BERNSTEIN-WALSH THEOREMS FOR ELLIPTIC OPERATORS

Józef Siciak

Jagiellonian University, Institute of Mathematics

January 3, 1997

IMUJ PREPRINT 1997/03

Abstract. Let L(D) be an elliptic operator on ℝ^{2} (identified with ℂ) given by the formula

L(D) := (a \frac{∂}{∂z} + b \frac{∂}{∂z̄})^{l}(ā \frac{∂}{∂z̄} + b̄\frac{∂}{∂z})^{k},

where k and l are nonnegative integers with k + l ≥ 1, and a and b are complex numbers with |a| ≠ |b|. Put T (z) := az̄ − bz.

We say that a compact set K ⊂ ℝ^{2} is L(D)-regular, if ℝ^{2} \ K is connected and for every b > 1 there exist a positive constant M and an open neighborhood Ω of K such that

∥p_{n}∥_{Ω} ≤ Mb^{n}∥p_{n}∥_{K}, n ≥ 1,

where p_{n} is any polynomial of two real variables of degree ≤ n such that L(D)p_{n} ≡ 0.

The aim of the paper is to improve some of the results of [2] concerning direct and inverse Bernstein-Walsh theorems for uniform approximation by polynomials satisfying L(D)p ≡ 0.

In particular, we prove that if a compact set K ⊂ ℂ is L(D)-regular then T (K) is regular in the sense of the logarithmic potential theory and for every R > 1 the sublevel set {z ∈ ℂ; g_{T (K)}(T (z)) < log R} is a Bernstein region for the triple (R, K, L(D)), where g_{K} is the Green function of ℂ \ K with pole at infinity. This theorem was earlier obtained in [2] for a narrower class K composed of compact sets K such that ℂ \ K is connected and int K is non-thin at every point a ∈ ∂K ( e.g. Cartesian product of two classical Cantor sets is L(D)-regular but is not a member of K.)

1991 Mathematics Subject Classification. Primary 41A10, 31B35, 30E10. Key words and phrases. Bernstein region, elliptic operator, multivariate polynomial approxi-

mation, Bernstein-Walsh inequality.

Research supported by the grant No 2 PO3A 057 08 from KBN (Committee for Scienfic Research)

1. Introduction

1.1. Let L(D) = L(\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{N}}) be an elliptic differential operator, where L(x) ≡ ∑_{|α|=m} a_{α}x^{α} is a nonconstant homogeneous polynomial on ℝ^{N}, with complex coefficients, which is never equal zero in ℝ^{N} \ {0}.

If N = 2, we identify ℝ^{2} with the complex plane ℂ. Put \frac{∂}{∂z̄} := \frac{1}{2}(\frac{∂}{∂x} + i\frac{∂}{∂y}) and \frac{∂}{∂z} := \frac{1}{2}(\frac{∂}{∂x} − i\frac{∂}{∂y}). A constant coefficient homogeneous differential operator L(D) of degree m on the complex plane is elliptic if and only if it can be written in the form L(D) = Π_{j=1}^{m}(a_{j} \frac{∂}{∂z} + b_{j} \frac{∂}{∂z̄}), where a_{j}, b_{j} are complex numbers with |a_{j}| ≠ |b_{j}|, j = 1, …, m.

1.2. For each integer n ≥ 0 we define the vector space P_{n}(ℝ^{N}, L(D)) of all complex polynomials p of degree ≤ n which satisfy L(D)p = 0 on ℝ^{N}, and if f is a complex valued continuous function on a compact set K ⊂ ℝ^{N} we set

d_{n}(f, K, L(D)) := inf{∥f − p∥_{K}; p ∈ P_{n}(ℝ^{N}, L(D))}.

1.3. A compact set K ⊂ ℝ^{N} is called L(D)-determining, if every function f ∈ C^{∞}(ℝ^{N}) such that f = 0 on K and L(D)f = 0 on a neighbourhood of K, vanishes on a neighbourhood of K.

1.4. We say that a compact subset K of ℝ^{N} is of positive L(D)-capacity, if c_{L(D)}(K) > 0, where

c_{L(D)}(K) := lim inf_{n→∞} \sqrt[n]{c_{n}(K)},

c_{n}(K) ≡ c_{n}(K, L(D)) := inf{∥p∥_{K}/∥p∥_{B}; p ∈ P_{n}(ℝ^{N}, L(D)), p ≢ 0},

B := {x ∈ ℝ^{N}; |x| ≤ 1}.

If c_{L(D)}(K) > 0 we also say that K has positive capacity with respect to polynomials p of N real variables satisfying L(D)p = 0.

1.5. Following [2], given R > 1 and a compact subset K of ℝ^{N} , an open set ω ⊃ K such that each connected component of ω intersects K is called Bernstein region for the triple (R, K, L(D)) provided that f ∈ C(K) satisfies

lim sup_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} ≤ \frac{1}{R}

if and only if f can be extended to F ∈ C^{∞}(ω) with L(D)F = 0 on ω.

1.6. If K is a compact subset of ℂ^{N} we define the extremal function Φ_{K} associated with K by the formula

Φ_{K}(z) := sup_{ν≥1} Φν^{\frac{1}{ν}} (z) ≡ lim_{ν→∞} Φν^{\frac{1}{ν}} (z), z ∈ ℂ^{N},

where

Φ_{ν}(z) := sup{|p(z)|},

the sup being taken over all holomorphic polynomials p of N complex variables with deg p ≤ ν and ∥p∥_{K} ≤ 1.

We put

V_{ν}(z) := \frac{1}{ν} log Φ_{ν}(z)

and

V_{K}(z) := lim_{ν→∞} V_{ν}(z) = log Φ_{K}(z), z ∈ ℂ^{N}.

It is well known (see e.g. [8]) that if K is a compact subset of ℂ with positive logarithmic capacity and ℂ \ K is connected then

V_{K}(z) = g_{K}(z) on ℂ \ K, V_{K}(z) = 0 on K,

where g_{K} is the Green function of ℂ \ K with pole at ∞.

1.7. If K is a compact subset of ℝ^{N} we define the extremal function χ_{K} associated with K (and with the operator L(D)) by the formula

χ_{K}(x) ≡ χ_{K}(x, L(D)) := lim sup_{n→∞} χ_{n}(x), x ∈ ℝ^{N},

where

χ_{n}(x) := sup\{\frac{1}{n} log |p(x)|; p ∈ P_{n}(ℝ^{N}, L(D)), ∥p∥_{K} ≤ 1\}.

For each p ∈ P_{n}(ℝ^{N}, L(D)) there exists a unique holomorphic polynomial p̃ of N complex variables such that p̃ = p on ℝ^{N}. Moreover,

L(D)p̃(z) := ∑_{|α|=n} a_{α}(\frac{∂}{∂z})^{α}p̃(z) = 0 on ℂ^{N}.

We can extend the extremal function χ_{K} from ℝ^{N} to ℂ^{N} by the formula

χ̃_{K}(z) ≡ χ̃_{K}(z, L(D)) := lim sup_{n→∞} χ̃_{n}(z), z ∈ ℂ^{N},

where

χ̃_{n}(z) := sup{|p̃(z)|; p ∈ P_{n}(ℝ^{N}, L(D)), ∥p∥_{K} ≤ 1}.

1.8. We say that a compact set K ⊂ ℝ^{N} is L(D)-regular ( or regular with respect to polynomials p with L(D)p ≡ 0), if ℝ^{N} \ K is connected and for every b > 1 there exist a positive constant M and an open neighborhood Ω of K such that

∥p_{n}∥_{Ω} ≤ Mb^{n}∥p_{n}∥_{K}, p_{n} ∈ P_{n}(ℝ^{N}, L(D)), n ≥ 1.

1.9. If L(D) = L_{1}(D)L_{2}(D) and K is L(D)-regular then K is L_{j}(D)-regular for j = 1, 2.

1.10. Lemma. If a compact set K ⊂ ℝ^{N} is L(D)-regular then ∀_{α∈ℤ_{+}^{N}} ∀_{b>1}∃_{M>0}

∥\frac{∂^{|α|}}{∂x^{α}} p_{n}∥_{K} ≤ Mb^{n}∥p_{n}∥_{K} , p_{n} ∈ P_{n}(ℝ^{N}, L(D)) , n ≥ 0.

Proof. A direct consequence of the definition of L(D)-regularity and of Theorem 5.3 in [1].

1.11. If a compact set K in ℝ^{N} is regular with respect to all polynomials of N real variables, then K is regular with respect to polynomials satisfying L(D)p = 0, where L(D) is any partial differential operator. It is known that a compact set K ⊂ ℝ^{N} is regular with respect to all polynomials of N real variables if and only if the extremal plurisubharmonic function V_{K} is continuous in ℂ^{N}. A compact subset K of ℂ^{N} such that V_{K} is continuous is called L − regular. Sufficient conditions for L-regularity were studied by many authors (see [4] for references).

1.12. The aim of the paper is to prove the following five theorems.

Theorem I. For a compact set K in ℝ^{N} the following conditions are equivalent. (a) c_{L(D)}(K) > 0;

(b) There exists a positive constant A such that

|p(x)| ≤ A^{n}∥p∥_{K}(1 + |x|)^{n}, x ∈ ℝ^{N}, p ∈ P_{n}(ℝ^{N}, L(D)), n ≥ 0;

(c) There exists a positive constant B such that

χ_{K}(x) ≤ B + log(1 + |x|), x ∈ ℝ^{N};

(d) K is determining for L(D)-entire functions and

\{f ∈ C(K); lim_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} = 0\} = \{f|_{K}; f ∈ C^{∞}(ℝ^{N}), L(D)f ≡ 0\}

(i.e. lim_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} = 0 for a function f ∈ C(K) if and only if f admits a unique extension to an L(D)-entire function).

Theorem I was earlier shown for pluriharmonic functions (see Theorem 3.9 [9]). The present theorem can be proven by the same method. To show the implications (c) ⟹ (d) ⟹ (a) we can now use Theorem 5.1 [1].

Theorem II. For a compact set K in ℝ^{N} such that ℝ^{N} \ K is connected the following conditions are equivalent:

(1) K is L(D)-regular;

(2) The extremal function χ_{K} is uniformly continuous in ℝ^{N}, i.e. ∀_{ϵ>0}∃_{δ>0}

|χ_{K}(x) − χ_{K}(y)| ≤ ϵ, if x, y ∈ ℝ^{N}, |x − y| ≤ δ;

(3) χ_{K}^{∗}(x) := lim sup_{y→x} χ_{K}(y) = 0 for all x ∈ K, i.e. χ_{K} is continuous at each point of K;

(4) K is L(D)-determining and \{f ∈ C(K); lim sup_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} < 1} = {f|_{K}; f ∈ C^{∞}(ℝ^{N}), L(D)f = 0 in a neighborhood of K}.

Theorem III. Let K be an L(D)-determining compact subset of ℝ^{N} with c_{L(D)}(K) > 0. Let R > 1 be a real number such that K ⊂ Ω_{R} := {x ∈ ℝ^{N}; χ_{K}^{∗}(x, L(D)) < log R}.

If ω is a Bernstein region for the triple (R, K, L(D)) then ω := ω_{R}, where ω_{R} denotes the union of all connected components of Ω_{R} intersecting K.

In particular, if K is an L(D)-regular compact set in ℝ^{N} and ω is a Bernstein region for a triple (R, K, L(D)) then ω = ω_{R}, where ω_{R} is the union of all connected components of the sublevel set Ω_{R} := {x ∈ ℝ^{N}; χ_{K}(x, L(D)) < log R}^{1}.

Theorem IV. Let L_{k,l}(D) be an elliptic operator on the complex plane, given by the formula L_{k,l}(D) = (\frac{∂}{∂z})^{k}(\frac{∂}{∂z̄})^{l}, where k, l are nonnegative real numbers with k + l ≥ 1. Let K be a compact subset of ℂ such that ℂ \ K is connected.

Then the following conditions are equivalent:

1^{∘}. K is L_{k,l}(D)-regular;

2^{∘}. K is \frac{∂}{∂z̄}-regular and ∀_{b>1}∃_{M>0}

|p_{n}(z)| ≤ Mb^{n}∥p_{n}∥_{K}Φ_{K}^{n}(z), z ∈ ℂ, p_{n} ∈ P_{n}(ℂ, L_{k,l}(D)), n ≥ 0;

3^{∘}. K is \frac{∂}{∂z̄}-regular and χ_{K}(z, L_{k,l}(D)) = V_{K}(z), z ∈ ℂ.

Theeorem IV and Lemma A [2] imply the following Theorem IVa.

Theorem IVa. Let L(D) be an elliptic operator on ℂ given by the formula

L(D) := (a\frac{∂}{∂z} + b\frac{∂}{∂z̄})^{l}(ā\frac{∂}{∂z̄} + b̄\frac{∂}{∂z})^{k},

where k and l are nonnegative integers with k + l ≥ 1, and a and b are complex numbers with |a| ≠ |b|. Put T (z) := az̄− bz. Let K be a compact subset of ℂ. Then the following are equivalent.

1^{∘}. K is L(D)-regular;

2^{∘}. T (K) is L_{k,l}(D)-regular, where L_{k,l}(D) := (\frac{∂}{∂z})^{k}(\frac{∂}{∂z̄})^{l}, and ∀_{b>1}∃_{M>0}∀_{n≥0}

|p_{n}(z)| ≤ Mb^{n}∥p_{n}∥_{K}Φ_{T (K)}^{n}(T (z)), z ∈ ℂ, p_{n} ∈ P_{n}(ℂ, L(D));

3^{∘}. T (K) is \frac{∂}{∂z̄}-regular and

χ_{K}(z; L(D)) = V_{T (K)}(T (z)), z ∈ ℂ.

Moreover, for every R > 1 and for every L(D)-regular compact set K ⊂ ℂ

ω_{R} := {z ∈ ℂ; V_{T (K)}(T (z)) < log R}

is a Bernstein region for (R, K, L(D)).

Let K denote the family of all compact sets on ℂ such that ℂ \ K is connected and intK is non-thin at every point of ∂K. We shall see that each compact set K of the class K is L(D)-regular for every operator L(D) of Theorem IVa. But there are compact sets K (e.g. K = C × C, where C is the classical Cantor set on ℝ) such that K is L(D)-regular for every L(D) but K ∉ K. Therefore Theorem IVa gives an improvement of Theorem 2 of [2] due to Bagby and Levenberg who proved the result only for K ∈ K.

1We do not know whether a sublevel set Ω_{R} can contain a component disjoined with K.

Theorem V. Given a compact subset K of the unit circle Γ = {z ∈ ℂ; |z| = 1}, let χ_{K} be the extremal function associated with K and the Laplace operator L(D) = \frac{∂^{2}}{∂z∂z̄}.

Then the following statements are true:

(i) K is of positive capacity with respect to harmonic polynomials of two real variables if and only if the logarithmic capacity (transfinite diameter) d(K) of K is positive;

(ii) χ_{K}(z) = V_{K}(z) for all z ∈ Γ;

(iii) V_{K}(z) ≤ χ_{K}(z) ≤ 2 sup_{ζ∈Γ} V_{K}(ζ) + log^{+} |z| on ℂ;

(iv) If K ≠ Γ, then χ_{K}^{∗}(a) > 0 for all a ∈ K. If K = Γ then χ_{K}(z) ≡ V_{K}(z) ≡ log^{+} |z| ;

(v) If K is a \frac{∂}{∂z̄}- regular (i.e. regular with respect to holomorphic polynomials), then for every R > 1 the open set

ω_{R} := {z ∈ ℂ; 2V_{K}(z) − log |z| < log R}

is a Bernstein region for the triple (R, K, \frac{∂^{2}}{∂z∂z̄}).

Let us observe that no compact subset K of Γ is determining for harmonic functions (e.g. log |z| is a non-constant harmonic function that vanishes on K). Therefore in the case of statement (v) we cannot apply Theorem III. Moreover, it follows from statement (iii) of Theorem V that if K ⊂ Γ is \frac{∂}{∂z̄}-regular and K ≠ Γ then {z ∈ ℂ; 2V_{K}(z) − log |z| < log R} ≠ {z ∈ ℂ; χ_{K}^{∗}(z) < log R} for all sufficiently large R. It follows that the assumption of L(D)-determinacy in Theorem III is essential.

2. Proof of Theorem II

It is enough to prove the following implications : (1) ⇒ (2) ⇒ (3) ⇒ (1) ⇒ (4) ⇒ (1).

(1) ⟹ (2). We need to show that if ∀_{b>1} there exist a positive constant M and an open neighborhood Ω of K such that

(1)  ∥p_{n}∥_{Ω} ≤ Mb^{n}∥p_{n}∥_{K}, p_{n} ∈ P_{n}(ℝ^{N}, L(D)), n ≥ 1,

then ∀_{ϵ>0}∃_{δ>0}

(2)  |χ_{K}(x) − χ_{K}(y)| ≤ ϵ, if x, y ∈ ℝ^{N}, |x − y| ≤ δ.

By (1) for every ϵ > 0 there exists δ > 0 such that \frac{1}{n} log |p_{n}(x)| ≤ \frac{1}{n} log M + ϵ for all p_{n} ∈ P_{n}(ℝ^{N}, L(D)) with ∥p_{n}∥_{K} ≤ 1, n ≥ 1 and for all x ∈ ℝ^{N} with dist(x, K) ≤ δ. Fix y ∈ ℝ^{N} with |y| ≤ δ . Then

\frac{1}{n} log |p_{n}(x + y)| ≤ \frac{1}{n} log M + ϵ + χ_{n}(x)

if x ∈ ℝ^{N}, p_{n} ∈ P_{n}(ℝ^{N}, L(D)), ∥p_{n}∥_{K} ≤ 1, n ≥ 1. Hence χ_{n}(x + y) ≤ \frac{1}{n} log M + ϵ + χ_{n}(x), n ≥ 1, x ∈ ℝ^{N}. It follows that χ_{K}(x + y) ≤ ϵ + χ_{K}(x), if x, y ∈ ℝ^{N} and |y| ≤ δ, which is equivalent to (2).

The implication (2) ⟹ (3) is obvious.

(3) ⟹ (1) follows from Leja’s polynomial Lemma [7].

(1) ⟹ (4). First we shall show that K is L(D)-determining. Fix f ∈ C^{∞}(ℝ^{N}) with L(D)f = 0 in a neighborhood U of K and f = 0 on K. Let E be a compact set such that K ⊂ int E, E ⊂ U and ℝ^{N} \ E is connected. By [1] there is ρ ∈ (0, 1) such that lim sup_{n→∞} \sqrt[n]{d_{n}(f, E, L(D))} ≤ ρ. Choose p_{n} ∈ P_{n}(ℝ^{N}, L(D)) with

d_{n}(f, E, L(D)) = ∥f − p_{n}||_{E}. Then lim sup_{n→∞} ∥p_{n}∥K^{\frac{1}{n}} ≤ ρ. Take ρ_{1} with ρ < ρ_{1} <

1. Then there exists a constant M_{1} > 0 such that ∥p_{n}∥_{K} ≤ M_{1}ρ_{1}^{n} for all n. Choose b > 1 so close to 1 that bρ_{1} < 1. By L(D)-regularity of K there exisit a neighborhood Ω of K and a positive constant M such that ∥p_{n}∥_{Ω} ≤ MM_{1}(bρ_{1})^{n}∥p_{n}∥_{K}, n ≥ 1, which implies that f = 0 in Ω ∩ int E that is a neighborhood of K.

The inclusion \{f ∈ C(K); lim sup_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} < 1\} ⊃ \{f|_{K}; f ∈ C^{∞}(ℝ^{N}), L(D)f = 0 in a neighborhood of K} follows from the main Theorem 1.2 of [1]. The opposite inclusion follows from the following lemma.

2.1. Lemma. Let K be an L(D)-determining compact subset of ℝ^{N} with positive L(D)-capacity. Let R > 1 be a real number such that K ⊂ Ω_{R} := {x ∈ ℝ^{N}; χ_{K}^{∗}(x, K, L(D)) < log R}. Then every function f ∈ C(K) with lim sup_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} ≤ \frac{1}{R} can be extended to a holomorphic solution F̃ of L(D)F̃ = 0 on Ω̃_{R} := {z ∈ ℂ^{N}; χ̃_{K}^{∗}(z, L(D)) < log R}.

In particular F := F̃|_{Ω_{R}} is a real-analytic solution of L(D)F = 0 on Ω_{R} with F = f on K.

Proof. Given f ∈ C(K) and R > 1 with lim sup \sqrt[n]{d_{n}(f, K, L(D))} ≤ \frac{1}{R}, choose p_{n} ∈ P_{n}(ℝ^{N}, L(D)) with ∥f − p_{n}∥_{K} = d_{n}(f, K, L(D)). Then for every b > 1 there exists M > 0 such that

∥p_{n} − p_{n−1}∥_{K} ≤ M (\frac{b}{R})^{n} , n ≥ 1.

Hence

|p̃_{n}(z) − p̃_{n−1}(z)| ≤ M (\frac{b}{R})^{n} exp(nχ̃_{n}(z), z ∈ ℂ^{N}, n ≥ 1.

Therefore

lim sup_{n→∞} \sqrt[n]{|p̃_{n}(z) − p̃_{n−1}(z)|} ≤ \frac{b}{R}e^{χ̃_{K}(z)}, z ∈ ^{N}.

By the arbitrary character of b > 1 we have

(*)  lim sup_{n→∞} \sqrt[n]{|p̃_{n}(z) − p̃_{n−1}(z)|} ≤ \frac{1}{R}e^{χ̃_{K}(z)}, z ∈ ℂ^{N}.

Given a compact subset E of Ω̃_{R} := {χ̃_{K}^{∗}(z) < log R}, choose r with 1 < r < R and b > 1 such that b^{2}r < R and E ⊂ Ω̃_{r}. By (*) for each z ∈ Ω̃_{r} there exists M(z) > 0 such that |p̃_{n}(z) − p̃_{n−1}(z)| ≤ M(z) (\frac{br}{R} )^{n} , n ≥ 2. By the Polynomial Lemma there exists a constant C > 0 such that

|p̃_{n}(z) − p̃_{n−1}(z)| ≤ C (\frac{b^{2}r}{R} )^{n} , n ≥ 1, z ∈ E.

It follows that the series p̃_{1} + ∑_{2}^{∞}(p̃_{n} − p̃_{n−1}) is uniformly convergent on each compact subset E of Ω̃_{R}. Its sum F̃ delivers the required extension of f.

(4) ⟹ (1) follows by Grothendieck open mapping theorem.

3. Proof of Theorem III

Given an open set Ω in ℝ^{N} ,

F(Ω) ≡ C^{∞}(Ω, L(D)) := {f ∈ C^{∞}(Ω); L(D)f = 0 on Ω}

is a Frechet space when endowed with the topology of locally uniform convergence. For every point a of the boundary of Ω there is a function f ∈ F(Ω)^{2} which has singularity at a (e.g. a fundamental solution for L(D)). By a standard method (using Banach open mapping theorem) one can show that for each open subset Ω of ℝ^{N} there is a function f ∈ F(Ω) for which Ω is a maximal domain of existence.

Let ω be a Bernstein region for the triple (R, K, L(D)) satisfying the assumptions of Theorem III. Let f be a function in F(ω) noncontinuable analytically beyond ω. By the definition of ω lim sup \sqrt[n]{d_{n}} ≤ \frac{1}{R}. Therefore by Lemma 2.1 there exists F ∈ F(ω_{R}) with F = f on K. But K is L(D)-determining, so F = f on ω_{R} and ω_{R} ⊂ ω.

We need to show that ω_{R} = ω. Suppose ω_{R} ≠ ω. Then there exists a point a ∈ ω ∩ ∂ω_{R}. To get a contradiction it suffices to construct a function f ∈ F(ω_{R}) such that lim sup_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} ≤ \frac{1}{R} and a is a singular point of f; by the definition of ω there exists f_{1} ∈ F(ω) with f_{1} = f on K, and consequently f_{1} = f on ω_{R}, which is an absurd.

We first show that χ̃_{K}^{∗}(x) = χ_{K}^{∗}(x) at each point x of ℝ^{N}. Fix x_{o} in ℝ^{N} and, given ϵ > 0, choose δ > 0 such that

χ_{K}^{∗}(x) < χ_{K}^{∗}(x_{o}) + ϵ, if |x − x_{o}| < δ.

By the Polynomial Lemma ∀_{b>1}∃_{M>0}∃_{δ_{1}>0}

|p_{n}(z)| ≤ Mb^{n}e^{n(χ_{K}^{∗}(x_{o})+2ϵ)}, |z − x_{o}| < δ_{1}, n ≥ 1,

where p_{n} is any polynomial in P_{n}(ℝ^{N}, L(D)) with ∥p_{n}∥_{K} ≤ 1. Therefore

χ̃_{K}(z) ≤ log b + 2ϵ + χ_{K}^{∗}(x_{o}), |z − x_{o}| ≤ δ_{1}, z ∈ ℂ^{N},

and consequently

χ̃_{K}^{∗}(x_{o}) ≤ log b + 2ϵ + χ_{K}^{∗}(x_{o}).

By the arbitrary character of b > 1 and ϵ > 0 , we get χ̃_{K}^{∗}(x_{o}) ≤ χ_{K}^{∗}(x_{o}). The opposite inequality being obvious, we obtain the required equality.

Now we shall construct the required function. Let a be a fixed point of ∂ω_{R}. The function χ̃_{K}^{∗}(z, L(D)) is plurisubharmonic in ℂ^{N} and χ̃_{K}^{∗}(z, L(D)) ≥ log R at all points z ∈ ∂ω_{R}. Hence, by the maximum principle, for each r > 0 the set {z ∈ ℂ^{N}; |z − a| < r, χ̃_{K}^{∗}(z, L(D)) > log R} is nonempty. Therefore, by Bedford-Taylor Theorem on negligible sets [3], for every n ≥ 1 there exists a point a_{n} ∈ ℂ^{N} with |z − a_{n}| < \frac{1}{n} such that χ̃_{K}(a_{n}) = χ̃_{K}^{∗}(a_{n}) > log R. Let \{b_{n}\} denote the sequence {a_{1}, a_{1}, a_{2}, a_{1}, a_{2}, a_{3}, …, } (each point of the set {a_{n}; n ≥ 1} is repeated infinitely many times in {b_{n}}).

2By Theorem 5.1 of [1] each function of F(Ω) is real-analytic in Ω.

Choose p_{n_{j}} ∈ P_{n_{j}}(ℝ^{N}, L(D)) such that n_{j} < n_{j+1}, ∥p_{n_{j}}∥_{K} ≤ 1 and

\frac{1}{n_{j}} log |p̃_{n_{j}}(b_{j})| ≥ χ̃_{K}(b_{j}) − \frac{1}{j}, j = 1, 2, ….

For every subset A of ℕ the series

f̃_{A}(z) := ∑_{j∈A}^{∞} \frac{p̃_{n_{j}}(z)}{R^{n_{j}}}

converges locally uniformly and absolutely in

ω̃_{R} := {z ∈ ℂ^{N}; χ̃_{K}^{∗}(z) < log R}

and diverges at some points of any neighborhood of a (e.g. the series diverges at a_{j}, j ≥ 1). By Saint Raymond Lemma 5 [6] there exists A ⊂ ℕ such that a is a singular point of f̃_{A}. The function f := f̃_{A}|_{ω_{R}} has the required properties.

4. Proof of Theorem IV

By the classical Bernstein-Walsh Inequality Theorem IV is true for k + l = 1. Suppose it is true for k + l = m ≥ 1. We need to prove that if k + l = m + 1 then 1^{∘} ⟹ 2^{∘} ⟹ 3^{∘} ⟹ 1^{∘}.

1^{∘} ⟹ 2^{∘}.

(i) First we shall show that this implication is true for k = 0, l = m + 1. To this order fix b > 1. Since K is (\frac{∂}{∂z̄})^{m+1}-regular, there exist an open neighborhood Ω of K and a positive constant M such that

(1)  ∥p_{n}∥_{Ω} ≤ Mb^{n}∥p_{n}∥_{K}, p_{n} ∈ P_{n}(ℂ, (\frac{∂}{∂z̄})^{m+1}), n ≥ 0.

If p_{n} ∈ P_{n}(ℂ, (\frac{∂}{∂z̄})^{m+1}) then p_{n}(z) = q_{0}(z) + q_{1}(z)z̄ + ⋯ + q_{m}(z)z̄^{m}, where q_{j} ∈ P_{n−j}(ℂ, \frac{∂}{∂z̄}) is a holomorphic polynomial of degree ≤ n − j. By Lemma 1.10 there is M_{1} > 0 that depends only on b such that

(2)  m!∥q_{m}∥_{K} ≤ M_{1}∥p_{n}∥_{Ω} ≤ M_{1}Mb^{n}∥p_{n}∥_{K}.

It is clear that p_{n}(z) − q_{m}(z)z̄^{m} ∈ P_{n}(ℂ, (\frac{∂}{∂z̄})^{m}) and

(3)  ∥p_{n}(z) − q_{m}(z)z̄^{m}∥_{K} ≤ ∥p_{n}∥_{K}(1 + r^{m}M_{1}Mb^{n}),

where r := max_{z∈K} |z|. By the induction assumption there exists M_{2} > 0 such that

(4)  |p_{n}(z) − q_{m}(z)z̄^{m}| ≤ M_{2}b^{n}(1 + r^{m}M_{1}Mb^{n})∥p_{n}∥_{K}Φ_{K}^{n}(z), z ∈ ℂ,

where M_{2} depends only on b. By the classical Bernstein-Walsh Inequality

(5)  |q_{m}(z)z̄^{m}| ≤ ∥q_{m}∥_{K}Φ_{K}^{n−m}(z) · r^{m}Φ_{K}^{n}(z) = r^{m}∥q_{m}∥_{K}Φ_{K}^{n}(z).

It follows from (1), (4) and (5) that

|p_{n}(z)| ≤ M_{3}b^{2n}Φ_{K}^{n}(z)∥p_{n}∥_{K}, z ∈ ℂ, p_{n} ∈ P_{n}(ℂ, (\frac{∂}{∂z̄})^{m+1}), n ≥ 0,

where M_{3} := M_{2}(1 + r^{m}M_{1}M) + r^{m}M_{1}M depends only on b > 1. By substituting \sqrt{b} for b one gets the required inequality of 2^{∘}.

(ii) If l = 0, k = m + 1 the proof of the implication 1^{∘} ⟹ 2^{∘} may be reduced to the case considered in (i) by observing that p_{n} ∈ P_{n}(ℂ, (\frac{∂}{∂z})^{m+1}) if and only if p̄_{n} ∈ P_{n}(ℂ, (\frac{∂}{∂z̄})^{m+1}).

(iii) Let k ≥ 1, l ≥ 1, k + l ≥ m + 1. Then by the induction assumption for every b > 1 there exist M_{1} = M_{1}(b) > 0 such that (6)

|\frac{∂}{∂z} p_{n}(z)| ≤ M_{1}b^{n}∥\frac{∂}{∂z} p_{n}∥_{K}Φ_{K}^{n−1}(z), z ∈ ℂ, p_{n} ∈ P_{n}(ℂ, L_{k,l}(D)), n ≥ 0,

(7)

|\frac{∂}{∂z̄}p_{n}(z)| ≤ M_{1}b^{n}∥\frac{∂}{∂z̄}p_{n}∥_{K}Φ_{K}^{n−1}(z), z ∈ ℂ, p_{n} ∈ P_{n}(ℂ, L_{k,l}(D)), n ≥ 0,

because K is L_{k−1,l}(D)- and L_{k,l−1}(D)-regular. By Lemma 1.10

(8) ∥\frac{∂}{∂z} p_{n}∥_{K} + ∥\frac{∂}{∂z̄}p_{n}∥_{K} ≤ M_{2}b^{n}∥p_{n}∥_{K}, p_{n} ∈ P_{n}(ℂ, L_{k,l}(D)), n ≥ 0,

where M_{2} depends only on b > 1. From (6), (7) and (8) we get (9)

|\frac{∂}{∂z} p_{n}(z)| + |\frac{∂}{∂z̄}p_{n}(z)| ≤ M_{3}b^{2n}∥p_{n}∥_{K}Φ_{K}^{n−1}(z), p_{n} ∈ P_{n}(ℂ, L_{k,l}(D)), n ≥ 0.

Now we shall show that

(10) |p_{n}(z)| ≤ M_{3}b^{n}Φ_{K}^{n}(z)∥p_{n}∥_{K}, |z − a| < r, p_{n} ∈ P_{n}(ℂ, L_{k,l}(D)), n ≥ 0.

By L_{k,l}(D)-regularity of K (10) is true for all a ∈ K. Fix a point a ∈ ℂ with Φ_{K}(a) =: ρ > 1. Given b > 1 take r > 0 so small that B(a, r) ⊂ {z ∈ ℂ; b^{−1}ρ < Φ_{K}(z) < bρ}. Then

(11)  b^{2}ρ ≤ b^{3}Φ_{K}(z), z ∈ B(a, r).

Observe that

p_{n}(z) = p(z_{o}) + ∫_{γ_{zoa}} dp_{n} + ∫_{a}^{z} dp_{n}, z ∈ ℂ,

where γ_{z_{o}a} is a path in K_{bρ} := {Φ_{K}(z) < bρ} joining a point z_{o} ∈ K with a. Hence by (9)

|p_{n}(z)| ≤ ∥p_{n}∥_{K} + M_{3}(b^{2}ρ)^{n}|γ_{z_{o}a}| · ∥p_{n}∥_{K} + M_{3}r(b^{3}ρ)^{n}∥p_{n}∥_{K}

for all z with |z − a| ≤ r and for n ≥ 1. Therefore

|p_{n}(z)| ≤ C∥p_{n}∥_{K}(b^{4}Φ_{K}(z))^{n}, |z − a| ≤ r, n ≥ 1,

where C = C(a, b), r = r(a, b) do not depend on p_{n}. By substituting \sqrt[4]{b} for b , we get the required inequality (10).

Let R_{o} be so large that the level curve {Φ_{K}(z) = R} is convex for all R ≥ R_{0}. It folows from (10) that there exists M_{0} > 0 such that

(12)  |p_{n}(z)| ≤ M_{o}b^{n}∥p_{n}∥_{K}Φ_{K}^{n}(z), if Φ_{K}(z) ≤ R_{o},

for all p_{n} ∈ P_{n}(ℂ, L_{k,l}(D)), n ≥ 1. Fix z with R = Φ_{K}(z) > R_{o}. Let z_{o} be a point of K such that |z−z_{o}| = dist(z, K). Then the interval [z_{o}, z] ⊂ {ζ ∈ ℂ; Φ_{K}(ζ) ≤ R}. Hence by (9)

(13)  |p_{n}(z)| ≤ |p_{n}(z_{o})| + | ∫_{z_{o}}^{z} dp_{n}| ≤ M_{1}b^{n}∥p_{n}∥_{K}Φ_{K}^{n−1}(z)|z − z_{o}|.

By [10] there is a sequence of holomorphic polynomials π_{n}(z) = (z−z_{n1}) … (z−z_{nn}) with zeros z_{nj} ∈ K (j = 1, …, n) such that \sqrt[n]{|π_{n}(z)|} ⟶ d(K)Φ_{K}(z) locally uniformly in ℂ, where d(K) is the transfinite diameter of K. Hence Φ_{K}(z)d(K) ≥ dist(z, K) = |z − z_{o}|. Therefore by (13)

|p_{n}(z)| ≤ d(K)M_{1}b^{n}Φ_{K}^{n}(z)

for all z with Φ_{K}(z) > R_{o}, which in view of (12) concludes the proof of 2^{∘}.

2^{∘} ⟹ 3^{∘}. It is clear that if p is a holomorphic polynomial then either L_{k,l}(D)p = 0 or L_{k,l}(D)p̄ = 0. Therefore

χ_{K}(z, L_{k,l}(D)) ≥ V_{K}(z), z ∈ ℂ.

By 2^{∘} we get the inverse inequality. Hence 3^{∘} follows from 2^{∘}.

3^{∘} ⟹ 1^{∘}. Follows from Theorem II. The proof of Theorem IV is concluded.

4a. Proof of Theorem IVa

By Lemma A [2]

p ∈ P_{n}(ℂ, L_{k,l}(D)) iff p ∘ T ∈ P_{n}(ℂ, L(D)).

Hence for every compact set K ⊂ ℂ

χ_{K}(z, L(D)) = χ_{T (K)}(T (z), L_{k,l}(D)),

which by Theorem II implies that a compact set K ⊂ ℂ is L(D)- regular if and only if T (K) is L_{k,l}(D)-regular (resp. K is L_{k,l}(D)-regular if and only if T^{−1}(K) is L(D)-regular). Moreover, if T (K) is L_{k,l}(D)-regular then, by Theorem IV,

χ_{T (K)}(T (z), L_{k,l}(D)) = V_{T (K)}(T (z)).

It remains to show the last statement of Theorem IVa. Fix R > 1, an L(D)- regular compact set K ⊂ ℂ and a function f ∈ C(K) with

lim sup_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} ≤ \frac{1}{R}

. We know , by Lemma 2.1, that f can be extended to a C^{∞} solution F of the equation L(D)F = 0 on ω_{R} = {z; χ_{K}(z, L(D)) ≤ log R} = {z; V_{T (K)}(T (z)) ≤ log R}.

If f ∈ C^{∞}(ω_{R}, L(D)) then by Lemma A [2] and Theorem 3 [2]

lim sup_{n→∞} \sqrt[n]{d_{n}(f, K, L(D))} ≤ \frac{1}{R},

which completes the proof.

We claim that Theorem IVa is an improvement of Theorem 2 [2] in the sense that each compact set of the class K is L(D)-regular but there are L(D)-regular compact sets that do not belong to the class K. Indeed, by the Wiener criterion, if K ∈ K then T (K) ∈ K. Given K ∈ K , let {E_{n}} be a sequence of compact sets such that E_{n} ⊂ E_{n+1}, ∪_{1}^{∞}E_{n} = intT (K) and for every n the set E_{n} is a finite union of compact squares. Then

χ_{T (K)}(w, L_{k,l}(D)) ≤ χ_{E_{n}}(w, L_{k,l}(D)) = V_{E_{n}}(w), w ∈ ℂ, n ≥ 1.

The funcftion U(w) := lim_{n→∞} V_{E_{n}}(w) is subharmonic on ℂ, vanishes on intT (K) and has a logarithmic pole at ∞. Since T (K) ∈ K, we get U(w) = 0 on T (K), which implies that U(w) ≤ V_{T (K)}(w) on ℂ. But V_{T (K)}(w) ≤ χ_{T (K)}(w, L_{k,l}(D)). Therefore

V_{T (K)}(T (z)) = χ_{T (K)}(T (z), L_{k,l}(D)) = χ_{K}(z, L(D)),

which , by Theorem II, shows that K is L(D)-regular.

5. Proof of Theorem V

The proofs of (i), (ii) and (iii) may be found in [9].

(iv). Given a compact subset K of Γ, K ≠ Γ, let f be a holomorphic branch of log z in a neighborhood of K. By Bernstein-Walsh theorem there exist R > 1 and a sequence of holomorphic polynomials f_{n} such that deg f_{n} ≤ n and ∥f − f_{n}∥_{K} ≤ MR^{−n}, n ≥ 1, where M is a positive constant. Then p_{n} := ℜf_{n} are harmonic polynomials with ∥p_{n}∥_{K} ≤ MR^{−n}. Therefore |p_{n}(z)| ≤ MR^{−n}e^{nχ_{n}(z)}, z ∈ ℂ, n ≥ 1, which implies that lim sup_{n→∞} \sqrt[n]{|p_{n}(z)|} ≤ \frac{1}{R}e^{χ_{K}^{∗}(z)} in ℂ. Suppose, contrary to our claim, that χ_{K}^{∗}(a) = 0 for some a ∈ K. Then lim p_{n}(z) = 0 in a neighborhood of a. But lim p_{n}(z) = log |z| in a neighborhood of K. We have got a contradiction which completes the proof of (iv).

(v). To begin with observe that each complex-valued harmonic polynomial p_{n} of degree ≤ n can be written in the form

(1)  p_{n}(z) = ∑_{j=1}^{n} c_{−j}z̄^{j} + ∑_{j=0}^{n} c_{j}z^{j},

where c_{j} (j = −n, −n + 1, …, n) are complex numbers. With each harmonic polynomial p_{n} we associate rational function p_{n}^{∗} given by the formula

(2)  p_{n}^{∗}(z) := ∑_{−n}^{n} c_{j}z^{j}.

The function p_{n}^{∗} is holomorphic in ℂ̂ \ {0, ∞} and p_{n}(z) = p_{n}^{∗}(z) for all z with |z| = 1. The mapping p_{n} ↦ p_{n}^{∗} is a one-to-one correspondence between complex valued harmonic polynomials of degree ≤ n and rational functions of the form (2).

Under the assumptions of (v) fix R > 1 and let f ∈ C(K) be a function such that

lim sup \sqrt[n]{d_{n}(f, K, \frac{∂^{2}}{∂z∂z̄})} ≤ \frac{1}{R}. We shall construct a function F harmonic in ω_{R} such

that F = f on K. To this order let p_{n} = ∑_{j=1}^{n} c_{−j}^{(n)}z̄^{j} + ∑_{j=0}^{n} c_{j}^{(n)}z^{j} be a harmonic polynomial of degree ≤ n with d_{n}(f, K, \frac{∂^{2}}{∂z∂z̄}) = ∥f − p_{n}∥_{K}. We claim that the function

(3)  F (z) := p_{1}^{∗} + ∑_{n=1}^{∞}(p_{n+1}^{∗} − p_{n}^{∗}), z ∈ ω_{R},

has the required properties. It is clear that F = f on K. It is sufficient to show that the series (3) is convergent locally uniformly in ω_{R}. Given ϵ > 0, we have

∥p_{n+1}^{∗} − p_{n}^{∗}∥_{K} = ∥p_{n+1} − p_{n}∥_{K} ≤ M(\frac{e^{ϵ}}{R})^{n+1}, n ≥ n_{o},

where M and n_{o} depend only on ϵ.

Observe that

u_{n}(z) := −2V_{K}(z) + log |z| + \frac{1}{n + 1} log |p_{n+1}^{∗}(z) − p_{n}^{∗}(z)|, z ∈ ℂ̂ \ \{0, ∞\},

u_{n}(0) := −2V_{K}(0) + \frac{1}{n + 1} log |c_{−n−1}^{(n+1)}| < +∞,

u_{n}(∞) := 2 log d(K) + \frac{1}{n + 1} log |c_{n+1}^{(n+1)}| < +∞,

is a subharmonic function on ℂ̂ \ K and sup_{K} u_{n} ≤ \frac{1}{n+1} log M + log \frac{e^{ϵ}}{R} . Here d(K) denotes the transfinite diameter of K. By the maximum principle u_{n}(z) ≤ \frac{1}{n+1} log M + log \frac{e^{ϵ}}{R} , z ∈ ℂ̂ \ K, n > n_{o}. Hence

\frac{1}{n + 1} log |p_{n+1}^{∗}(z) − p_{n}^{∗}(z)|

≤ 2V_{K}(z) − log |z| + \frac{1}{n + 1} log M + log \frac{e^{ϵ}}{R} , z ∈ ℂ̂ \ K, n > n_{o},

which implies that the series (2) is convergent uniformly on the set ω_{Re^{−ϵ}}. By the arbitrary character of ϵ > 0 the series converges locally uniformly in ω_{R}. Observe that ∀_{c∈ℝ} the function F_{c}(z) = F (z) + c log |z| is harmonic in ω_{R} and F_{c}(z) = f(z) on K. So there are many harmonic continuations of f from K to ω_{R}.

Let now u be a harmonic function in ω_{R}. We need to show that

lim sup \sqrt[n]{d_{n}(u, K, \frac{∂^{2}}{∂z∂z̄})} ≤ \frac{1}{R}.

Without loss of generality we may assume that u is real-valued. If D is a connected component of ω_{R} then D is either simply-connected or doubly-connected. In the second case the bounded component of ℂ \ D is a neighborhood of 0. It follows that there exist a real number α and a holomorphic function f such that ℜf(z) = u(z) + α log |z| in ω_{R}.

Let z^{2n} = {z_{o}, …, z_{2n}} be a 2n-th system of Fekete extremal points of K. Put

F_{n}(z) := (z − z_{o}) … (z − z_{2n})z^{−n}.

We know [10] that ||F_{n}∥K^{\frac{1}{n}} → d^{2}(K) and

\frac{1}{n + 1} log |F_{n}(z)| → 2V_{K}(z) − log |z| + 2 log d(K)

locally uniformly in ℂ \ (K ∪ {0}).

Put

p_{n}^{∗}(z) := ∑_{j=0}^{2n} f(z_{j})(\frac{z_{j}}{z})^{n}L_{2n}^{(j)}(z),

where L_{2n}^{(j)}(z) is the unique holomorphic polynomial of degree 2n such that L_{2n}^{(j)}(z_{k}) = δ_{j,k}. By the residuum theorem

f(z) − p_{n}^{∗}(z) = \frac{1}{2πi} ∫_{∂ω_{r}} \frac{F_{n}(z)}{F_{n}(ζ)} \frac{f(ζ)}{ζ − z} dζ, z ∈ ω_{r}, 1 < r < R.

Hence by the arbitrary character of r one gets lim sup_{n→∞} \sqrt[n]{d_{n}(u, K, \frac{∂^{2}}{∂z∂z̄})} ≤ lim sup_{n→∞} ∥f − p_{n}^{∗}∥_{K} ≤ \frac{1}{R}. The proof is complete.

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