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Pluricapacity and approximation numbers of composition operators Daniel Li, Hervé Queffélec, L. Rodríguez-Piazza September 25, 2018

Abstract. For suitable bounded hyperconvex sets Ω in CN , in particular the ball or the polydisk, we give estimates for the approximation numbers of composition operators Cϕ : H 2 (Ω) → H 2 (Ω) when ϕ(Ω) is relatively compact in Ω, involving the Monge-Ampère capacity of ϕ(Ω). Key-words: approximation numbers ; composition operator; Hardy space ; hyperconvex domain ; Monge-Ampère capacity ; pluricapacity ; pluripotential theory ; Zakharyuta conjecture MSC 2010 numbers – Primary: 47B33 – Secondary: 30H10 – 31B15 – 32A35 – 32U20 – 41A25 – 41A35 – 46B28 – 46E20 – 47B06

1

Introduction

Let D be the unit disk in C, H 2 (D) the corresponding Hardy space, ϕ a non-constant analytic self-map of D and Cϕ : H 2 (D) → H 2 (D) the associated composition operator. In [40], we proved a formula connecting the approximation numbers an (Cϕ ) of Cϕ , and the Green capacity of the image ϕ(D) in D, namely, when [ϕ(D)] ⊂ D, we have: (1.1) β(Cϕ ) := lim [an (Cϕ )]1/n = exp − 1/Cap [ϕ(D)] , n→∞

where Cap [ϕ(D)] is the Green capacity of ϕ(D). A non-trivial consequence of that formula was the following: (1.2)

kϕk∞ = 1

=⇒

an (Cϕ ) ≥ δ e−nεn where εn → 0+ .

In other terms, as soon as kϕk∞ = 1, we cannot hope better for the numbers an (Cϕ ) than a subexponential decay, however slowly εn tends to 0. In [41], we pursued that line of investigation in dimension N ≥ 2, namely on H 2 (DN ), and showed that in some cases the implication (1.2) still holds ([41, Theorem 3.1]): (1.3)

kϕk∞ = 1

=⇒

1/N

an (Cϕ ) ≥ δ e−n 1

εn

where εn → 0+ ,

(the substitution of n by n1/N is mandatory as shown by the results of [4]). We show in this paper that, in general, for non-degenerate symbols, we have similar formulas to (1.1) at our disposal for the parameters: − (1.4) βN (Cϕ ) = lim inf [anN (Cϕ )]1/n n→∞

+ and βN (Cϕ ) = lim sup[anN (Cϕ )]1/n . n→∞

These bounds are given in terms of the Monge-Ampère (or Bedford-Taylor) capacity of ϕ(DN ) in DN , a notion which is the natural multidimensional extension of the Green capacity when the dimension N is ≥ 2 ([41, Theorem 6.4]). We − + show that we have βN (Cϕ ) = βN (Cϕ ) for well-behaved symbols.

2 2.1

Notations and background Complex analysis

Let Ω be a domain in CN ; a function u : Ω → R ∪ {−∞} is said plurisubharmonic (psh) if it is u.s.c. and if for every complex line L = {a + zw ; z ∈ C} (a ∈ Ω, w ∈ CN ), the function z 7→ u(a + zw) is subharmonic in Ω ∩ L. We denote PSH(Ω) the set of plurisubharmonic functions in Ω. If f : Ω → C is holomorphic, then log |f | and |f |α , α > 0, are psh. Every real-valued convex function is psh (convex functions are those whose composition with all R-linear isomorphisms are subharmonic, though plurisubharmonic functions are those whose composition with all C-linear isomorphisms are subharmonic: see [30, Theorem 2.9.12]). ¯ and (ddc )N = ddc ∧· · ·∧ddc (N times). When u ∈ PSH(Ω)∩ Let ddc = 2i∂ ∂, 2 C (Ω), we have: 2 ∂ u (ddc u)N = 4N N ! det dλ2N (z) , ∂zj ∂ z¯k where dλ2N (z) = (i/2)N dz1 ∧d¯ z1 ∧· · ·∧dzN ∧d¯ zN is the usual volume in CN . In c N general, the current (dd u) can be deﬁned for all locally bounded u ∈ PSH(Ω) and is actually a positive measure on Ω ([5]). Given p1 , . . . , pJ ∈ Ω, the pluricomplex Green function with poles p1 , . . . , pJ and weights c1 , . . . , cN > 0 is deﬁned as: g(z) = g(z, p1 , . . . , pJ ) = sup{v(z) ; v ∈ PSH(Ω) , v ≤ 0 and v(z) ≤ cj log |z − pj | + O (1) , ∀j = 1, . . . , J} . In particular, for J = 1 and p1 = a, c1 = 1, g( · , a) is the pluricomplex Green function of Ω with pole a ∈ Ω. If 0 ∈ Ω and a = 0, we denote it by gΩ and call it the pluricomplex Green function of Ω; hence: ga (z) = g(z, a) = sup{u(z) ; u ∈ PSH(Ω) , u ≤ 0 and u(z) ≤ log |z−a|+O (1)} . 2

Let Ω be an open subset of CN . A continuous function ρ : Ω → R is an exhaustion function if there exists a ∈ (−∞, +∞] such that ρ(z) < a for all z ∈ Ω, and the set Ωc = {z ∈ Ω ; ρ(z) < c} is relatively compact in Ω for every c < a. A domain Ω in CN is said hyperconvex if there exists a continuous psh exhaustion function ρ : Ω → (−∞, 0) (see [30, p. 80]). We may of course replace the upper bound 0 by any other real number. Without this upper bound, Ω is said pseudoconvex. Let Ω be a hyperconvex domain, with negative continuous psh exhaustion function ρ and µρ,r the associated Demailly-Monge-Ampère measures, deﬁned as: (2.1)

µu,r = (ddc ur )N − 1Ω\BΩ,u (r) (ddc u)N ,

for r < 0, where ur = max(u, r) and: BΩ,u (r) = {z ∈ Ω ; u(z) < r} . The nonnegative measure µu,r is supported by SΩ,u (r) := {z ∈ Ω ; u(z) = r}. If: Z (ddc ρ)N < ∞ , Ω

these measures, considered as measures on Ω, weak-∗ converge, as r Rgoes to 0, to a positive measure µ = µΩ,ρ supported by ∂Ω and with total mass Ω (ddc ρ)N ([16, Théorème 3.1], or [30, Lemma 6.5.10]). For the pluricomplex Green function ga with pole a, we have (ddc ga )N = (2π)N δa ([16, Théorème 4.3]) and ga (a) = −∞, so a ∈ BΩ,ga (r) for every r < 0 and 1Ω\BΩ,ga (r) (ddc ga )N = 0. Hence the Demailly-Monge-Ampère mesure µga ,r N N = is equal to ddc (ga )r . By [51, Lemma 1], we have (1/|r|) ddc (ga )r ¯Ω,ga (r) = {z ∈ Ω ; ga (z) ≤ r} uB¯Ω,ga (r),Ω , the relative extremal function of B in Ω (see (3.2) for the deﬁnition), and this measure is supported, not only by ¯Ω,ga (r) (see Section 2.2.1 for SΩ,ga (r), but merely by the Shilov boundary of B the deﬁnition). Since (ddc ga )N = (2π)N δa has mass (2π)N < ∞, these measures weak-∗ converge, as r goes to 0, to a positive measure µ = µΩ,ga supported by ∂Ω 1 with mass (2π)N . Demailly ([16, Déﬁnition 5.2] call the measure (2π) N µΩ,ga the pluriharmonic measure of a. When Ω is balanced (az ∈ Ω for every z ∈ Ω and |a| = 1), the support of this pluriharmonic measure is the Shilov boundary of Ω ([51, very end of the paper]). A bounded symmetric domain of CN is a bounded open and convex subset Ω of CN which is circled (az ∈ Ω for z ∈ Ω and |a| ≤ 1) and such that for every point a ∈ Ω, there is an involutive bi-holomorphic map γ : Ω → Ω such that a is an isolated ﬁxed point of γ (equivalently, γ(a) = a and γ ′ (a) = −id: see [52, Proposition 3.1.1]). For this deﬁnition, see [13, Deﬁnition 16 and Theorem 17], or [14, Deﬁnition 5 and Theorem 4]. Note that the convexity is automatic 3

(Hermann Convexity Theorem; see [27, p. 503 and Corollary 4.10]). É. Cartan showed that every bounded symmetric domain of CN is homogeneous, i.e. the group Γ of automorphisms of Ω acts transitively on Ω: for every a, b ∈ Ω, there is an automorphism γ of Ω such that γ(a) = b (see [52, p. 250]). Conversely, every homogeneous bounded convex domain is symmetric, since σ(z) = −z is a symmetry about 0 (see [52, p. 250] or [26, Remark 2.1.2 (e)]). Moreover, each automorphism extends continuously to Ω (see [22]). The unit ball BN and the polydisk DN are examples of bounded symmetric domains. Another example is, for N = p q, bi-holomorphic to the open unit ball of M(p, q) = L(Cq , Cp ) for the operator norm (see [27, Theorem 4.9]). Every product of bounded symmetric domains is still a bounded symmetric domain. In particular, every product of balls Ω = Bl1 × · · · × Blm , l1 + · · · + lm = N , is a bounded symmetric domain. If Ω is a bounded symmetric domain, its gauge is a norm k . k on CN whose open unit ball is Ω. Hence every bounded symmetric domain is hyperconvex (take ρ(z) = kzk − 1).

2.2 2.2.1

Hardy spaces on hyperconvex domains Hardy spaces on bounded symmetric domains

We begin by deﬁning the Hardy space on a bounded symmetric domain, because this is easier. The Shilov boundary (also called the Bergman-Shilov boundary or the distinguished boundary) ∂S Ω of a bounded domain Ω is the smallest closed set F ⊆ ∂Ω such that supz∈Ω |f (z)| = supz∈F |f (z)| for every function f holomorphic in some neighborhood of Ω (see [13, § 4.1]). When Ω is a bounded symmetric domain, it is also, since Ω is convex, the Shilov boundary of the algebra A(Ω) of the continuous functions on Ω which are holomorphic in Ω (because every function f ∈ A(Ω) can be approximated by fε with fε (z) = f εz0 + (1 − ε)z , where z0 ∈ Ω is given: see [20, pp. 152–154]). The Shilov boundary of the ball BN is equal to its topological boundary, but the Shilov boundary of the bidisk is ∂S D2 = {(z1 , z2 ) ∈ C2 ; |z1 | = |z2 | = 1}, whereas, its usual boundary ∂D2 is (T × D) ∪ (D × T); for the unit ball BN , the Shilov boundary is equal to the usual boundary SN −1 ([13, § 4.1]). Another example of a bounded symmetric domain, in C3 , is the set Ω = {(z1 , z2 , z3 ) ∈ C3 ; |z1 |2 + |z2 |2 < 1 , |z3 | < 1} and its Shilov boundary is ∂S Ω = {(z1 , z2 , z3 ) ; |z1 |2 + |z2 |2 = 1 , |z3 | = 1}. For p ≥ q, the matrix A is in the topological boundary of M(p, q) if and only if kAk = 1, but A is in the Shilov boundary if and only if A∗ A = Iq ; therefore the two boundaries coincide if and only if q = 1, i.e. Ω = BN (see [14, Example 2, p. 30]). Equivalently (see [24, Corollary 9], or [13, Theorem 33], [14, Theorem 10]), ∂S Ω is the set of the extreme points of the convex set Ω. The Shilov boundary ∂S Ω is invariant by the group Γ of automorphisms of Ω and the subgroup Γ0 = {γ ∈ Γ ; γ(0) = 0} act transitively on ∂S Ω (see [22]). A theorem of H. Cartan states that the elements of Γ0 are linear trans4

formations of CN and commute with the rotations (see [24, Theorem 1] or [26, Proposition 2.1.8]). It follows that the Shilov boundary of a bounded symmetric domain Ω coincides with its topological boundary only for Ω = BN (see [35, p. 572] or [36, p. 367]); in particular the open unit ball of CN for the norm k . kp , 1 < p < ∞, is never a bounded symmetric domain, unless p = 2. The unique Γ0 -invariant probability measure σ on ∂S Ω is the normalized surface area (see [22]). Then the Hardy space H 2 (Ω) is the space of all complexvalued holomorphic functions f on Ω such that: kf kH 2 (Ω) :=

sup 0

Z

∂S Ω

1/2 |f (rξ)|2 dσ(ξ)

is ﬁnite (see [22] and [23]). It is known that the integrals in this formula are non-decreasing as r increases to 1, so we can replace the supremum by a limit. The same deﬁnition can be given when Ω is a bounded complete Reinhardt domain (see [1]). The space H 2 (Ω) is a Hilbert space (see [22, Theorem 5]) and for every z ∈ Ω, the evaluation map f ∈ H 2 (Ω) 7→ f (z) is uniformly bounded on compacts subsets of Ω, by a depending only on that compact set, and of Ω ([22, Lemma 3]). For every f ∈ H 2 (Ω), there exists a boundary values function f ∗ such that kfr − f ∗ kL2 (∂S Ω) −→ 0, where fr (z) = f (rz) ([9, Theorem 3]), and the map r→1

f ∈ H 2 (Ω) 7→ f ∗ ∈ L2 (∂S Ω) is an isometric embedding ([22, Theorem 6]).

2.2.2

Hardy spaces on hyperconvex domains

For hyperconvex domains, the deﬁnition of Hardy spaces is more involved. It was done by E. Poletsky and M. Stessin ([47, Theorem 6]). Those domains are associated to a continuous negative psh exhaustion function u on Ω and the deﬁnition of the Hardy spaces uses the Demailly-Monge-Ampère measures. The space Hu2 (Ω) is the space of all holomorphic functions f : Ω → C such that: Z sup |f |2 dµu,r < ∞ r<0

SΩ,u (Ω)

and its norm is deﬁned by: kf kHu2 (Ω) = sup r<0

1 (2π)N

Z

2

SΩ,u (Ω)

|f | dµu,r

1/2

.

We can replace the supremum by a limit since the integrals are non-decreasing as r increases to 0 ([16, Corollaire 1.9]. The space H ∞ (Ω) of bounded holomorphic functions in Ω is contained in 2 Hu (Ω) (see [47], remark before Lemma 3.4). These spaces Hu2 (Ω) are Hilbert spaces ([47, Theorem 4.1]), but depends on the exhaustion function u (even when N = 1: see for instance [49]). Nevertheless, they all coincide, with equivalent norms, for the functions u for which the measure (ddc u)N is compactly supported ([47, Lemma 3.4]); this is the case 5

when u(z) = g(z, a) is the pluricomplex Green function with pole a ∈ Ω (because then (ddc u)N = (2π)N δa : see [16, Théorème 4.3], or [30, Theorem 6.3.6]). When Ω is the ball BN and u(z) = log kzk2, then (ddc u)N = C δ0 and µu,r = (2π)N dσt , where dσt is the normalized surface area on the sphere of radius t := er (see [47, Section 4] or [17, Example 3.3]). When Ω is the polydisk DN and u(z) = log kzk∞, then (ddc u)N = (2π)N δ0 ([18, Corollary 5.4]) and 1 N (see [17, Example 3.10]). (2π)N µu,r is the Lebesgue measure of the torus rT i ¯ c Note that in [17] and [18], the operator d is deﬁned as 2π (∂ − ∂) instead of ¯ i(∂ − ∂), as usually used. In these two cases, the Hardy spaces are the same as the usual ones (see [2, Remark 5.2.1]). In the sequel, we only consider the exhausting function u = gΩ ; hence we will write BΩ (r), SΩ (r) and H 2 (Ω) instead of BΩ,u (r), SΩ,u (r) and Hu2 (Ω). The two notions of Hardy spaces for a bounded symmetric domain are the same: Proposition 2.1. Let Ω be a bounded symmetric domain in CN . Then the Hardy space H 2 (Ω) coincides with the subspace of the Poletski-Stessin Hardy space HgΩ (Ω), with equality of the norms. Proof. First let us note that if k . k is the norm whose open unit ball is Ω, then gΩ (z) = log kzk (see [7, Proposition 3.3.2]). Let µΩ be the measure which is the ∗-weak limit of the Demailly-MongeN Ampère measures µr = ddc (gΩ )r . We saw that it is supported by ∂S Ω. By the remark made in [16, pp. 536-537], since the automorphisms of Ω continuously extend on ∂Ω, the measure µΩ is Γ-invariant. By unicity, the harmonic measure µ eΩ = (2π)−N µΩ at 0 hence coincides with the normalized area measure on ∂S Ω. We have, for f : Ω → C holomorphic and 0 < s < 1: Z Z Z 1 |f (sz)|2 dµr (z) , |f (sz)|2 de µΩ (z) = lim |f (sz)|2 de µΩ (z) = r→0 (2π)N S (r) ∂Ω ∂S Ω Ω because z 7→ |f (sz)|2 is continuous on Ω. Now, since gΩ (z) = log kzk, we have SΩ (r) = er ∂Ω and (gΩ )r (z)+t = (gΩ )r+t (sz); hence µr (sA) = µr+t (A) for every Borel subset A of ∂Ω, where t = log s. It follows that: Z Z |f (ζ)|2 dµr+t (ζ) . |f (sz)|2 dµr (z) = SΩ (r+t)

SΩ (r)

By letting r and t going to 0, we get: Z 1 kf k2H 2 (Ω) = lim |f (ζ)|2 dµr+t (ζ) = kf k2Hg2 ; r,t→0 (2π)N S (r+t) Ω Ω hence f ∈ H 2 (Ω) if and only if f ∈ Hg20 (Ω), with the same norms. We have ([47, Theorem 3.6]): 6

Proposition 2.2 (Poletsky-Stessin). For every z ∈ Ω, the evaluation map f ∈ H 2 (Ω) 7→ f (z) is uniformly bounded on compacts subsets of Ω, by a constant depending only on that compact set, and of Ω. Hence H 2 (Ω) has a reproducing kernel, deﬁned by: (2.2)

for f ∈ H 2 (Ω) ,

f (a) = hf, Ka i ,

and for each r < 0: (2.3)

Lr :=

sup kKa k2 < ∞ .

a∈BΩ (r)

2.3

Composition operators

A Schur map, associated with the bounded hyperconvex domain Ω, is a non-constant analytic map of Ω into itself. It is said non degenerate if its Jacobian is not identically null. It is equivalent to say that the diﬀerential ϕ′ (a) : CN → CN is an invertible linear map for at least one point a ∈ Ω. In [4], we used the terminology truly N -dimensional. Then, by the implicit function theorem, this is equivalent to saying that ϕ(Ω) has non-void interior. We say that the Schur map ϕ is a symbol if it deﬁnes a bounded composition operator Cϕ : H 2 (Ω) → H 2 (Ω) by Cϕ (f ) = f ◦ ϕ. Let us recall that although any Schur function generates a bounded composition operator on H 2 (D), this is no longer the case on H 2 (DN ) as soon as N ≥ 2, as shown for example by the PnSchur map ϕ(z1 , z2 ) = (z1 , z1 ). Indeed (see [3]), if say N = 2, taking f (z) = j=0 z1j z2n−j , we see that: √ kf k2 = n + 1 while kCϕ f k2 = k(n + 1)z1n k2 = n + 1 . The same phenomenon occurs on H 2 (BN ) ([43]; see also [11], [12], and [15]; see also [47]).

2.4

s-numbers of operators on a Hilbert space

We begin by recalling a few operator-theoretic facts. Let H be a Hilbert space. The approximation numbers an (T ) = an of an operator T : H → H are deﬁned as: (2.4)

an =

inf

rank R

kT − Rk ,

n = 1, 2, . . .

The operator T is compact if and only if limn→∞ an (T ) = 0. According to a result of Allahverdiev [10, p. 155], √ an = sn , the n-th singular number of T , i.e. the n-th eigenvalue of |T | := T ∗ T when those eigenvalues are rearranged in non-increasing order. The n-th width dn (K) of a subset K of a Banach space Y measures the defect of ﬂatness of K and is by deﬁnition: sup dist (f, E) , (2.5) dn (K) = inf dim E

f ∈K

7

where E runs over all subspaces of Y with dimension < n and where dist (f, E) denotes the distance of f to E. If T : X → Y is an operator between Banach spaces, the n-th Kolmogorov number dn (T ) of T is the nth-width in Y of T (BX ) where BX is the closed unit ball of X, namely: sup dist (T f, E) . (2.6) dn (T ) = inf dim E

f ∈BX

In the case where X = Y = H, a Hilbert space, we have: (2.7)

an (T ) = dn (T ) for all n ≥ 1 ,

and ([40]) the following alternative deﬁnition of an (T ): sup dist (T f, T E) . (2.8) an (T ) = inf dim E

f ∈BH

In this work, we use, for an operator T : H → H, the following notation: − βN (T ) = lim inf [anN (T )]1/n

(2.9)

n→∞

and: (2.10)

+ βN (T ) = lim sup[anN (T )]1/n . n→∞

When these two quantities are equal, we write them βN (T ).

3 3.1

Pluripotential theory Monge-Ampère capacity

Let K be a compact subset of an open subset Ω of CN . The Monge-Ampère capacity of K has been deﬁned by Bedford and Taylor ([5]; see also [30, Part II, Chapter 1]) as: Z c N Cap (K) = sup (dd u) ; u ∈ PSH(Ω) and 0 ≤ u ≤ 1 on Ω . K

When Ω is bounded and hyperconvex, we have a more convenient formula ([5, Proposition 5.3], [30, Proposition 4.6.1]): Z Z c ∗ N (3.1) Cap (K) = (dd uK ) = (ddc u∗K )N , Ω

K

(the positive measure (ddc u∗K )N is supported by K; actually by ∂K: see [17, Properties 8.1 (c)]), where uK = uK,Ω is the relative extremal function of K, deﬁned, for any subset E ⊆ Ω, as: (3.2)

uE,Ω = sup{v ∈ PSH(Ω) ; v ≤ 0 and v ≤ −1 on E} , 8

and u∗E,Ω is its upper semi-continuous regularization: u∗E,Ω (z) = lim sup uE,Ω (ζ) , ζ→z

z ∈ Ω,

called the regularized relative extremal function of E. For an open subset ω of Ω, its capacity is deﬁned as: Cap (ω) = sup{Cap (K) ; K is a compact subset of ω} . When ω ⊂ Ω is a compact subset of Ω, we have ([5, equation (6.2)], [30, Corollary 4.6.2]): Z (3.3) Cap (ω) = (ddc uω )N . Ω

The outer capacity of a subset E ⊆ Ω is: Cap ∗ (E) = inf{Cap (ω) ; ω ⊇ E and ω open} . If Ω is hyperconvex and E relatively compact in Ω, then ([30, Proposition 4.7.2]): Z ∗ Cap (E) = (ddc u∗E,Ω )N . Ω

Remark. A. Zeriahi ([57]) pointed out to us the following result. Proposition 3.1. Let K be a compact subset of Ω. Then: Cap (K) = Cap (∂K) . Proof. Of course uK ≤ u∂K since ∂K ⊆ K. Conversely, let v ∈ PSH(Ω) non-positive such that v ≤ −1 on ∂K. By the maximum principle (see [30, Corollary 2.9.6]), we get that v ≤ −1 on K. Hence v ≤ uK . Taking the supremum over all those v, we obtain u∂K ≤ uK , and therefore u∂K = uK . By (3.1), it follows that: Z Z (3.4) Cap (K) = (ddc u∗K )N = (ddc u∗∂K )N = Cap (∂K) . Ω

Ω

3.2

Regular sets

Let E ⊆ CN be bounded. Recall that the polynomial convex hull of E is: b = {z ∈ C ; |P (z)| ≤ sup |P | for every polynomial P } . E E

b is called regular if u∗ (a) = −1 for an open set Ω ⊇ E b A point a ∈ E E,Ω (note that we always have uE,Ω = uE,Ω = −1 on the interior of E: see [17, b are Properties 8.1 (c)]). The set E is said to be regular if all points of E regular. 9

The pluricomplex Green function of E, also called the L-extremal function of E, is deﬁned, for z ∈ CN , as: VE (z) = sup{v(z) ; v ∈ L ,

v ≤ 0 on E} ,

where L is the Lelong class of all functions v ∈ PSH(CN ) such that, for some constant C > 0: v(z) ≤ C + log(1 + |z|) for all z ∈ CN .

b is called L-regular if V ∗ (a) = 0, where V ∗ is the upper semiA point a ∈ E E E b are continuous regularization of VE . The set E is L-regular if all points of E L-regular. By [28, Proposition 2.2] (see also [30, Proposition 5.3.3, and Corollary 5.3.4]), b for E bounded and non pluripolar, and Ω a bounded open neigbourhood of E, we have: (3.5)

m(uE,Ω + 1) ≤ VE ≤ M (uE,Ω + 1)

b is equivalent for some positive constants m, M . Hence the regularity of a ∈ E to its L-regularity. Recall that E is pluripolar if there exists an open set Ω containing E and v ∈ PSH(Ω) such that E ⊆ {v = −∞}. This is equivalent to say that there exists a hyperconvex domain Ω of CN containing E such that u∗E,Ω ≡ 0 (see [30, Corollary 4.7.3 and Theorem 4.7.5]). By Josefson’s theorem ([30, Theorem 4.7.4]), E is pluripolar if and only if there exists v ∈ PSH(CN ) such that E ⊆ {v = −∞}. Recall also that E is pluripolar if and only if its outer capacity Cap ∗ (E) is null ([30, Theorem 4.7.5]). When Ω is hyperconvex and E is compact, non pluripolar, the regularity of E implies that uE,Ω and VE are continuous, on Ω and CN respectively ([30, Proposition 4.5.3 and Corollary 5.1.4]). Conversely, if uE,Ω is continuous, for some hyperconvex neighbourhood Ω of E, then uE,Ω (z) = −1 for all z ∈ E; hence VE (z) = 0 for all z ∈ E, by (3.5); but VE = VEb when E is compact b by (3.5) again, we obtain ([30, Theorem 5.1.7]), so VE (z) = 0 for all z ∈ E; b that uE,Ω (z) = −1 for all z ∈ E; therefore E is regular. In the same way, the continuity of VE implies the regularity of E. These results are due to Siciak ([50, Proposition 6.1 and Proposition 6.2]). Every closed ball B = B(a, r) of an arbitrary norm k . k on CN is regular since its L-extremal function is: VB (z) = log+ kz − ak/r) ([50, p. 179, § 2.6]).

3.3

Zakharyuta’s formula

We will need a formula that Zakharyuta, in order to solve a problem raised by Kolmogorov, proved, conditionally to a conjecture, called Zakharyuta’s conjecture, on the uniform approximation of the relative extremal function uK,Ω 10

by pluricomplex Green functions. This conjecture has been proved by Nivoche ([45, Theorem A]), in a more general setting that we state below: Theorem 3.2 (Nivoche). Let K be a regular compact subset of a bounded hyperconvex domain Ω of CN . Then for every ε > 0 and δ small enough, there exists a pluricomplex Green function g on Ω with a finite number of logarithmic poles such that: 1) the poles of g lie in W = {z ∈ Ω ; uK (z) < −1 + δ}; 2) we have, for every z ∈ Ω \ W : (1 + ε) g(z) ≤ uK (z) ≤ (1 − ε) g(z) . In order to state Zakharyuta’s formula, we need some additional notations. Let K be a compact subset of Ω with non-empty interior, and AK the set of restrictions to K of those functions that are analytic and bounded by 1, i.e. those functions belonging to the unit ball BH ∞ (Ω) of the space H ∞ (Ω) of the bounded analytic functions in Ω, considered as a subset of the space C(K) of complex functions deﬁned on K, equipped with the sup-norm on K. Let dn (AK ) be the nth-width of AK in C(K), namely: (3.6) dn (AK ) = inf sup dist (f, L) , L

f ∈AK

where L runs over all k-dimensional subspaces of C(K), with k < n. Equivalently, dn (AK ) is the nth-Kolmogorov number of the natural injection J of H ∞ (Ω) into C(K) (recall that K has non-empty interior). It is convenient to set, as in [56]: (3.7) and: (3.8)

τN (K) =

1 Cap (K) (2π)N "

ΓN (K) = exp −

N! τN (K)

1/N #

,

i.e.: (3.9)

ΓN (K) = exp − 2π

N! Cap (K)

1/N

.

Observe that Cap (K) > 0 since we assumed that K has non-empty interior. Now, we have ([56, Theorem 5.6]; see also [55, Theorem 5] or [54, pages 30–32], for a detailed proof): Theorem 3.3 (Zakharyuta-Nivoche). Let Ω be a bounded hyperconvex domain and K a regular compact subset of Ω with non-empty interior, which is holoe Ω ). Then: morphically convex in Ω (i.e. K = K 1/N N! (3.10) − log dn (AK ) ∼ n1/N . τN (K) 11

e Ω is the holomorphic convex hull of K in Ω, that is: Here K

e Ω = {z ∈ Ω ; |f (z)| ≤ sup |f | for every f ∈ O(Ω)} , K K

where O(Ω) is the set of all functions holomorphic in Ω. Relying on that theorem, which may be seen as the extension of a result of Erokhin, proved in 1958 (see [19]; see also Widom [53] which proved a more general result, with a diﬀerent proof), to dimension N > 1, and as a result on the approximation of functions, we will give an application to the study of approximation numbers of a composition operator on H 2 (Ω) for a bounded symmetric domain of CN .

4

The spectral radius type formula In [41, Section 6.2], we proved the following result.

Theorem 4.1. Let ϕ : DN → DN be given by ϕ(z1 , . . . , zN ) = (r1 z1 , . . . , rN zN ) where 0 < rj < 1. Then: βN (Cϕ ) = ΓN ϕ(DN ) = ΓN ϕ(DN ) .

The proof was simple, based on result of Blocki [8] on the Monge-Ampère capacity of a cartesian product, and on the estimation, when A → ∞, of the number νA of N -tuples α = (α1 , . . . , αN ) of non-negative integers αj such that PN j=1 αj σj ≤ A, where the numbers σj > 0 are ﬁxed. The estimation was: (4.1)

νA ∼

AN · N ! σ1 · · · σN

As J. F. Burnol pointed out to us, this is a consequence of the following elementary fact. Let λN be the Lebesgue measure on RN , and let E be a compact subset of RN such that λN (∂E) = 0. Then: λN (E) = lim A−N |(A × E) ∩ ZN | . A→∞

PN Then, just take E = {(x1 , . . . , xN ) ; xj ≥ 0 and j=1 xj σj ≤ 1}. In any case, this lets us suspect that the formula of Theorem 4.1 holds in much more general cases. This is not quite true, as evidenced by our counterexample of [41, Theorem 5.12]. Nevertheless, in good cases, this formula holds, as we will see in the next sections. In remaining of this section, we consider functions ϕ : Ω → Ω such that ϕ(Ω) ⊆ Ω. If ρ is an exhaustion function for Ω, there is some R0 < 0 such that ϕ(Ω) ⊆ BΩ (R0 ), and that implies that Cϕ maps H 2 (Ω) into itself and is a compact operator (see [47, Theorem 8.3], since, with their notations, for r > R0 , we have T (r) = ∅ and hence δϕ (r) = 0) . 12

4.1

Minoration

Recall that every hyperconvex domain Ω is pseudoconvex. By H. CartanThullen and Oka-Bremermann-Norguet theorems, being pseudoconvex is equivalent to being a domain of holomorphy, and equivalent to being holomorphically convex (meaning that if K is a compact subset in Ω, then its holomorphic hull e is also contained in Ω): see [33, Corollaire 7.7]. Now (see [32, Chapter 5, K Exercise 11], a domain of holomorphy Ω is said a Runge domain if every holomorphic function in Ω can be approximated uniformly on its compact subsets by polynomials, and that is equivalent to saying that the polynomial hull and the holomorphic hull of every compact subset of Ω agree. By [32, Chapter 5, Exercise 13], every circled domain (in particular every bounded symmetric domain) is a Runge domain. Definition 4.2. A hyperconvex domain Ω is said strongly regular if there exists a continuous psh exhaustion function ρ such that all the sub-level sets: Ωc = {z ∈ Ω ; ρ(z) < c} (c < 0) have a regular closure. For example, every bounded symmetric domain Ω is strongly regular since if k . k is the associated norm, its sub-level sets Ωc (with ρ(z) = log kzk) are the open balls B(0, ec ), and the closed balls are regular, as said above. Theorem 4.3. Let Ω be a strongly regular bounded hyperconvex and Runge domain in CN , and let ϕ : Ω → Ω be an analytic function such that ϕ(Ω) ⊆ Ω, and which is non-degenerate. Then: − (Cϕ ) . (4.2) ΓN ϕ(Ω) ≤ βN

Recall that if Ω is a domain in CN , a holomorphic function ϕ : Ω → CM (M ≤ N ) is non-degenerate if there exists a ∈ Ω such that ranka ϕ = M . Then ϕ(Ω) has a non-empty interior.

Proof. Let (rj )j≥1 be an increasing sequence of negative numbers tending to 0. The set Hj = Ωrj is a regular compact subset of Ω, with non-void interior cj its polynomial convex hull; this compact set is (hence non pluripolar). Let H cj = H fj , and since H fj ⊆ contained in Ω, since Ω being a Runge domain, we have H cj is Ω, because Ω is holomorphically convex (being hyperconvex). Moreover H regular since VE = VEb for every compact subset of CN ([50, Corollary 4.14]). cj and let G be a subspace of H 2 (Ω) with dimension < nN . Let Kj = ϕ H

The set Kj is regular because of the following result (see [30, Theorem 5.3.9], [46, top of page 40], [29, Theorem 1.3], or [44, Theorem 4], with a detailed proof). Theorem 4.4 (Pleśniak). Let E be a compact, polynomially convex, regular and non pluripolar, subset of CN . Then if Ω is a hyperconvex domain such that E ⊆ Ω and if ϕ : Ω → CN is a non-degenerate holomorphic function, the set ϕ(E) is regular. 13

cj of Kj is contained in Ω and is also As before, the polynomial convex hull K cj also. regular. Since ϕ is non-degenerate, Kj has a non-void interior; hence K cj . We can hence use Zakharyuta’s formula (Theorem 3.3) for the compact set K cj ). By By restriction, the subspace G can be viewed as a subspace of C(K Zakharyuta’s formula, for 0 < ε < 1, there is nε ≥ 1 such that, for n ≥ nε : 1/N N! ) ≥ exp − (1 + ε) (2π) n dnN (AK · cj cj ) Cap (K Hence, there exists f ∈ BH ∞ ⊆ BH 2 such that, for all g ∈ G: 1/N N! · kg − f kC(K cj ) ≥ (1 − ε) exp − (1 + ε) (2π) n cj ) Cap (K cj = K fj and, by deﬁnition k . k f = k . kC(K ) , we have: Since K j C(Kj )

kg − f kC(K cj ) = kg − f kC(Kj ) = kCϕ (g) − Cϕ (f )kC(H fj ) .

Equivalently, since, by deﬁnition k . kC(H fj ) = k . kC(Hj ) , we have, for all g ∈ G:

kCϕ (g) − Cϕ (f )kC(Hj ) ≥ (1 − ε) exp − (1 + ε) (2π) n

N! cj ) Cap (K

This implies, thanks to (2.3), that, for all g ∈ G: (1 − ε) exp − (1 + ε) (2π) n kCϕ (g) − Cϕ (f )kH 2 (Ω) ≥ L−1 rj Using (2.8), we get, since the subspace G is arbitrary: (1 − ε) exp − (1 + ε) (2π) n anN (Cϕ ) ≥ Lr−1 j

1/N

N! cj ) Cap (K

N! cj ) Cap (K

1/N

·

1/N

·

·

Taking the nth-roots and passing to the limit, we obtain: 1/N N! − βN (Cϕ ) ≥ exp − (1 + ε) (2π) · cj ) Cap (K and then, letting ε go to 0: − βN (Cϕ )

≥ exp − (2π)

N! cj ) Cap (K

1/N

cj ) . = ΓN (K

S c cj )j≥1 is increasing and Now, the sequence (K j≥1 Kj ⊇ ϕ(Ω); hence, by [5, S cj ≥ Cap [ϕ(Ω)], so: cj ) −→ Cap K Theorem 8.2 (8.3)], we have Cap (K j→∞

− βN (Cϕ ) ≥ ΓN [ϕ(Ω)] ,

and the proof of Theorem 4.3 is ﬁnished. 14

j≥1

4.2

Majorization

For the majorization, we assume diﬀerent hypotheses on the domain Ω. Nevertheless these assumptions agree with that of Theorem 4.3 when Ω is a bounded symmetric domain. 4.2.1

Preliminaries

Recall that a domain Ω of CN is a Reinhardt domain (resp. complete Reinhardt domain) if z = (z1 , . . . , zN ) ∈ Ω implies that (ζ1 z1 , . . . , ζN zN ) ∈ Ω for all complex numbers ζ1 , . . . , ζN of modulus 1 (resp. of modulus ≤ 1). A complete bounded Reinhardt domain is hyperconvex if and only if log jΩ is psh and continuous in CN \ {0}, where jΩ is the Minkowski functional of Ω (see [7, Exercise following Proposition 3.3.3]). In general, the Minkowski functional jΩ of a bounded complete Reinhardt domain Ω is usc and log jΩ is psh if and only if Ω is pseudoconvex ([7, Theorem 1.4.8]). Other conditions for a bounded complete Reinhardt domain to being hyperconvex can found in [34, Theorem 3.10]. For a bounded hyperconvex and complete Reinhardt domain Ω, its pluricomplex Green function with pole 0 is gΩ (z) = log jΩ (z), where jΩ is the Minkowski functional of Ω ([7, Proposition 3.3.2]), and SΩ (r) = er ∂Ω. Since ∂Ω is in particular invariant by the pluri-rotations z = (z1 , . . . , zN ) 7→ (eiθ1 z1 , . . . , eiθN zN ), with θ1 , . . . , θN ∈ R, the harmonic measure µ eΩ at 0 (see the proof of Proposition 2.1) is also invariant by the pluri-rotations (note that it is supported by the Shilov boundary of Ω: see [51, very end of the paper]). We have, as in the proof of Proposition 2.1, for f ∈ H 2 (Ω): Z sup |f (sz)|2 de µΩ (z) = kf k2H 2 (Ω) < ∞ . 0

∂Ω

Since µ eΩ is in particular invariant by the rotations z 7→ eiθ z, θ ∈ R, there exists, by [9, Theorem 3], a function f ∗ ∈ L2 (∂Ω, µ eΩ ) such that: Z |f (sz) − f ∗ (z)|2 de µΩ (z) −→ 0 . s→1

∂Ω

It ensues that the map f ∈ H 2 (Ω) 7→ f ∗ ∈ L2 (∂Ω, µ eΩ ) is an isometric embedding (in fact, f ∗ is the radial limit of f : see [21, Lemma 2]). Therefore, we can consider H 2 (Ω) as a complemented subspace of L2 (∂Ω, µ eΩ ), and we call P the orthogonal projection of L2 (∂Ω, µ eΩ ) onto H 2 (Ω). Every holomorphic function f in a Reinhardt domain Ω containing 0 (in particular if Ω is a complete Reinhardt domain) has a power series expansion about 0: X f (z) = bα z α α

which converges normally on compact subsets of Ω ([32, Proposition 2.3.14]). αN Recall that if z = (z1 , . . . , zN ) and α = (α1 , . . . , αN ), then z α = z1α1 · · · zN , |α| = α1 + · · · + αN , and α! = α1 ! · · · αN !. We have: 15

Proposition 4.5. Let Ω be a bounded hyperconvex and complete Reinhardt domain, and set eα (z) = z α . Then the system (eα )α is orthogonal in H 2 (Ω). Proof. We use the fact that the level sets S(r) and the Demailly-Monge-Ampère N are pluri-rotation invariant. For α 6= β, we choose measures µr = ddc (gΩ )r θ1 , . . . , θN ∈ R such that 1, (θ1 /2π), . . . , (θN /2π) are rationally independent. PN 6= 1. Hence, as in [25, p. 78], we have, making Then exp i j=1 (αj − βj )θj iθ1 the change of variables z = (e w1 , . . . , eiθN wN ): Z X Z N α β z z dµr (z) = exp i wα wβ dµr (w) , (αj − βj )θj S(r)

S(r)

j=1

which implies that:

Z

and hence:

z α z β dµr (z) = 0 , S(r)

(z α | z β ) := lim

r→0

Z

z α z β dµr (z) = 0 .

S(r)

For the polydisk, we have keα kH 2 (DN ) = 1, and for the ball (see [48, Proposition 1.4.9]): (N − 1)! α! · keα k2H 2 (BN ) = (N − 1 + |α|)!

Definition 4.6. We say that Ω is a good complete Reinhardt domain if, for some positive constant CN and some positive integer c, we have, for all p ≥ 0: X |z α |2 ≤ CN pcN [jΩ (z)]2p , keα k2H 2 (Ω) |α|=p

where jΩ is the Minkowski functional of Ω.

Examples 1. The polydisk DN is a good Reinhardt domain because keα kH 2 (DN ) = 1, |α| |z α | ≤ kzk∞ , and the number of indices α such that |α| = p is N −1+p ≤ CN pN p (see [35, p. 498] or [37, pp. 213–214]). 2. The ball BN is a good Reinhardt domain. In fact, observe that: (p + 1)(p + 2) · · · (p + N − 1) (N − 1 + p)! = p! ≤ p! (p + 1)N −1 ≤ p! (p + 1)N ; (N − 1)! 1 × 2 × · · · × (N − 1)

hence:

X

|α|=p

X |z α |2 (N − 1 + |α|)! |z α |2 = 2 keα kH 2 (BN ) (N − 1)! α! |α|=p

≤ (p + 1)N

X |α|! |z1 |2α1 · · · |zN |2αN α!

|α|=p

N

= (p + 1) (|z1 |2 + · · · + |zN |2 )p , 16

by the multinomial formula, so: X

|α|=p

|z α |2 2p N N ≤ (p + 1)N kzk2p 2 ≤ 2 p kzk2 . keα k2H 2 (BN )

3. More generally, if Ω = Bl1 × · · · × Blm , l1 + · · · + lm = N , is a product of balls, we have, writing α = (β1 , . . . , βm ), where each βj is an lj -tuple: Z keα k2H 2 (Ω) = |uβ1 1 |2 . . . |uβmm |2 dσl1 (u1 ) . . . dσlm (um ) Sl1 ×···×Sl2

=

m Y

j=1

(lj − 1)! βj ! , (lj − 1 + |βj |)!

and, writing z = (z1 , . . . , zm ), with zj ∈ Blj : X

|α|=p

|z α |2 ≤ keα k2H 2 (Ω) p

X

1 +···+pm =p

m Y

j=1

2pj

(pj + 1)lj kzj k2

≤ Cm p (p + 1)l1 +···+lm [jΩ (z)]2(p1 +···+pm ) , m

since jΩ (z) = max{kz1 k2 , . . . , kzm k2 }. Hence: X

|α|=p

4.2.2

|z α |2 ≤ CN p2N [jΩ (z)]2p . keα k2H 2 (Ω)

The result

Theorem 4.7. Let Ω be a bounded hyperconvex domain which is a good complete Reinhardt domain in CN , and let ϕ : Ω → Ω be an analytic function such that ϕ(Ω) ⊆ Ω. Then, for every compact subset K ⊇ ϕ(Ω) of Ω with non void interior, we have: (4.3)

+ βN (Cϕ ) ≤ ΓN (K) .

In particular, if ϕ is moreover non-degenerate, we have: + (4.4) βN (Cϕ ) ≤ ΓN ϕ(Ω) .

The last assertion holds because ϕ(Ω) is open if ϕ is non-degenerate.

Corollary 4.8. Let Ω be a good complete bounded symmetric domain in CN , and ϕ : Ω → Ω a non-degenerate analytic map such that ϕ(Ω) ⊆ Ω. Then: − + (Cϕ ) ≤ βN (Cϕ ) ≤ ΓN ϕ(Ω) . ΓN ϕ(Ω) ≤ βN

For the proof of Theorem 4.7, we will use the following result ([56, Proposition 6.1]), which do not need any regularity condition on the compact set (because it may be written as a decreasing sequence of regular compact sets). 17

Proposition 4.9 (Zakharyuta). If K is any compact subset of a bounded hyperconvex domain Ω of CN with non-empty interior, we have: 1/N log dn (AK ) N! lim sup . ≤ − τN (K) n1/N n→∞ Proof of Theorem 4.7. In the sequel we write k . kH 2 for k . kH 2 (Ω) . We set: ΛN = lim sup[dn (AK )]n

−1/N

.

n→∞

Changing n into nN , Proposition 4.9 means that for every ε > 0, there exists, for n large enough, an (nN − 1)-dimensional subspace F of C(K) such that, for any g ∈ H ∞ (Ω), there exists h ∈ F such that: (4.5)

kg − hkC(K) ≤ (1 + ε)n ΛnN kgk∞ .

Let l be an integer to be adjusted later, and X f (z) = bα z α ∈ H 2 (Ω) with kf kH 2 ≤ 1 . α

By Proposition 4.5, we have: kf k2H 2 =

X α

We set: g(z) =

|bα |2 keα k2H 2 .

X

bα z α .

|α|≤l

By the Cauchy-Schwarz inequality: X X X |z α |2 |z α |2 2 2 2 ≤ · |bα | keα kH 2 |g(z)| ≤ 2 keα kH 2 keα k2H 2 |α|≤l

|α|≤l

|α|≤l

Since Ω is a good complete Reinhardt domain and since jΩ (z) < 1 for z ∈ Ω, we have: l X 2 |g(z)| ≤ pcN [jΩ (z)]2p ≤ (l + 1)cN +1 . p=0

It follows from (4.5) that there exists h ∈ F such that:

kg − hkC(K) ≤ (1 + ε)n ΛnN (l + 1)(cN +1)/2 . Since Cϕ f (z) − Cϕ g(z) = f ϕ(z) − g ϕ(z) and ϕ(Ω) ⊆ K, we have kCϕ f − Cϕ gk∞ ≤ kf − gkC(K) ; therefore: (4.6)

kg ◦ ϕ − h ◦ ϕkH 2 ≤ kg ◦ ϕ − h ◦ ϕk∞ ≤ kg − hkC(K) ≤ (1 + ε)n ΛnN (l + 1)(cN +1)/2 . 18

Now, the subspace Fe formed by functions v ◦ ϕ, for v ∈ F , can be viewed as a subspace of L∞ (∂Ω, µ eΩ ) ⊆ L2 (∂Ω, µ eΩ ) (indeed, since v is continuous, we can ∗ ∗ write (v ◦ ϕ) = v ◦ ϕ , where ϕ∗ denotes the almost everywhere existing radial limits of ϕ(rz), which belong to K). Let ﬁnally E = P (Fe ) ⊆ H 2 (Ω) where P : L2 (∂Ω, µ eΩ ) → H 2 (Ω) is the orthogonal projection. This is a subspace of H 2 (Ω) with dimension < nN , and we have dist (Cϕ g, E) ≤ kg ◦ ϕ − P (h ◦ ϕ)kH 2 ; hence, by (4.6): (4.7)

dist (Cϕ g, E) ≤ (1 + ε)n ΛnN (l + 1)(cN +1)/2 .

Now, the same calculations give that: X |f (z) − g(z)|2 ≤ pcN [jΩ (z)]2p ; p>l

hence, for some positive constant MN : |f (z) − g(z)| ≤ MN (l + 1)(cN +1)/2

[jΩ (z)]l , (1 − [jΩ (z)]2 )(cN +1)/2

by using the following lemma, whose proof is postponed. Lemma 4.10. For every non-negative integer m, there exists a positive constant Am such that, for all integers l ≥ 0 and all 0 < x < 1, we have: X p≥l

pm xp ≤ Am lm

xl . (1 − x)m+1

Since K is a compact subset of Ω, there is a positive number r0 < 1 such that jΩ (z) ≤ r0 for z ∈ K. Since Cϕ f (z) − Cϕ g(z) = f ϕ(z) − g ϕ(z) and ϕ(Ω) ⊆ K, we have kCϕ f − Cϕ gk∞ ≤ kf − gkC(K) , and we get: (4.8) kCϕ f −Cϕ gkH 2 ≤ kCϕ f −Cϕ gk∞ ≤ MN (l+1)(cN +1)/2

r0l · (1 − r02 )(cN +1)/2

Now, (4.7) and (4.8) give: dist (Cϕ f, E) ≤ MN (l + 1)(cN +1)/2

r0l + (1 + ε)n ΛnN 2 (1 − r0 )(cN +1)/2

.

It ensues, thanks to (2.7), that:

1/n anN (Cϕ ) ≤ [MN (l + 1)(cN +1)/2 ]1/n

l/n r0 + (1 + ε) ΛN . (1 − r02 )(cN +1)/2n

Taking now for l the integer part of n log n, and passing to the upper limit as n → ∞, we obtain (since l/n → ∞ and (log l)/n → 0): + βN (Cϕ ) ≤ (1 + ε) ΛN ,

19

and therefore, since ε > 0 is arbitrary: + βN (Cϕ ) ≤ ΛN .

That ends the proof, by using Proposition 4.9. Proof of Lemma 4.10. We make the proof by induction on m. We set: X Sm = pm xp p≥l

P xl · The result is obvious for m = 0, with A0 = 1, since then S0 = p≥l xp = 1−x Let us assume that it holds till m − 1 and prove it for m. We observe that, since pm − (p − 1)m ≤ mpm−1 , we have: X X X X (1 − x)Sm = pm xp − pm xp+1 = pm xp − (p − 1)m xp p≥l

X

=

p≥l+1

≤

X p≥l

p≥l

m

p≥l

m

p

m l

(p − (p − 1) )x + l x ≤

p≥l+1

X

p≥l+1

mpm−1 xp + lm xl ≤ mAm−1 lm−1

≤ (mAm−1 + 1) lm

mpm−1 xp + lm xl

xl + lm xl (1 − x)m

xl , (1 − x)m

giving the result, with Am = mAm−1 + 1.

4.3

Equality

Proposition 4.11. Let Ω be a bounded hyperconvex domain and ω a relatively compact open subset of Ω. Assume that: (4.9)

For every a ∈ ∂ω, except on a pluripolar set E ⊆ ∂ω, there exists z0 ∈ ω such that the open segment (z0 , a) is contained in ω.

Then: Cap (ω) = Cap (ω) . In particular, if ϕ : Ω → Ω a non-degenerate holomorphic map such that ϕ(Ω) ⊆ Ω and ω = ϕ(Ω) satisfies (4.9), we have: Cap ϕ(Ω) = Cap ϕ(Ω) . Before proving Proposition 4.11, let us give an example of such a situation.

Proposition 4.12. Let Ω be a bounded hyperconvex domain with C 1 boundary. Let U be an open neighbourhood of Ω and ϕ : U → CN be a non-degenerate holomorphic function such that ϕ(Ω) ⊆ Ω. Then the condition (4.9) is satisfied. 20

Proof. Let ω = ϕ(Ω). We may assume that U is connected, hyperconvex and bounded. Let Bϕ be the set of points z ∈ U such that the complex Jacobian Jϕ is null. Since Jϕ is holomorphic in Ω, we have log |Jϕ | ∈ PSH(U ) and hence (see [31, proof of Lemma 10.2]): Bϕ = {z ∈ U ; Jϕ (z) = 0} = {z ∈ U ; log |Jϕ (z)| = −∞} is pluripolar . Therefore (see [5, Theorem 6.9]), Cap (Bϕ , U ) = 0. It follows (see [5, page 2, line -8]) that Cap [ϕ(Bϕ )] := Cap [ϕ(Bϕ ), Ω] = 0. Now, for every a ∈ ∂ω ∩ [ϕ(U \ Bϕ )], there is a tangent hyperplane Ha to ω, and hence an inward normal to ∂ω (note that ∂ω ⊆ ϕ(∂Ω) ⊆ ϕ(U )). It follows that there is z0 ∈ ω such that the open interval (z0 , a) is contained in ω. Proof of Proposition 4.11. Let a ∈ ∂ω and L be a complex line containing (z0 , a); we have a ∈ ω ∩ L. Assume now that this point a is a fine (“effilé”) point of ω, i.e. that there exists u ∈ PSH(V ), for V a neighbourhood of a, such that: lim sup u(z) < u(a) . z→a ,z∈ω

By deﬁnition, the restriction u e of u to ω ∩ L is subharmonic and we keep the inequality: lim sup u e(z) < u e(a) = u(a) . z→a ,z∈ω∩L

That means that a is a ﬁne point of ω ∩L. But a ∈ ω ∩ L and ω ∩L is connected, so this is not possible, by [40, Lemma 2.4]. Hence no point of ∂ω \ E is ﬁne. Let now ω f be the closure of ω for the ﬁne topology (i.e. the coarsest topology on U for which all the functions in PSH(U ) are continuous; it is known: see [6, comment after Theorem 2.3], that it is the trace on U of the ﬁne topology on CN ). It is also known (see [30, Corollary 4.8.10]) that ω f is the set of points of ω which are not ﬁne. By the above reasoning, we thus have: ω \ ωf ⊆ E . Since Cap (E) = 0, we have: Cap (ω \ ω f ) = 0 , and it follows that: Cap (ω) = Cap [ω f ∪ (ω \ ω f )] ≤ Cap (ω f ) + Cap (ω \ ω f ) = Cap (ω f ) ,

and hence Cap (ω f ) = Cap (ω). But, since, by deﬁnition, the psh functions are continuous for the ﬁne topology, it is clear, that the relative extremal functions uω,Ω and uωf ,Ω are equal; hence we have, by [30, Proposition 4.7.2]: Z Z c ∗ N Cap (ω) = (dd uω,Ω ) = (ddc u∗ωf ,Ω )N = Cap (ω f ) . Ω

Ω

Hence Cap (ω) = Cap (ω).

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4.4

Consequences of the spectral radius type formula

Theorem 4.3 has the following consequence. Proposition 4.13. Let Ω be a regular bounded symmetric domain in CN , and let ϕ : Ω → Ω be a non-degenerate analytic function inducing a bounded composition operator Cϕ on H 2 (Ω). Then, if Cap [ϕ(Ω)] = ∞, we have βN (Cϕ ) = 1. 1/N In other words, if, for some constants C, c > 0, we have an (Cϕ ) ≤ C e−cn for all n ≥ 1, then Cap [ϕ(Ω)] < ∞. As a corollary, we can give a new proof of [41, Theorem 3.1]. Corollary 4.14. Let τ : D → D be an analytic map such that kτ k∞ = 1 and ψ : DN −1 → DN −1 such that the map ϕ : DN → DN , defined as: ϕ(z1 , z2 , . . . , zN ) = τ (z1 ), ψ(z2 , . . . , zN ) , is non-degenerate. Then βN (Cϕ ) = 1.

Proof. Since the map ϕ is non-degenerate, ψ is also non-degenerate. Hence (see [44, Proposition 2] ψ(DN −1 ) is not pluripolar, i.e. CapN −1 [ψ(DN −1 )] > 0. On the other hand, it follows from [40, Theorem 3.13 and Theorem 3.14] that Cap1 [τ (D)] = +∞. Then, by [8, Theorem 3], we have: CapN [ϕ(DN )] = CapN [τ (D) × ψ(DN −1 )]

= Cap1 [τ (D)] × CapN −1 [ψ(DN −1 )] = +∞ .

It follows from Proposition 4.13 that βN (Cϕ ) = 1. Proof of Proposition 4.13. If R : H 2 (Ω) → H 2 (Ω) is a ﬁnite-rank operator, we set, for t < 0: (Rt f )(w) = (Rf )(et w) ,

f ∈ H 2 (Ω) .

Then the rank of the operator Rt is less or equal to that of R. Recall that if k . k is the norm whose unit ball is Ω, then the pluricomplex Green function of Ω is gΩ (z) = log kzk, and hence the level set S(r) is the sphere S(0, er ) = er ∂Ω for this norm. Since: Z Z t t 2 |f [ϕ(z)] − (Rf )(z)|2 dµr+t (z) , |f [ϕ(e w)] − (Rf )(e w)| dµr (w) = S(r+t)

S(r)

we have, setting ϕt (w) = ϕ(et w): kCϕt (f ) − Rt (f )kH 2 ≤ kCϕ (f ) − R(f )kH 2 . − (Cϕt ) ≤ It follows that an (Cϕt ) ≤ an (Cϕ ) for every n ≥ 1. Therefore βN − βN (Cϕ ).

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By Theorem 4.3, we have: " exp − 2π

N! Cap [ϕt (Ω)]

1/N #

− ≤ βN (Cϕt ) .

Since ϕt (Ω) = ϕ(et Ω) increases to ϕ(Ω) as t ↑ 0, we have (see [30, Corollary 4.7.11]): Cap [ϕ(Ω)] = lim Cap [ϕt (Ω)] . t→0

As Cap [ϕ(Ω)] = ∞, we get: − − βN (Cϕ ) ≥ lim sup βN (Cϕt ) = 1 . t→0

Remark 1. In [41, Theorem 5.12], we construct a non-degenerate analytic function ϕ : D2 → D2 such that ϕ(D2 ) ∩ ∂D2 6= ∅ and for which β2+ (Cϕ ) < 1. We hence have Cap [ϕ(D2 )] < ∞. Remark 2. The capacity cannot tend to inﬁnity too fast when the compact set approaches the boundary of Ω; in fact, we have the following result, that we state for the ball, but which holds more generally. Proposition 4.15. For every compact set K of BN , we have, for some constant CN : CN · Cap (K) ≤ [dist (K, SN )]N Proof. We know that: Cap (K) =

Z

BN

(ddc u∗K )N .

Let ρ(z) = |z|2 − 1 and aK := minz∈K [−ρ(z)] = − maxz∈K ρ(z). Then ρ is in PSH and is non-positive. Since aK > 0, the function: v(z) =

ρ(z) aK

is in PSH, non-positive on BN , and v ≤ −1 on K. Hence v ≤ uK ≤ u∗K . Since v(w) = 0 for all w ∈ SN and (see [5, Proposition 6.2 (iv)], or [30, Proposition 4.5.2]): lim u∗K (z) = 0 , z→w

for all w ∈ SN , the comparison theorem of Bedford and Taylor ([5, Theorem 4.1]; [30, Theorem 3.7.1] gives, since v ≤ u∗K and v, u∗K ∈ PSH: Z Z Z 1 c N c ∗ N (dd v) = N (dd uK ) ≤ (ddc ρ)N . a BN BN K BN As (ddc ρ)N = 4N N ! dλ2N , we get, with CN := 4N N ! λ2N (BN ): Cap (K) ≤ 23

CN · aN K

That ends the proof since: aK = min(1 − |z|2 ) ≥ min(1 − |z|) = dist (K, SN ) z∈K

z∈K

We have assumed that the symbol ϕ is non-degenerate. For a degenerate symbol ϕ, we have: Proposition 4.16. Let Ω be a bounded hyperconvex and good complete Reinhardt domain in CN , and let ϕ : Ω → Ω be an analytic function such that ϕ(Ω) ⊆ Ω is pluripolar. Then βN (Cϕ ) = 0. Recall that ϕ(Ω) is pluripolar when ϕ is degenerate (see [44, Proposition 2]); its closure is also pluripolar if it satisﬁes the condition (4.9). Proof. Let K = ϕ(Ω). By hypothesis, we have Cap (K) = 0. For every ε > 0, + let Kε = {z ∈ Ω ; dist (z, K) ≤ ε}. By Theorem 4.7, we have βN (Cϕ ) ≤ ΓN (Kε ). As limε→0 Cap (Kε ) = Cap (K) = 0 ([30, Proposition 4.7.1(iv)]), we get βN (Cϕ ) = 0. Remark 1. In [41, Section 4], we construct a degenerate symbol ϕ on the bi-disk D2 , deﬁned by ϕ(z1 , z2 ) = λθ (z1 ), λθ (z1 ) , where λθ is a lens map, for which β − (Cϕ ) > 0. For this function ϕ(D2 ) ∩ ∂D2 6= ∅ and hence ϕ(D2 ) is not a compact subset of D2 . Remark 2. In the one dimensional case, for any (non constant) analytic map ϕ : D → D, the parameter β(Cϕ ) = β1 (Cϕ ) is determined by its range ϕ(D), as shown by the formula: β(Cϕ ) = e−1/Cap [ϕ(D)] proved in [40]. This is no longer true in dimension N ≥ 2. In [42], we construct pairs of (degenerate) symbols ϕ1 , ϕ2 : D2 → D2 , such that ϕ1 (D2 ) = ϕ2 (D2 ) and: 1) Cϕ1 is not bounded, but Cϕ2 is compact, and even β2 (Cϕ2 ) = 0; 2) Cϕ1 is bounded but not compact, so β2 (Cϕ1 ) = 1, and Cϕ2 is compact, with β2 (Cϕ2 ) = 0; 3) Cϕ1 is compact, with 0 < β2 (Cϕ1 ) < 1, and Cϕ2 is compact, with β2 (Cϕ2 ) = 0. Acknowledgements. We thank S. Nivoche and A. Zeriahi for useful discussions and informations, and Y. Tiba, who send us his paper [51]. We than specially S. Nivoche, who carefully read a preliminary version of this paper. The third-named author is partially supported by the project MTM201563699-P (Spanish MINECO and FEDER funds).

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