dμ2)×(∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1)×(1+((∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2)×(∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1))-1)=1.
Consequently, (24), (26), and (27) give
(28)limn→∞∫Fn,ε‍(2apcp|fnfn′|)p/2dμ2∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2=1.
Furthermore, since(29)0≤1(1+ε)p/2∫Fn,ε‍(cp|fn′|2)p/2dμ2≤1∫Fn,ε‍(apcp|fnfn′|)p/2dμ2,
we obtain
(30)0≤∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2(1+ε)p/2∫Fn,ε‍(cp|fn′|2)p/2dμ2≤∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2∫Fn,ε‍(apcp|fnfn′|)p/2dμ2.
Therefore, (24), (28), and (30) give
(31)limn→∞∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2∫Fn,ε‍(cp|fn′|2)p/2dμ2=0.
Similar arguments allow us to show(32)limn→∞∫Fn,εc‍(ap|fn|2+cp|fn′|2)p/2dμ2∫Fn,ε‍(ap|fn|2)p/2dμ2=0.
From  (31) and (32) we obtain
(33)limn→∞∫Fn,εc‍(ap|fn|2)p/2dμ2∫Fn,ε‍(cp|fn′|2)p/2dμ2=0=limn→∞∫Fn,εc‍(cp|fn′|2)p/2dμ2∫Fn,ε‍(cp|fn′|2)p/2dμ2,(34)limn→∞∫Fn,εc‍(ap|fn|2)p/2dμ2∫Fn,ε‍(ap|fn|2)p/2dμ2=0=limn→∞∫Fn,εc‍(cp|fn′|2)p/2dμ2∫Fn,ε‍(ap|fn|2)p/2dμ2.
As a consequence of (33) we have
(35)limsupn→∞∫‍(ap|fn|2)p/2dμ2∫‍(cp|fn′|2)p/2dμ2=limsupn→∞((((∫Fn,ε‍(cp|fn′|2)p/2dμ2)-1(∫Fn,ε‍(ap|fn|2)p/2dμ2)×(∫Fn,ε‍(cp|fn′|2)p/2dμ2)-1)+((∫Fn,ε‍(cp|fn′|2)p/2dμ2)-1(∫Fn,εc‍(ap|fn|2)p/2dμ2)×(∫Fn,ε‍(cp|fn′|2)p/2dμ2)-1))×(1+((∫Fn,ε‍(cp|fn′|2)p/2dμ2)-1(∫Fn,εc‍(cp|fn′|2)p/2dμ2)×(∫Fn,ε‍(cp|fn′|2)p/2dμ2)-1))-1)=limsupn→∞∫Fn,ε‍(ap|fn|2)p/2dμ2∫Fn,ε‍(cp|fn′|2)p/2dμ2≤(1+ε)p.
In a similar way we obtain(36)1(1+ε)p≤liminfn→∞∫‍(ap|fn|2)p/2dμ2∫‍(cp|fn′|2)p/2dμ2.
Since these inequalities hold for every ε>0, we conclude that (18) holds. Applying now Lemma 6 we obtain (16).
Using Lemma4, (18), and (34) we obtain that for every ε,η>0 there exists N such that for every n≥N the following holds:
(37)1≤2p/2-1∫‍((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2∫‍(ap|fn|2+cp|fn′|2)p/2dμ2≤2p/2-1∫‍((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2∫Fn,ε‍(ap|fn|2+cp|fn′|2)p/2dμ2≤2p/2-1∫‍((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2(1+(1/(1+ε)2))p/2∫Fn,ε‍(ap|fn|2)p/2dμ2≤2p/2(1+η)∫‍(ap|fn|2)p/2dμ2(1+(1/(1+ε)2))p/2∫Fn,ε‍(ap|fn|2)p/2dμ2≤2p/2(1+η)2(1+(1/(1+ε)2))p/2·∫Fn,ε‍(ap|fn|2)p/2dμ2∫Fn,ε‍(ap|fn|2)p/2dμ2=2p/2(1+η)2(1+(1/(1+ε)2))p/2.
Then (17) follows from the previous inequalities, since ε,η>0 are arbitrary.
This completes the proof.Definition 9.
For each0<ε<1, we define the sets Aε and Aεc as
(38)Aε∶={z∈S(μ2):|bp|>(1-ε)apcp},Aεc∶=S(μ2)∖Aε.Lemma 10.
If1≤p<∞ and {fn}n is an extremal sequence for p and ε is small enough, then
(39)limn→∞∫Aε|bpfnfn′|p/2dμ2∫|bpfnfn′|p/2dμ2=1.Remark 11.
The statement of the lemma might seem strange, because we could have a prioriμ2(Aε)=0; however, the existence of the fundamental sequence implies μ2(Aε)>0.Proof.
If1≤p≤2, then the result follows from [22, Lemma 3.8]. For the case p>2 it suffices to follow the proof of [22, Lemma 3.8] applying Lemma 8 to conclude the result.Lemma 12.
If1≤p<∞, {fn}n is an extremal sequence for p and ε is small enough, then
(40)limn→∞∫Aε((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2∫((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2=1,limn→∞∫Aεc((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2∫Aε((ap|fn|2)p/2+(cp|fn′|2)p/2)dμ2=0.Proof.
If1≤p≤2, then the result follows from [22, Lemma 3.10]. For the case p>2 it suffices to follow the proof of [22, Lemma 3.10] applying Lemmas 8 and 10 to conclude the result.Lemma 13.
If1≤p<∞, {fn}n is an extremal sequence for p and ε is small enough, then
(41)limn→∞∫Aε(ap|fn|2)p/2dμ2∫(ap|fn|2)p/2dμ2=1,limn→∞∫Aε(cp|fn′|2)p/2dμ2∫(cp|fn′|2)p/2dμ2=1.Proof.
If1≤p≤2, then the result follows from [22, Lemma 3.11]. For the case p>2 it suffices to follow the proof of [22, Lemma 3.11] applying Lemmas 8, 10, and 12 to conclude the result.Lemma 14.
If1≤p<∞ and {fn}n is an extremal sequence for p, then for every ε>0 small enough with μ2(Aεc)>0 and for every t∈(0,1) there exists N such that infz∈Aεc|fn(z)|<t for every n≥N.Proof.
If1≤p≤2, then the result follows from [22, Lemma 3.12]. For the case p>2 it is sufficient to follow the proof of [22, Lemma 3.12] applying Lemma 13 to conclude the result.Definition 15.
Iff is a continuous function on γ, we define the oscillation of f on γ, and we denote it by osc(f), as
(42)osc(f)∶=supz,w∈γ|f(z)-f(w)|.Lemma 16 (see [22, Lemma 3.14]).
For1≤p<∞, let us assume that γ is connected and (cpp/2dμ2/ds)-1∈L1/(p-1)(γ), where dμ2/ds is the Radon-Nykodim derivative of μ2 with respect to the Euclidean length in γ. (According to one’s notation, if p=1 then 1/(p-1)=∞.) Then
(43)∫γ|f′|pcpp/2dμ2≥k·oscp(f),with1k=∥1(cpp/2(dμ2/ds))∥L1/(p-1)(γ),
for every polynomial f.


## 4. Equivalent Norms

Now we prove the announced result about the equivalence of norms for1≤p<∞.Theorem 17.
Let one consider1≤p<∞ and (V,μ) a p-admissible pair. Then the norms W1,p(app/2μ,cpp/2μ), W1,p(Dμ), and W1,p(Vμ) defined as in (3) are equivalent on the space of polynomials ℙ.Proof.
The equivalence of the two first norms is straightforward, by Lemmas4 and 5. We prove now the equivalence of the two last norms.
Let us prove that there exists a positive constantC:=C(V,μ,p) such that
(44)C∥f∥W1,p(Dμ)≤∥f∥W1,p(Vμ)≤2∥f∥W1,p(Dμ),foreveryf∈ℙ.
Let us prove first the second inequality∥f∥W1,p(Vμ)≤2∥f∥W1,p(Dμ).
Note that|2ℜ(bpff′¯)|≤|2bpff′|≤2apcp|ff′|≤ap|f|2+cp|f′|2; therefore, for every polynomial f it holds that
(45)∥f∥W1,p(Vμ)p=∫(ap|f|2+cp|f′|2+2ℜ(bpff′¯)|f′|2)p/2dμ≤2p/2∫(ap|f|2+cp|f′|2)p/2dμ=2p/2∥f∥W1,p(Dμ)p.
In order to prove the first inequality,C∥f∥W1,p(Dμ)≤∥f∥W1,p(Vμ), note that
(46)∥f∥W1,p(Vμ1)p=∫(ap|f|2+cp|f′|2+2ℜ(bpff′¯))p/2dμ1≥∫(ap|f|2+cp|f′|2-2|bpff′|))p/2dμ1≥∫(ap|f|2+cp|f′|2-21-ε0apcp|ff′|)p/2dμ1≥(1-1-ε0)p/2∫(ap|f|2+cp|f′|2)p/2dμ1=(1-1-ε0)p/2∥f∥W1,p(Dμ1)p.
Ifγ=∅ (i.e., μ=μ1), then we have finished the proof. Assume that γ≠∅ then we prove C∥f∥W1,p(Dμ2)≤∥f∥W1,p(Vμ2), seeking for a contradiction. It is clear that it suffices to prove it when γ is connected, that is, when γ is a rectifiable compact curve. Let us assume that there exists a sequence {fn}n⊂ℙ such that
(47)limn→∞∫‍(ap|fn|2+cp|fn′|2+2ℜ(bpfnfn′¯))p/2dμ2∫‍(ap|fn|2+cp|fn′|2)p/2dμ2=0.
If1≤p≤2, then [22, Lemma 3.1] (with α=p/2) gives
(48)∫‍(ap|fn|2+cp|fn′|2+2ℜ(bpfnfn′¯))p/2dμ2≥∫(ap|fn|2+cp|fn′|2-|2bpfnfn′|)p/2dμ2≥∫(ap|fn|2+cp|fn′|2)p/2dμ2-∫|2bpfnfn′|p/2dμ2.
This right-hand side of the inequality is positive, because|2bpff′|≤ap|f|2+cp|f′|2μ2-almost everywhere. This implies
(49)limn→∞((∫‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1(∫‍(ap|fn|2+cp|fn′|2)p/2dμ2-∫‍|2bpfnfn′|p/2dμ2)×(∫‍(ap|fn|2+cp|fn′|2)p/2dμ2)-1)=0,
and hence
(50)limn→∞∫‍|2bpfnfn′|p/2dμ2∫‍(ap|fn|2+cp|fn′|2)p/2dμ2=1.
Ifp>2, then
(51)(∫‍(ap|fn|2+cp|fn′|2+2ℜ(bpfnfn′¯))p/2dμ2)2/p≥(∫‍(ap|fn|2+cp|fn′|2)p/2dμ2)2/p-(∫‍|2bpfnfn′|p/2dμ2)2/p≥0,
since |2bpff′|≤ap|f|2+cp|f′|2μ2-almost everywhere. Therefore,
(52)limn→∞((×((∫‍(ap|fn|2+cp|fn′|2)p/2dμ2)2/p)-1(∫‍(ap|fn|2+cp|fn′|2)p/2dμ2)2/p-(∫‍|2bpfnfn′|p/2dμ2)2/p)×((∫‍(ap|fn|2+cp|fn′|2)p/2dμ2)2/p)-1)=0,
or, equivalently,
(53)limn→∞(∫‍|2bpfnfn′|p/2dμ2)2/p(∫‍(ap|fn|2+cp|fn′|2)p/2dμ2)2/p=1,
and (50) also holds for p>2.
Iffn is constant for some n, then ∫‍|2bpfnfn′|p/2dμ2=0; therefore, taking a subsequence if it is necessary, without loss of generality we can assume that fn is nonconstant and ∥fn∥L∞(μ2)=1 for every n. Then {fn}n is an extremal sequence for p. Applying Lemma 8,
(54)limn→∞∫‍(ap|fn|2)p/2dμ2∫‍(cp|fn′|2)p/2dμ2=1.
By Lemma14, there exists {zn}n⊂S(μ2) such that |fn(zn)|≤1/2 for every n≥N1. Now, taking into account that ∥fn∥L∞(μ2)=1 and that γ is connected, we can apply Lemma 16, and then
(55)∫‍|fn′|pcpp/2dμ2≥k·oscp(fn)≥k(∥fn∥L∞(μ2)-|fn(zn)|)p≥k(1-12)p=k2p>0
for every n≥N1, with 1/k=∥1/(cpp/2dμ2/ds)∥L1/(p-1)(γ).
Let us fixε small enough. On the one hand, by Lemma 13 it holds that
(56)∫(ap|fn|2)p/2dμ2≤2∫Aε(ap|fn|2)p/2dμ2≤2∥fn∥L∞(μ2)p∫Aε‍app/2dμ2=2∫Aεapp/2dμ2
for every n≥N2=N2(ε).
On the other hand, we have(57)limε→0+μ2(Aε)=limε→0+μ2({|bp|>(1-ε)apcp})=μ2({|bp|≥apcp})=0.
This implies
(58)limε→0+∫Aεapp/2dμ2=0.
Given any δ>0 there exists ε1 with ∫Aε1‍app/2dμ2<δ. Hence, ∫‍(ap|fn|2)p/2dμ2<2δ for every n≥N2(ε1). Therefore, limn→∞∫‍(ap|fn|2)p/2dμ2=0, which is a contradiction with (54) and (55).The following result is a direct consequence of Theorems3 and 17.Theorem 18.
Let one consider1≤p<∞ and (V,μ) a p-admissible pair. Then the multiplication operator is bounded in W1,p(Vμ) if and only if the following condition holds:
(59)thenormsinW1,p((app/2+cpp/2)μ,cpp/2μ)andW1,p(app/2μ,cpp/2μ)areequivalentonℙ.This latter theorem and Theorem2 give the following result.Theorem 19.
Let one consider1≤p<∞ and (V,μ) a p-admissible pair such that (59) takes place. Let {qn}n≥0 be a sequence of extremal polynomials with respect to (2). Then the multiplication operator is bounded and the zeros of {qn}n≥0 lie in the bounded disk {z:|z|≤2∥M∥}.In general, it is not difficult to check wether or not (59) holds. It is clear that if there exists a constant C such that cp≤Capμ-almost everywhere, then (59) holds. In [8, 13] some other very simple conditions implying (59) are shown.The following is a direct consequence of Theorem19.Corollary 20.
Let one consider1≤p<∞ and (V,μ) a p-admissible pair. Assume that cp≤Cap,μ-almost everywhere for some constant C. Let {qn}n≥0 be a sequence of extremal polynomials with respect to (2). Then the zeros of {qn}n≥0 are uniformly bounded in the complex plane.Finally, we have the following particular consequence for Sobolev orthogonal polynomials.Corollary 21.
Let(V,μ) be a 2-admissible pair. Assume that there exists a constant C such that c2≤Ca2,μ-almost everywhere. Let {qn}n≥0 be the sequence of Sobolev orthogonal polynomials with respect to Vμ. Then the zeros of the polynomials in {qn}n≥0 are uniformly bounded in the complex plane.


## 5. Asymptotic of Extremal Polynomials

We start this section by setting some notation. LetQn, ∥·∥L2(μ), cap(S(μ)), and ωS(μ) denote, respectively, the nth monic orthogonal polynomial with respect to L2(μ), the usual norm in the space L2(μ), the logarithmic capacity of S(μ), and the equilibrium measure of S(μ). Furthermore, in order to analyze the asymptotic behavior for extremal polynomials we will use a special class of measures, “regular measures,” denoted by Reg and defined in [24]. In that work, the authors proved (see Theorem 3.1.1) that, for measures supported on a compact set of the complex plane, μ∈Reg if and only if
(60)limn→∞∥Qn∥L2(μ)1/n=cap(S(μ)).Finally, ifz1,z2,…,zn denote the zeros, repeated according to their multiplicity, of a polynomial q whose degree is exactly n, and δzj is the Dirac measure with mass one at the point zj, the expression
(61)ν(q):=1n∑j=1n‍δzj
defines the normalized zero counting measure of q.We can already state the first result in this section.Theorem 22.
Let one consider1≤p<∞, (V,μ) a p-admissible pair and {qn}n≥0 the sequence of extremal polynomials with respect to ∥·∥W1,p(Vμ). Assume that the following conditions hold:  (i)
app/2μ∈Reg;(ii)
S(μ) is regular with respect to the Dirichlet problem;(iii)
condition (59) takes place. Then,
(62)limn→∞∥qn(j)∥S(μ)1/n=cap(S(μ)),j≥0.
Furthermore, if the complement of S(μ) is connected, then
(63)limn→∞ν(qn(j))=ωS(μ),j≥0.
in the weak star topology of measures.Proof.
Note that, in our context, the hypothesis removed with respect to [5, Theorem 2] is equivalent to the following two facts: on the one hand, the multiplication operator is bounded (see Theorem 3), and on the other hand, the norms of W1,p(app/2μ,cpp/2μ) and W1,p(Vμ) defined as in (3) are equivalent (see Theorem 18). With this in mind, we just need to follow the proof of [5, Theorem 2] to conclude the result.In the following theorem, we usegΩ(z;∞) to denote the Green's function for Ω with logarithmic singularity at ∞, where Ω is the unbounded component of the complement of S(μ). Notice that, if S(μ) is regular with respect to the Dirichlet problem, then gΩ(z;∞) is continuous up to the boundary and it can be extended continuously to all ℂ, with value zero on ℂ∖Ω.Theorem 2323.
Let one consider1≤p<∞, (V,μ) a p-admissible pair and {qn}n≥0 the sequence of extremal polynomials with respect to ∥·∥W1,p(Vμ). Assume that the following conditions hold:  (i)
app/2μ∈Reg;(ii)
S(μ) is regular with respect to the Dirichlet problem;(iii)
condition (59) takes place. Then, for each j≥0,
(64)limsupn→∞|qn(j)(z)|1/n≤cap(S(μ))egΩ(z;∞),
uniformly on compact subsets of ℂ. Furthermore, for each j≥0,
(65)limn→∞|qn(j)(z)|1/n=cap(S(μ))egΩ(z;∞),
uniformly on each compact subset of {z:|z|>2∥M∥}∩Ω. Finally, if the complement of S(μ) is connected, one has equality in (64) for all z∈ℂ, except for a set of capacity zero, S(ωS(μ))⊂{z:|z|≤2∥M∥} and
(66)limn→∞qn(j+1)(z)nqn(j)(z)=∫‍dωS(μ)(x)z-x
uniformly on each compact subset of {z:|z|>2∥M∥}.Proof.
Note that, in our context, the multiplication operator is bounded (see Theorem3) and the norms of W1,p(app/2μ,cpp/2μ) and W1,p(Vμ) defined as in (3) are equivalent (see Theorem 18). This is the crucial fact in the proof of this theorem; once we know this, we just need to follow the proof given in [5, Theorem 6] point by point to conclude the result.



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*Source: 628031-2013-04-09.xml*