Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

Abstract

Let ℙ be the space of polynomials with complex coefficients endowed with a nondiagonal Sobolev norm ‖⋅‖_𝑊^1,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 ‖⋅‖_𝑊^1,𝑝_(𝑉𝜇), stating hypothesis on the matrix 𝑉 rather than on the diagonal matrix appearing in its unitary factorization.

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 𝑞_{𝑊𝑁,𝑝(𝜇0,𝜇1,…,𝜇𝑁)}∶=^𝑁_𝑘=0𝑞^(𝑘)_{𝐿𝑝(𝜇𝑘)}^1/𝑝 (1) (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.

Let ℙ 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(𝜆_𝑗),𝑗=0,…,𝑁, with 𝜆_𝑗 being positive 𝜇-almost everywhere measurable functions, and 𝑈∶=(𝑢_𝑗𝑘),0≤𝑗,𝑘≤𝑁, a matrix of measurable functions such that the matrix  𝑈(𝑥)=(𝑢_𝑗𝑘(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 ℙ𝑞_{𝑊𝑁,𝑝(𝑉𝜇)}∶=𝑞,𝑞^′,…,𝑞^(𝑁)𝑉^2/𝑝×𝑞,𝑞^′,…,𝑞^{(𝑁)∗𝑝/2}𝑑𝜇^1/𝑝∶=𝑞,𝑞^′,…,𝑞^(𝑁)𝑈Λ^2/𝑝𝑈^∗×𝑞,𝑞^′,…,𝑞^{(𝑁)∗𝑝/2}𝑑𝜇^1/𝑝. (2)

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.

If 𝑈 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 𝑞_𝑛(𝑧)=𝑧^𝑛+𝑎_𝑛−1𝑧^𝑛−1+⋯+𝑎₁𝑧+𝑎₀ is an 𝑛th monic extremal polynomial with respect to the norm (2) if 𝑞_{𝑛𝑊𝑁,𝑝(𝑉𝜇)}=inf𝑞_{𝑊𝑁,𝑝(𝑉𝜇)}∶𝑞(𝑧)=𝑧^𝑛+𝑏_𝑛−1𝑧^𝑛−1+⋯+𝑏₁𝑧+𝑏₀,𝑏_𝑗∈ℂ. (3)

It is clear that there exists at least an 𝑛th 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 {𝑞_𝑛}_𝑛≥0 are uniformly bounded in the complex plane, whenever there exists a constant 𝐶 such that 𝜆_𝑗≤𝐶𝜆_𝑘, 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].

Taking 𝑁=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]). Let 𝛾 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 𝑉^2/𝑝=𝑎_𝑝𝑏_𝑝𝑏_𝑝𝑐_𝑝. (4) Assume that 1≤𝑝≤2, (𝑐_𝑝^𝑝/2𝑑𝜇/𝑑𝑠)⁻¹∈𝐿^1/(𝑝−1)(𝛾), and the norms in 𝑊^1,𝑝((𝑎_𝑝^𝑝/2+𝑐_𝑝^𝑝/2)𝜇,𝑐_𝑝^𝑝/2𝜇) and 𝑊^1,𝑝(𝑎_𝑝^𝑝/2𝜇,𝑐_𝑝^𝑝/2𝜇) are equivalent on ℙ. Let {𝑞_𝑛}_𝑛≥0 be a sequence of extremal polynomials with respect to (2). Then the multiplication operator is bounded with the norm 𝑊^1,𝑝(𝑉𝜇) and the zeros of {𝑞_𝑛}_𝑛≥0 lie in the bounded disk {𝑧∶|𝑧|≤2‖𝑀‖}.

In this paper we improve Theorem 1 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 |𝑏_𝑝|²≤(1−𝜀₀)𝑎_𝑝𝑐_𝑝,   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 (𝑐_𝑝^𝑝/2𝑑𝜇₂/𝑑𝑠)⁻¹∈𝐿^1/(𝑝−1)(𝛾) if 𝛾≠∅, 𝑉^2/𝑝∶=^{𝑎𝑝𝑏𝑝}_{𝑏𝑝𝑐𝑝} 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, since 𝑉=^𝑎2𝑏2_𝑏2𝑐2 is a positive definite matrix 𝜇-almost everywhere, 𝑉^2/𝑝 also has this property and hence |𝑏_𝑝|²<𝑎_𝑝𝑐_𝑝, 𝜇-almost everywhere.

In order to obtain (𝑐_𝑝^𝑝/2𝑑𝜇₂/𝑑𝑠)⁻¹∈𝐿^1/(𝑝−1)(𝛾) the best choice for 𝜇₂ is the restriction of 𝜇 to 𝛾.

Note that the support of 𝜇 is an arbitrary compact set: we just require that 𝑆(𝜇₂) (the part of 𝑆(𝜇) in which 𝑉^2/𝑝 is about to be a degenerated quadratic form, when |𝑏_𝑝|² is very close to 𝑎_𝑝𝑐_𝑝) is a union of curves.

Therefore, with the results on 𝑝-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 in 𝐿² 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 Theorem 23 we find the following asymptotic relation: lim_𝑛→∞𝑞_𝑛^(𝑗+1)(𝑧)𝑛𝑞_𝑛^(𝑗)(𝑧)=𝑑𝜔_𝑆(𝜇)(𝑥)𝑧−𝑥 (5) 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 (𝑎_𝑝,𝑏_𝑝,𝑐_𝑝), but in the simple case of two derivatives (𝑁=2) we have 𝑎𝑉∶=¹¹𝑎₁₂𝑎₁₃𝑎₁₂𝑎₂₂𝑎₂₃𝑎₁₃𝑎₂₃𝑎₃₃, (6) 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 Section 2 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, given 1≤𝑝<∞ we define 𝑓_{𝑊1,𝑝(𝑎𝑝}^𝑝/2_{𝜇,𝑐𝑝}^𝑝/2_𝜇)∶=𝑎_𝑝^𝑝/2𝑓^𝑝+𝑐_𝑝^𝑝/2𝑓^′𝑝𝑑𝜇^1/𝑝,𝑓_{𝑊1,𝑝(𝐷𝜇)}∶=𝑎_𝑝𝑓²+𝑐_𝑝𝑓^′2𝑝/2𝑑𝜇^1/𝑝,𝑓_{𝑊1,𝑝(𝑉𝜇)}∶=𝑎_𝑝𝑓²+𝑐_𝑝𝑓^′22ℜ𝑏_𝑝𝑓𝑓^′𝑝/2𝑑𝜇^1/𝑝, (7) for every polynomial 𝑓.

It is obviously much easier to deal with the norms ‖⋅‖_{𝑊1,𝑝(𝑎𝑝}^𝑝/2_{𝜇,𝑐𝑝}^𝑝/2_𝜇) and ‖⋅‖_𝑊^1,𝑝_(𝐷𝜇) than with the one ‖⋅‖_𝑊^1,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 variable 𝑀𝑓(𝑧)=𝑧𝑓(𝑧), where ‖𝑀‖∶=sup𝑧𝑞(𝑧)_{𝑊1,𝑝(𝑉𝜇)}∶𝑞_{𝑊1,𝑝(𝑉𝜇)}=1. (8) Regarding this issue, the following result is known.

Theorem 2 (see [5, Theorem 3]). Let 𝜇 be a finite Borel measure in ℂ with compact support let and 1≤𝑝<∞. Let {𝑞_𝑛}_𝑛≥0 be a sequence of extremal polynomials with respect to (2). Then the zeros of {𝑞_𝑛}_𝑛≥0 lie in the disk {𝑧∶|𝑧|≤2‖𝑀‖}.

It is also known the following simple characterization of the boundedness of 𝑀.

Theorem 3 (see [3, Theorem 8.1]). Let 𝜇 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 𝑊^1,𝑝(𝛼𝜇,𝛽𝜇) if and only if the following condition holds: thenormsin𝑊^1,𝑝𝛼+𝛽𝜇,𝛽𝜇and𝑊^1,𝑝𝛼𝜇,𝛽𝜇areequivalentonℙ. (9)

It is clear that if there exists a constant 𝐶 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 a 𝑝-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 Theorem 17 much more understandable.

The following result is well known.

Lemma 4. Let us consider 1≤𝛼<∞. Then 𝑥+𝑦^𝛼≤2^𝛼−1𝑥^𝛼+𝑦^𝛼forevery𝑥,𝑦≥0. (10)

Lemma 5 (see [22, Lemma 3.1]). Let us consider 0<𝛼≤1. Then (1)|𝑦|^𝛼−|𝑥|^𝛼≤|𝑦−𝑥|^𝛼 for every 𝑥,𝑦∈ℝ; (2)2^𝛼−1(𝑦^𝛼+𝑥^𝛼)≤(𝑦+𝑥)^𝛼≤𝑦^𝛼+𝑥^𝛼 for every 𝑥,𝑦≥0.

Lemma 6 (see [22, Lemma 3.2]). Let {𝑠_𝑛}_𝑛 and {𝑡_𝑛}_𝑛 be two sequences of positive numbers. Then lim_𝑛→∞2𝑠_𝑛𝑡_𝑛𝑠²_𝑛+𝑡²_𝑛=1iiflim_𝑛→∞𝑠_𝑛𝑡_𝑛=1. (11)

In what follows 𝑎_𝑝,𝑏_𝑝, and 𝑐_𝑝 refer to the coefficients of the fixed matrix 𝑉^2/𝑝.

Definition 7. We say that {𝑓_𝑛}_𝑛⊂ℙ is an extremal sequence for 𝑝 if, for every 𝑛, ‖𝑓_𝑛‖_𝐿^∞_(𝜇2)=1 and lim_𝑛→∞∫2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (12)

Lemma 8. If 1≤𝑝<∞ and {𝑓_𝑛}_𝑛 is an extremal sequence for 𝑝, then lim_𝑛→∞∫𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂=1,lim_𝑛→∞∫𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂^1/2∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^1/2=1,lim_𝑛→∞2∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂^1/2∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^1/2∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1,lim_𝑛→∞2^𝑝/2−1∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1,lim_𝑛→∞∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (13)

Proof. The case 1≤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: lim_𝑛→∞∫𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂⋅∫𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂^1/2∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^1/2⋅2∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂^1/2∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^1/2∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⋅2^𝑝/2−1∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (14)
Since the limit of the product is 1, if we prove that the first, third, and fourth factors tend to 1 as 𝑛 tends to infinity, then the limit of the second factor must also be 1.
So, our problem is reduced to show lim_𝑛→∞∫𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂=1,lim_𝑛→∞2∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂^1/2∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^1/2∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1,lim_𝑛→∞2^𝑝/2−1∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1,lim_𝑛→∞∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (15)(16)(17)(18)
Again, we can rewrite the limit in the definition of extremal sequence as the limit of the following product: lim_𝑛→∞∫𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂⋅∫2𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (19)
The two factors above are nonnegative and less than or equal to 1 using, respectively, that |𝑏_𝑝|²<𝑎_𝑝𝑐_𝑝𝜇₂-almost everywhere and 2𝑥𝑦≤𝑥²+𝑦². Thus, lim_𝑛→∞∫𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂=1=lim_𝑛→∞∫2𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂, (20) and (15) holds.
Given 𝜀>0, for each 𝑛 let us define the following sets: 𝐹_𝑛,𝜀∶=𝑧∈𝑆𝜇₂∶1≤1+𝜀𝑎_𝑝𝑓_𝑛𝑐_𝑝𝑓^′_𝑛,𝐹≤1+𝜀_{𝑐𝑛,𝜀}∶=𝑆𝜇₂⧵𝐹_𝑛,𝜀. (21)
Let us consider the strictly decreasing function 𝐴(𝑡)∶=2𝑡/(𝑡²+1) on [1,∞). If 𝑡≥1+𝜀, then 𝐴(𝑡)≤𝐴(1+𝜀)=∶𝐶_𝜀<𝐴(1)=1. Consequently, if 𝑥/𝑦≥1+𝜀, then 2𝑥𝑦/(𝑥²+𝑦²)=𝐴(𝑥/𝑦)≤𝐶_𝜀, and if 𝑥/𝑦≤1/(1+𝜀), then 𝑦/𝑥≥1+𝜀 and 2𝑥𝑦/(𝑥²+𝑦²)=𝐴(𝑦/𝑥)≤𝐶_𝜀. Therefore, ∫_{𝐹𝑐𝑛,𝜀}2𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂≤𝐶_𝜀^𝑝/2<1. (22)
Using this fact and (20), we have 1=lim_𝑛→∞∫_{𝐹𝑛,𝜀}2𝑎_𝑝c_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂×∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹+∫_{𝐹𝑐𝑛,𝜀}2𝑎_𝑝c_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂×∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹×1+∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂×∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇⁻¹⁻¹≤𝐶_𝜀^𝑝/2+liminf_𝑛→∞∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂×∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹×1+liminf_𝑛→∞∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂×∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹⁻¹. (23)
If we assume that liminf_𝑛→∞((∫_{𝐹𝑛,𝜀}(𝑎_𝑝|𝑓_𝑛|²+𝑐_𝑝|𝑓^′_𝑛|²)^𝑝/2𝑑𝜇₂)(∫_{𝐹𝑐𝑛,𝜀}(𝑎_𝑝|𝑓_𝑛|²+𝑐_𝑝|𝑓^′_𝑛|²)^𝑝/2𝑑𝜇₂)⁻¹)=𝑙<∞, then from the previous inequality we have 1≤((𝐶_𝜀^𝑝/2+𝑙)/(1+𝑙))<1, and this is a contradiction. Hence, lim_𝑛→∞((∫_{𝐹𝑛,𝜀}(𝑎_𝑝|𝑓_𝑛|²+𝑐_𝑝|𝑓^′_𝑛|²)^𝑝/2𝑑𝜇₂)(∫_{𝐹𝑐𝑛,𝜀}(𝑎_𝑝|𝑓_𝑛|²+𝑐_𝑝|𝑓^′_𝑛|²)^𝑝/2𝑑𝜇₂)⁻¹)=∞ and consequently, lim_𝑛→∞∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=0. (24)
Since for each 𝑛, we have ∫_{𝐹𝑐𝑛,𝜀}2𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂≤∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂, (25) then (24) implies that lim_𝑛→∞∫_{𝐹𝑐𝑛,𝜀}2𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=0. (26)
On the other hand, using (20) it is easy to deduce that lim_𝑛→∞∫_{𝐹𝑛,𝜀}2𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂×∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹+∫_{𝐹𝑐𝑛,𝜀}2𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂×∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹×1+∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂×∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹⁻¹=1. (27)
Consequently, (24), (26), and (27) give lim_𝑛→∞∫_{𝐹𝑛,𝜀}2𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (28)
Furthermore, since 10≤(1+𝜀)^𝑝/2∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂≤1∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂, (29) we obtain ∫0≤^{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂(1+𝜀)^𝑝/2∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂≤∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑐_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂. (30)
Therefore, (24), (28), and (30) give lim_𝑛→∞∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=0. (31)
Similar arguments allow us to show lim_𝑛→∞∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂=0. (32)
From (31) and (32) we obtain lim_𝑛→∞∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=0=lim_𝑛→∞∫_{𝐹𝑐𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂,lim_𝑛→∞∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂=0=lim_𝑛→∞∫_{𝐹𝑐𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂. (33)(34)
As a consequence of (33) we have limsup_𝑛→∞∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=limsup_𝑛→∞∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂×∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹+∫_{𝐹𝑐𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂×∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹×1+∫_{𝐹𝑐𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂×∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹⁻¹=limsup_𝑛→∞∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂≤(1+𝜀)^𝑝. (35)
In a similar way we obtain 1(1+𝜀)^𝑝≤liminf_𝑛→∞∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂. (36) Since these inequalities hold for every 𝜀>0, we conclude that (18) holds. Applying now Lemma 6 we obtain (16).
Using Lemma 4, (18), and (34) we obtain that for every 𝜀,𝜂>0 there exists 𝑁 such that for every 𝑛≥𝑁 the following holds: 21≤^𝑝/2−1∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂≤2^𝑝/2−1∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂≤2^𝑝/2−1∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂1+1/(1+𝜀)^2𝑝/2∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂≤2^𝑝/21+𝜂∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂1+1/(1+𝜀)^2𝑝/2∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂≤2^𝑝/21+𝜂²1+1/(1+𝜀)^2𝑝/2⋅∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫_{𝐹𝑛,𝜀}𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂=2^𝑝/21+𝜂²1+1/(1+𝜀)^2𝑝/2. (37)
Then (17) follows from the previous inequalities, since 𝜀,𝜂>0 are arbitrary.
This completes the proof.

Definition 9. For each 0<𝜀<1, we define the sets 𝐴_𝜀 and 𝐴^𝑐_𝜀 as 𝐴_𝜀∶=𝑧∈𝑆𝜇₂∶𝑏_𝑝>(1−𝜀)𝑎_𝑝𝑐_𝑝,𝐴^𝑐_𝜀∶=𝑆𝜇₂⧵𝐴_𝜀. (38)

Lemma 10. If 1≤𝑝<∞ and {𝑓_𝑛}_𝑛 is an extremal sequence for 𝑝 and 𝜀 is small enough, then lim_𝑛→∞∫_𝐴𝜀𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂=1. (39)

Remark 11. The statement of the lemma might seem strange, because we could have a priori 𝜇₂(𝐴_𝜀)=0; however, the existence of the fundamental sequence implies 𝜇₂(𝐴_𝜀)>0.

Proof. If 1≤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. If 1≤𝑝<∞, {𝑓_𝑛}_𝑛 is an extremal sequence for 𝑝 and 𝜀 is small enough, then lim_𝑛→∞∫_𝐴𝜀𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1,lim_𝑛→∞∫_𝐴𝑐𝜀𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫_𝐴𝜀𝑎_𝑝𝑓_𝑛^2𝑝/2+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=0. (40)

Proof. If 1≤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. If 1≤𝑝<∞, {𝑓_𝑛}_𝑛 is an extremal sequence for 𝑝 and 𝜀 is small enough, then 𝑙𝑖𝑚_𝑛→∞∫_𝐴𝜀𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂=1,lim_𝑛→∞∫_𝐴𝜀𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (41)

Proof. If 1≤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. If 1≤𝑝<∞ and {𝑓_𝑛}_𝑛 is an extremal sequence for 𝑝, then for every 𝜀>0 small enough with 𝜇₂(𝐴^𝑐_𝜀)>0 and for every 𝑡∈(0,1) there exists 𝑁 such that inf_𝑧∈𝐴^𝑐_𝜀|𝑓_𝑛(𝑧)|<𝑡 for every 𝑛≥𝑁.

Proof. If 1≤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. If 𝑓 is a continuous function on 𝛾, we define the oscillation of 𝑓 on 𝛾, and we denote it by osc(𝑓), as osc𝑓∶=sup_{𝑧,𝑤∈𝛾}𝑓(𝑧)−𝑓(𝑤). (42)

Lemma 16 (see [22, Lemma 3.14]). For 1≤𝑝<∞, let us assume that 𝛾 is connected and (𝑐_𝑝^𝑝/2𝑑𝜇₂/𝑑𝑠)⁻¹∈𝐿^1/(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 _𝛾𝑓^′𝑝𝑐_𝑝^𝑝/2𝑑𝜇₂≥𝑘⋅𝑜𝑠𝑐^𝑝𝑓,1with𝑘=1𝑐_𝑝^𝑝/2𝑑𝜇₂/𝑑𝑠_{𝐿1/(𝑝−1)(𝛾)}, (43) for every polynomial 𝑓.

4. Equivalent Norms

Now we prove the announced result about the equivalence of norms for 1≤𝑝<∞.

Theorem 17. Let one consider 1≤𝑝<∞ and (𝑉,𝜇) a 𝑝-admissible pair. Then the norms 𝑊^1,𝑝(𝑎_𝑝^𝑝/2𝜇,𝑐_𝑝^𝑝/2𝜇), 𝑊^1,𝑝(𝐷𝜇), and 𝑊^1,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 Lemmas 4 and 5. We prove now the equivalence of the two last norms.
Let us prove that there exists a positive constant 𝐶∶=𝐶(𝑉,𝜇,𝑝) such that 𝐶𝑓_{𝑊1,𝑝(𝐷𝜇)}≤𝑓_{𝑊1,𝑝(𝑉𝜇)}≤√2𝑓_{𝑊1,𝑝(𝐷𝜇)},forevery𝑓∈ℙ. (44)
Let us prove first the second inequality ‖𝑓‖_{𝑊1,𝑝(𝑉𝜇)}≤√2‖𝑓‖_{𝑊1,𝑝(𝐷𝜇)}.
Note that |2ℜ(𝑏_𝑝𝑓𝑓^′)|≤|2𝑏_𝑝𝑓𝑓^′|≤2𝑎_𝑝𝑐_𝑝|𝑓𝑓^′|≤𝑎_𝑝|𝑓|²+𝑐_𝑝|𝑓^′|²; therefore, for every polynomial 𝑓 it holds that 𝑓^𝑝_{𝑊1,𝑝𝑉𝜇}=𝑎_𝑝𝑓²+𝑐_𝑝𝑓^′2+2ℜ𝑏_𝑝𝑓𝑓^′𝑝/2𝑑𝜇≤2^𝑝/2𝑎_𝑝𝑓²+𝑐_𝑝𝑓^′2𝑝/2𝑑𝜇=2^𝑝/2𝑓^𝑝_{𝑊1,𝑝(𝐷𝜇)}. (45)
In order to prove the first inequality, 𝐶‖𝑓‖_{𝑊1,𝑝(𝐷𝜇)}≤‖𝑓‖_{𝑊1,𝑝(𝑉𝜇)}, note that 𝑓^𝑝_{𝑊1,𝑝(𝑉𝜇1)}=𝑎_𝑝𝑓²+𝑐_𝑝𝑓^′2+2ℜ𝑏_𝑝𝑓𝑓^′𝑝/2𝑑𝜇₁≥𝑎_𝑝𝑓²+𝑐_𝑝𝑓^′2−2𝑏_𝑝𝑓𝑓^′𝑝/2𝑑𝜇₁≥𝑎_𝑝𝑓²+𝑐_𝑝𝑓^′2−21−𝜀₀𝑎_𝑝𝑐_𝑝𝑓𝑓^′𝑝/2𝑑𝜇₁≥1−1−𝜀₀^𝑝/2𝑎_𝑝𝑓²+𝑐_𝑝𝑓^′2𝑝/2𝑑𝜇₁=1−1−𝜀₀^𝑝/2𝑓^𝑝_{𝑊1,𝑝𝐷𝜇1}. (46)
If 𝛾=∅ (i.e., 𝜇=𝜇₁), then we have finished the proof. Assume that 𝛾≠∅ then we prove 𝐶‖𝑓‖_𝑊^1,𝑝_(𝐷𝜇2)≤‖𝑓‖_𝑊^1,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 {𝑓_𝑛}_𝑛⊂ℙ such that lim_𝑛→∞∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛²+2ℜ𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=0. (47)
If 1≤𝑝≤2, then [22, Lemma 3.1] (with 𝛼=𝑝/2) gives 𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛²+2ℜ𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂≥𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛²−2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂≥𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂−2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂. (48)
This right-hand side of the inequality is positive, because |2𝑏_𝑝𝑓𝑓^′|≤𝑎_𝑝|𝑓|²+𝑐_𝑝|𝑓^′|²𝜇₂-almost everywhere. This implies lim_𝑛→∞𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂−2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂×𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂⁻¹=0, (49) and hence lim_𝑛→∞∫2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (50)
If 𝑝>2, then 𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛²+2ℜ𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂^2/𝑝≥𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^2/𝑝−2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂^2/𝑝≥0, (51) since |2𝑏_𝑝𝑓𝑓^′|≤𝑎_𝑝|𝑓|²+𝑐_𝑝|𝑓^′|²𝜇₂-almost everywhere. Therefore, lim_𝑛→∞𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^2/𝑝−2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂^2/𝑝×𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^2/𝑝−1=0, (52) or, equivalently, lim_𝑛→∞∫2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛^𝑝/2𝑑𝜇₂^2/𝑝∫𝑎_𝑝𝑓_𝑛²+𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂^2/𝑝=1, (53) and (50) also holds for 𝑝>2.
If 𝑓_𝑛 is constant for some 𝑛, then ∫|2𝑏_𝑝𝑓_𝑛𝑓^′_𝑛|^𝑝/2𝑑𝜇₂=0; therefore, taking a subsequence if it is necessary, without loss of generality we can assume that 𝑓_𝑛 is nonconstant and ‖𝑓_𝑛‖_{𝐿∞(𝜇2)}=1 for every 𝑛. Then {𝑓_𝑛}_𝑛 is an extremal sequence for 𝑝. Applying Lemma 8, lim_𝑛→∞∫𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂∫𝑐_𝑝𝑓^′_𝑛^2𝑝/2𝑑𝜇₂=1. (54)
By Lemma 14, there exists {𝑧_𝑛}_𝑛⊂𝑆(𝜇₂) such that |𝑓_𝑛(𝑧_𝑛)|≤1/2 for every 𝑛≥𝑁₁. Now, taking into account that ‖𝑓_𝑛‖_{𝐿∞(𝜇2)}=1 and that 𝛾 is connected, we can apply Lemma 16, and then 𝑓^′_𝑛^𝑝𝑐_𝑝^𝑝/2𝑑𝜇₂≥𝑘⋅osc^𝑝𝑓_𝑛≥𝑘𝑓_{𝑛𝐿∞(𝜇2)}−𝑓_𝑛𝑧_𝑛^𝑝1≥𝑘1−2^𝑝=𝑘2^𝑝>0 (55) for every 𝑛≥𝑁₁, with 1/𝑘=‖1/(𝑐_𝑝^𝑝/2𝑑𝜇₂/𝑑𝑠)‖_{𝐿1/(𝑝−1)(𝛾)}.
Let us fix 𝜀 small enough. On the one hand, by Lemma 13 it holds that 𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂≤2_𝐴𝜀𝑎_𝑝𝑓_𝑛^2𝑝/2𝑑𝜇₂≤2𝑓_𝑛^𝑝_{𝐿∞𝜇2𝐴𝜀}𝑎_𝑝^𝑝/2𝑑𝜇₂=2_𝐴𝜀𝑎_𝑝^𝑝/2𝑑𝜇₂ (56) for every 𝑛≥𝑁₂=𝑁₂(𝜀).
On the other hand, we have lim_𝜀→0+𝜇₂𝐴_𝜀=lim_𝜀→0+𝜇₂𝑏_𝑝>(1−𝜀)𝑎_𝑝𝑐_𝑝=𝜇₂𝑏_𝑝≥𝑎_𝑝𝑐_𝑝=0. (57) This implies lim_{𝜀→0+𝐴𝜀}𝑎_𝑝^𝑝/2𝑑𝜇₂=0. (58) Given any 𝛿>0 there exists 𝜀₁ with ∫_𝐴𝜀1𝑎_𝑝^𝑝/2𝑑𝜇₂<𝛿. Hence, ∫(𝑎_𝑝|𝑓_𝑛|²)^𝑝/2𝑑𝜇₂<2𝛿 for every 𝑛≥𝑁₂(𝜀₁). Therefore, lim_𝑛→∞∫(𝑎_𝑝|𝑓_𝑛|²)^𝑝/2𝑑𝜇₂=0, which is a contradiction with (54) and (55).

The following result is a direct consequence of Theorems 3 and 17.

Theorem 18. Let one consider 1≤𝑝<∞ and (𝑉,𝜇) a 𝑝-admissible pair. Then the multiplication operator is bounded in 𝑊^1,𝑝(𝑉𝜇) if and only if the following condition holds: thenormsin𝑊^1,𝑝𝑎_𝑝^𝑝/2+𝑐_𝑝^𝑝/2𝜇,𝑐_𝑝^𝑝/2𝜇and𝑊^1,𝑝𝑎_𝑝^𝑝/2𝜇,𝑐_𝑝^𝑝/2𝜇areequivalentonℙ. (59)

This latter theorem and Theorem 2 give the following result.

Theorem 19. Let one consider 1≤𝑝<∞ and (𝑉,𝜇) a 𝑝-admissible pair such that (59) takes place. Let {𝑞_𝑛}_𝑛≥0 be a sequence of extremal polynomials with respect to (2). Then the multiplication operator is bounded and the zeros of {𝑞_𝑛}_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 𝑐_𝑝≤𝐶𝑎_𝑝𝜇-almost everywhere, then (59) holds. In [8, 13] some other very simple conditions implying (59) are shown.

The following is a direct consequence of Theorem 19.

Corollary 20. Let one consider 1≤𝑝<∞ and (𝑉,𝜇) a 𝑝-admissible pair. Assume that 𝑐_𝑝≤𝐶𝑎_𝑝,𝜇-almost everywhere for some constant 𝐶. Let {𝑞_𝑛}_𝑛≥0 be a sequence of extremal polynomials with respect to (2). Then the zeros of {𝑞_𝑛}_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 {𝑞_𝑛}_𝑛≥0 be the sequence of Sobolev orthogonal polynomials with respect to 𝑉𝜇. Then the zeros of the polynomials in {𝑞_𝑛}_𝑛≥0 are uniformly bounded in the complex plane.

5. Asymptotic of Extremal Polynomials

We start this section by setting some notation. Let 𝑄_𝑛, ‖⋅‖_𝐿²_(𝜇), cap(𝑆(𝜇)), and 𝜔_𝑆(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 lim_𝑛→∞𝑄_𝑛𝐿^1/𝑛_2(𝜇)=cap𝑆𝜇. (60)

Finally, if 𝑧₁,𝑧₂,…,𝑧_𝑛 denote the zeros, repeated according to their multiplicity, of a polynomial 𝑞 whose degree is exactly 𝑛, and 𝛿_𝑧𝑗 is the Dirac measure with mass one at the point 𝑧_𝑗, the expression 𝜈𝑞1∶=𝑛^𝑛_𝑗=1𝛿_𝑧𝑗 (61) defines the normalized zero counting measure of 𝑞.

We can already state the first result in this section.

Theorem 22. Let one consider 1≤𝑝<∞, (𝑉,𝜇) a 𝑝-admissible pair and {𝑞_𝑛}_𝑛≥0 the sequence of extremal polynomials with respect to ‖⋅‖_𝑊^1,𝑝_(𝑉𝜇). Assume that the following conditions hold: (i)𝑎_𝑝^𝑝/2𝜇∈𝐑𝐞𝐠;(ii)𝑆(𝜇) is regular with respect to the Dirichlet problem; (iii)condition (59) takes place. Then, lim_𝑛→∞𝑞_𝑛^(𝑗)1/𝑛_𝑆(𝜇)=𝑐𝑎𝑝𝑆𝜇,𝑗≥0. (62) Furthermore, if the complement of 𝑆(𝜇) is connected, then lim_𝑛→∞𝜈𝑞_𝑛^(𝑗)=𝜔_𝑆(𝜇),𝑗≥0. (63) 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 𝑊^1,𝑝(𝑎_𝑝^𝑝/2𝜇,𝑐_𝑝^𝑝/2𝜇) and 𝑊^1,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 use 𝑔_Ω(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 𝑔_Ω(𝑧;∞) is continuous up to the boundary and it can be extended continuously to all ℂ, with value zero on ℂ⧵Ω.

Theorem 23 23. Let one consider 1≤𝑝<∞, (𝑉,𝜇) a 𝑝-admissible pair and {𝑞_𝑛}_𝑛≥0 the sequence of extremal polynomials with respect to ‖⋅‖_𝑊^1,𝑝_(𝑉𝜇). Assume that the following conditions hold: (i)𝑎_𝑝^𝑝/2𝜇∈𝐑𝐞𝐠;(ii)𝑆(𝜇) is regular with respect to the Dirichlet problem;(iii)condition (59) takes place. Then, for each 𝑗≥0, limsup_𝑛→∞𝑞_𝑛^(𝑗)(𝑧)^1/𝑛≤𝑐𝑎𝑝𝑆𝜇𝑒^{𝑔Ω(𝑧;∞)}, (64) uniformly on compact subsets of ℂ. Furthermore, for each 𝑗≥0, lim_𝑛→∞𝑞_𝑛^(𝑗)(𝑧)^1/𝑛=𝑐𝑎𝑝𝑆𝜇𝑒^{𝑔Ω(𝑧;∞)}, (65) 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(𝜔_𝑆(𝜇))⊂{𝑧∶|𝑧|≤2‖𝑀‖} and lim_𝑛→∞𝑞_𝑛^(𝑗+1)(𝑧)𝑛𝑞_𝑛^(𝑗)(𝑧)=𝑑𝜔_𝑆(𝜇)(𝑥)𝑧−𝑥 (66) uniformly on each compact subset of {𝑧∶|𝑧|>2‖𝑀‖}.

Proof. Note that, in our context, the multiplication operator is bounded (see Theorem 3) and the norms of 𝑊^1,𝑝(𝑎_𝑝^𝑝/2𝜇,𝑐_𝑝^𝑝/2𝜇) and 𝑊^1,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.

Acknowledgments

Ana Portilla and Eva Tourís are supported in part by a grant from Ministerio de Ciencia e Innovación (MTM 2009-12740-C03-01), Spain. Yamilet Quintana is supported in part by the Research Sabbatical Fellowship Program (2011-2012) from Universidad Simón Bolvar, Venezuela. Ana Portilla, José M. Rodríguez, and Eva Tourís are supported in part by two grants from Ministerio de Ciencia e Innovación (MTM 2009-07800 and MTM 2008-02829-E), Spain. José M. Rodríguez is supported in part by a grant from CONACYT (CONACYT-UAG I0110/62/10 FON.INST.8/10), México. This word is dedicated to Francisco Marcellán Español on his 60th birthday.