Geometry of holomorphic mappings and Hölder continuity of the pluricomplex Green function

We provide a solution to a long-standing open problem that lives in the interface of pluripotential theory and multivariate approximation theory. The problem is to characterize the holomorphic maps which preserve Hölder continuity of the pluricomplex Green function associated with a compact subset of CN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^N$$\end{document}. We also prove, under mild restrictions, that nondegenerate holomorphic maps preserve Markov’s inequality for polynomials.


Introduction
For each compact set ∅ = K ⊂ C N , the following function V K (z) := sup φ(z) : φ ∈ L(C N ), φ ≤ 0 on K (z ∈ C N ), where L(C N ) denotes the class of plurisubharmonic functions φ in C N (see [23] for the definition and basic properties of plurisubharmonic functions) satisfying the logarithmic growth condition Rafal.Pierzchala@im.uj.edu.pl 1 Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland is called the pluricomplex Green function of K (with pole at infinity) or the Siciak-Zakharyuta extremal function; see for example [4,23,25,39,43,44] and the bibliography therein. Here and subsequently, | | denotes the maximum norm in C N .
If N = 1, K ⊂ C is nonpolar (that is, of positive logarithmic capacity), and K ∞ denotes the unbounded component of C\K , then V K is harmonic in C\K and the restriction of V K to K ∞ is the Green function of K ∞ with pole at infinity; see [44, 7.1 and 7.2]. Definition 1.1 (see [23]) A set A ⊂ C N is said to be pluripolar if, for each point a ∈ A, there exists an open neighbourhood U of a such that A ∩ U ⊂ {z ∈ U : ϕ(z) = −∞} for some plurisubharmonic function ϕ : U → [−∞, +∞).
If K ⊂ C N is a nonpluripolar compact set, then the upper semicontinuous regularization V * K (z) := lim sup ζ →z V K (ζ ) of V K is plurisubharmonic in C N and satisfies the complex Monge-Ampère equation: see [4,23,26] for more details. Moreover, for each compact set ∅ = K ⊂ C N , see [44,Corollary 3.9 and Theorem 3.10]). We should also recall that the pluricomplex Green function is essentially equivalent to the Siciak extremal function. The latter is defined by the formula K (z) := sup |Q(z)| 1/deg Q : Q ∈ C[z 1 , . . . , z N ] , deg Q > 0 and Q K ≤ 1 (z ∈ C N ), where Q K := sup z∈K |Q(z)|. Strictly speaking, for each compact set ∅ = K ⊂ C N , (1.2) see [23,Theorem 5.1.7]. It may be worth noting that, for each Q ∈ C[z 1 , . . . , z N ] with Q K > 0 and each z ∈ C N , (We adopt here the convention that, for each r ≥ 0, (+∞) r := +∞. Moreover, for K being nonpluripolar, the additional assumption that Q K > 0 is superfluous.) This trivial (but useful) estimate is called the Bernstein-Walsh inequality.
The pluricomplex Green function has been used to study various problems in (real and complex) analysis, functional analysis, pluripotential theory, complex dynamics and in approximation theory. From the point of view of applications, the most desirable property of this function is the HCP property. Definition 1.2 (see [29]) We say that a compact set ∅ = K ⊂ C N has the HCP property if there exist , μ > 0 such that, for each z ∈ K (1) , (1.4) D(a, r ) := z ∈ C N : |z − a| ≤ r (a ∈ C N , r > 0), the HCP property holds with the exponent μ = 1. Indeed, since V D(a,r ) (z) = max 0, log |z − a| r (see [23,Example 5.1.1]), it follows that V D(a,r ) (z) ≤ dist z, D(a, r ) r for all z ∈ C N . On the other hand, however, for the cube [−1, 1] N ⊂ R N ⊂ C N , the exponent μ = 1/2 is the best possible, which follows from the formula valid for all z = (z 1 , . . . , z N ) ∈ C N ; see [23,Corollary 5.4.5]. For each j ≤ N , the square root is so chosen that |z j + z j 2 − 1| ≥ 1.
There have been several significant advances in understanding the HCP property for compact subsets of C. In particular, we should mention here a very interesting work of Carleson and Totik [16], in which they give a sufficient Wiener-type criterion for a compact set K ⊂ C to have the HCP property.
Incomparably less has been done so far in the multivariate case (that is, for N > 1). However, Pawłucki and Pleśniak in a seminal paper [29] give a sufficient geometric condition (UPC condition) for a compact set K ⊂ R N to have the HCP property. Furthermore, in [29,30,32] large and natural classes of compact sets in R N satisfying the UPC condition (and hence with the HCP property) are provided. More precisely, these classes consist of all compact, fat (a set E is said to be fat if E = IntE) and definable sets in certain o-minimal structures; see [18] for the definition of an ominimal structure. Each compact, fat and semianalytic subset of R N is an explicit example of such a set. Definition 1.4 (see [5,27]) Let ⊂ R N be an open set. A set A ⊂ is said to be a semianalytic subset of if, for each point in , we can find a neighbourhood W such that A ∩ W is a finite union of sets of the form where ξ, ξ 1 , . . . , ξ m are real analytic functions in W .
One of the long-standing open problems concerning the HCP property is the following. Problem 1.5 (Pleśniak, 1988)  Recall thatK denotes the polynomially convex hull of K : (We set∅ := ∅.) IfK = K , then we say that K is polynomially convex. Occasionally, we will write K instead ofK . It is well known that:  6 Note that, in Problem 1.5, the assumption thatK ⊂ U is quite natural. Indeed, consider the simplest example: take a ∈ C with |a| > 1, and put Moreover, for each n ∈ N, set Q n (z) := z n (z − a) , a n := a + 1 n .
SinceK = {|z| ≤ 1}, it follows that for all z ∈ C; see Example 1.3. In particular, the set K has the HCP property. On the other hand, and hence h(K ) (and V h(K ) ) is not even continuous.
Problem 1.5 is well known to specialists in the field and, at least since the 1980s, a number of attempts have been made to give a solution. We should mention here a result due to Pleśniak [37], which reads as follows. Let h : U → C N , where U ⊂ C N is an open set, be a holomorphic map (N ∈ N). Assume that a compact, polynomially convex set ∅ = K ⊂ U has the HCP property and h is nonsingular (see Definition 1.7) on K . Then h(K ) has the HCP property as well. To my knowledge, except for this result of Pleśniak, which goes back to 1988, there has been no satisfactory progress on Problem 1.5. In this paper, we prove Theorem 1.8, which gives a complete solution of this problem. Before we state it, however, we set up terminology. • We say that h is nondegenerate if, for each connected component U ι of U , there exists ζ ι ∈ U ι such that rank d ζ ι h = N . • Let K ⊂ U . We say that h is nonsingular on K if N = N and, for each ζ ∈ K , we have rank d ζ h = N .
where {U ι } ι∈I is the family of all connected components of U . Assume that a compact set ∅ = K ⊂ C N has the HCP property andK ⊂ U . Then the following three statements are equivalent: In particular, condition (iii) is the answer to Problem 1.5. Obviously, this condition is automatically satisfied if h nondegenerate. Hence, we get the following result. One of the most important applications of the HCP property concerns multivariate polynomial inequalities. More precisely, the HCP property is a sufficient condition for Markov's inequality. Definition 1. 10 We say that a compact set ∅ = K ⊂ C N satisfies Markov's inequality (or: is a Markov set) if there exist ε, C > 0 such that, for each polynomial Q ∈ C[z 1 , . . . , z N ] and each α = (α 1 , . . . , α N ) ∈ N N 0 , where This is a generalization of the classical inequality due to Markov: If Q is a polynomial of one variable, then It is perhaps worth remarking that, for a compact set ∅ = K ⊂ C N , Markov's inequality (Definition 1.10) is equivalent to the following condition: there exist ε, D, M > 0 such that, for each polynomial Q ∈ C[z 1 , . . . , z N ] with deg Q ≤ n (n ∈ N), This follows easily from Cauchy's inequalities and Taylor's formula.
Assume that a compact set K ⊂ C N satisfies Markov's inequality,K ⊂ U , and h(K ) is a nonpluripolar subset of C N . Then h(K ) satisfies Markov's inequality as well.
Theorem 1.12 "almost" solves Problem 1.11. The only issue here is the nonpluripolarity assumption on the set h(K ). However, this assumption is really weak (in particular, pluripolar sets have Lebesgue measure zero; see [23, Corollary 2.9.10]). Furthermore, it is conjectured that all (nonempty) Markov sets are nonpluripolar. If this conjecture is true, then the set K of Theorem 1.12 is nonpluripolar and an elementary argument shows (cf. the proof of [36, Lemma 2.5]) that h(K ) is nonpluripolar as well.
The proofs of Theorems 1.8 and 1.12 involve several ingredients. One of them is common to both proofs. It is the following result describing the geometry of nondegenerate holomorphic mappings.
is an open set, be a nondegenerate holomorphic map (N , N ∈ N). Assume that K ⊂ U is a compact set. Then there exist κ, θ, t * > 0 and q ∈ N such that, for each a ∈ K , D(a, t * ) ⊂ U and we can choose a polynomial map Q a : Recall that in our notation D(a, t * ) := z ∈ C N : |z − a| ≤ t * .
2 Proof of Theorem 1.13

Key lemmas
We have divided the proof into a sequence of lemmas. The last lemma is the desired conclusion.
Take an open and bounded set ⊂ C N such that K ⊂ , ⊂ U and is a semianalytic subset of R 2N . Moreover, set Lemma 2.1 There exist θ 1 , θ 2 , υ > 0 and d ∈ N such that, for each a ∈ E, we can choose a polynomial map R a : C → C N with deg R a ≤ d, satisfying the following conditions:

Lemma 2.4
There exist θ, t * > 0 and q ∈ N such that, for each a ∈ K , D(a, t * ) ⊂ U and we can choose a polynomial map Q a :

Proofs of key lemmas
Proof of Lemma 2.1 Set Note that It follows that Y is a closed and semianalytic subset of U ⊂ R 2N , which implies that E = \Y is an open, bounded and semianalytic subset of R 2N . By [29, Theorem 6.4], there exist θ 1 , υ > 0 and d ∈ N such that, for each x ∈ E, we can choose a polynomial map P x : R → R 2N with deg P x ≤ d, satisfying the following conditions: Take ϕ 1 , . . . , ϕ d : E → R 2N such that, for each x ∈ E and each t ∈ R, By [30, Lemma 3.1], the maps ϕ 1 , . . . , ϕ d are bounded, and hence For each a ∈ E and t ∈ R, set R a (t) := χ N P a (t) , where It is straightforward to see that conditions (S1) and (S2) hold with θ 2 := √ 2C.
Proof of Lemma 2.2 Note that, for each a ∈ E and each t ∈ [0, 1], R a (t) ∈ E, and hence the first required estimate holds with the constant Let Y be as in the proof of Lemma 2.1. There are two cases to consider.
Note that, for each ζ ∈ E, Thus, we have in particular the following estimate: M N η , Note that θ 5 and θ 6 depend neither on a nor t. Moreover, (2.4) gives Clearly, d b H σ ≤ max{η 1 , 1}, and hence, by (2.5), which gives (2.6). Consider the map and hence But, for each z ∈ C N and τ > 0, Combining this with (2.8), we get The above inclusion implies that and hence 2) and (2.4)) and r ≤ θ 5 t (see (2.2)), we get which is the desired conclusion.

Proof of Theorem 1.9
Proof of Theorem 1. 9 Take an open, bounded set ⊂ C N such thatK ⊂ and ⊂ U . Furthermore, take a compact and polynomially convex set E ⊂ C N such that K ⊂ IntE and E ⊂ ; cf. [22, Proof of Lemma 2.7.4]. By the uniform version of the Bernstein-Walsh-Siciak theorem (see [36,41] and see also [33]), there exist C 1 > 0 and ρ ∈ (0, 1) with the following property: for each holomorphic and bounded function f : By Theorem 1.13, there exist κ, θ, t * > 0 and q ∈ N such that, for each a ∈ K , D(a, t * ) ⊂ U and we can choose a polynomial map Q a : C → C N with deg Q a ≤ q such that We can clearly assume that κ ∈ N and K (t * ) ⊂ E.

.4 and Corollary 5.2.5] imply that
and hence , where E := K ∩U * . We can, clearly, assume that U * = U , because otherwise Theorem 1.9 immediately completes the proof. In brief outline, the idea of the proof is the following: • we use the hypothesis that K has the HCP property (4.1) to show that E satisfies Markov's inequality (4.8); • then we show, with the aid of (4.1) and (4.8), that E has the HCP property (4.9); • finally, we apply Theorem 1.9 to the set E and the map h| U * to deduce that h(E) has the HCP property (4.11); • the proof of (i) concludes by observing, via the hypothesis (iii), that The HCP property of K means that there exist θ 1 , μ > 0 such that, for each z ∈ K (1) , (4.1) By [46,Remark 3.7], μ ≤ 1. Consider f : U → C given by Also, take a compact and polynomially convex set ⊂ C N such thatK ⊂ Int and ⊂ U ; cf. [22, Proof of Lemma 2.7.4]. By [44,Theorem 8.5(1)], there exist θ 2 > 0, ρ ∈ (0, 1), and a sequence of polynomials W ν ∈ C[z 1 , . . . , z N ] (ν ∈ N) with deg W ν ≤ ν such that, for each ν ∈ N, We can, clearly, assume that θ 2 ≥ 1. Take ν 0 ∈ N such that 2θ 2 ρ ν 0 < 1. We now show that To this end, take c ∈Ê, and suppose, towards a contradiction, that c / ∈ U * . Since c ∈K \U * ⊂ U \U * , we have from (4.2) Hence, which implies that c / ∈Ê, a contradiction. Similarly, we show that E is nonpluripolar. Indeed, striving for a contradiction, assume that E is pluripolar. Note that ∅ = h(K ) ⊂ h(E) , which implies that E = ∅. Takec ∈ E. Then, from (4.2), Hence,c / ∈ (K \E) , and by [23, Theorem 5.2.4 and Corollary 5.2.5], which is a contradiction.
We are now in a position to show that, for each a ∈ E (θ 3 ) , To prove this, fix a ∈ E (θ 3 ) and Q ∈ C[z 1 , . . . , z N ] such that d := deg Q > 0. We need to check that where δ := dist(a, E).
To complete the proof, note that h(E) has the HCP property. Indeed, sinceÊ ⊂ U * (see (4.3)) and E has the HCP property (see (4.9)), we can apply Theorem 1.9 to the set E and the map h| U * . Therefore, there exist θ, γ > 0 such that, for all y, y ∈ C N , (4.11) Recall also that we are now working under the assumption that (iii) holds, and so Consequently, On the other hand, [23, Theorem 5.
. Together with (4.11), this completes the proof of (i) under the assumption that (iii) holds.

Proof of Theorem 1.12
Proof of Theorem 1. 12 Take an open, bounded set ⊂ C N such thatK ⊂ and ⊂ U . Since h(K ) is a nonpluripolar subset of C N and h( ) is bounded, see (1.1) and (1.2).
As in the proof of Theorem 1.9, we show that there exist a set E ⊂ C N and constants C 2 > 0, ρ ∈ (0, 1) such that • K ⊂ IntE and E ⊂ , • for each holomorphic and bounded function f : → C and each ν ∈ N, there exists a polynomial R ν ∈ C[z 1 , . . . , z N ] with deg R ν ≤ ν and such that Furthermore, take k ∈ N and t 1 > 0 such that and Let ε, C > 0 be of the definition of Markov's inequality for the set K. That is, for each polynomial R ∈ C[z 1 , . . . , z N ] and each β ∈ N N 0 , Moreover, let κ, θ, t * > 0 and q ∈ N be of Theorem 1.13. Set κ := κ(2 + ε) , Ck ε , Fix a polynomial P ∈ C[y 1 , . . . , y N ] with deg P ≤ n (n ∈ N), and also fix y ∈ h(K ) (C 5 n −κ ) . We now show that |P(y)| ≤ C 6 P h(K ) . (5.5) Set l = l(n) := kn. By (5.1), there exists a polynomial R l ∈ C[z 1 , . . . , z N ] with deg R l ≤ l and such that Indeed, which proves (5.7).

A refinement of Theorems 1.9 and 1.12
We conclude this paper with one more result, which is a slight strengthening of Theorems 1.9 and 1.12.  (N , N ∈ N). Assume that Z ⊂ C N is a compact set withẐ ⊂ U . Then there exist η 1 , η 2 > 0 such that: (1) For each compact set ∅ = K ⊂ Z having the HCP property with the exponent μ > 0 (that is, (1.4) holds with some > 0), the set h(K ) has the HCP property with the exponent η 1 μ. (2) For each compact set K ⊂ Z satisfying Markov's inequality with the exponent ε > 0 (that is, (1.5) holds with some C > 0) and such that h(K ) is a nonpluripolar subset of C N , the set h(K ) satisfies Markov's inequality with the exponent η 2 ε.