Complex geodesics in convex tube domains II

Complex geodesics are fundamental constructs for complex analysis and as such constitute one of the most vital research objects within this discipline. In this paper, we formulate a rigorous description, expressed in terms of geometric properties of a domain, of all complex geodesics in a convex tube domain in 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} containing no complex affine lines. Next, we illustrate the obtained result by establishing a set of formulas stipulating a necessary condition for extremal mappings with respect to the Lempert function and the Kobayashi–Royden metric in a large class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^2$$\end{document} and for those in 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} whose logarithmic image is strictly convex in the geometric sense.


Introduction
A non-empty open set D ⊂ C n is a tube domain if for some domain ⊂ R n one has that D = + iR n . We call the base of D and, in this paper, denote it by Re D. In [16], convex tube domains were investigated from the holomorphically invariant distances theory viewpoint with a particular focus on the notion of complex geodesics. Given a convex domain D ⊂ C n , we call a holomorphic map ϕ : D → D a complex geodesic for D if there exists a Complex geodesics of D are precisely the holomorphic isometries between the unit disc D ⊂ C equipped with the Poincaré distance and the domain D equipped with the Carathéodory pseudodistance (see [12][13][14]). Lempert's theorem yields that if D ⊂ C n is a taut convex domain, then for any pair of points in D there exists a complex geodesic passing through them (see [9] or [5,Chapter 8] and also [11,15]).
The problem of finding an explicit description of all complex geodesics in a given domain is fundamental to the geometric function theory, and so far it has been solved completely only for a few classes of domains (see e.g. [1,4,6,10]). This study, which may be considered as a continuation of the recent paper [16], centres on the family of convex tube domains in C n containing no complex affine lines. As the main result of this paper, we establish an exhaustive characterization of all complex geodesics for any domain of this type.
The restriction to convex tube domains with no complex lines is made for several reasons, among which the most important is that it results in every holomorphic map with the image in such a domain admitting the boundary measure ([16, Observation 2.5]). Furthermore, from [16,Observation 2.4] it follows that narrowing research to this family does not cause the loss of generality. In [16], we demonstrated an equivalent condition for a holomorphic map ϕ : D → D to be a complex geodesic in a convex tube domain D which is taut (what in this case is equivalent to containing no complex affine lines; see e.g. [2]). The condition is expressed in terms of the measure theory and formulated using the boundary measures of coordinates of ϕ. For the sake of clarity, we shall follow the notation of Zając [16] and refer to the n-tuple of these measures as the boundary measure of the map ϕ.
The derivation of the main result of this paper, Theorem 3.1, draws upon the equivalent condition from Zając [16] and the following, 'spherical' decomposition of n-tuples of measures (Lemma 2.1): given real Borel measures μ 1 , . . . , μ n on T, there exist a finite positive Borel measure ν on T singular to the Lebesgue measure L T on T, a Borel-measurable map from T to the unit sphere ∂B n and a map g : T → R n with components in L 1 (T, L T ) such that (μ 1 , . . . , μ n ) = g dL T + dν.
The objects ν, and g are correspondingly unique. Theorem 3.1 states that a holomorphic map with the boundary measure μ = (μ 1 , . . . , μ n ) is a complex geodesic for D if and only if the parts ν, and g of the decomposition of μ satisfy a number of geometric conditions. So, strictly speaking, in Theorem 3.1 we describe every n-tuple of measures which defines a complex geodesic for D. Then, the complex geodesic itself can be easily recovered, up to an imaginary constant, from its boundary measure employing the Schwarz formula (1).
Only thereafter we apply Theorem 3.1 to obtain more detailed form of complex geodesics for special classes of convex tube domains. In Sect. 3.1, this is done for, inter alia, convex tubes D ⊂ C n with the base being bounded from above on each coordinate and satisfying the equality Re D + (−∞, 0] n = Re D. Considering these domains is beneficial when studying extremal mappings with respect to the Lempert function and the Kobayashi-Royden metric in bounded, pseudoconvex, complete Reinhardt domains in C n . We treat this topic in Sect. 5, where we postulate formulas for extremal mappings for all such domains in C 2 and for those in C n with strictly convex, in the geometric sense, logarithmic image. Next, in Sect. 3.2 we investigate complex geodesics in convex tube domains in C 2 . The obtained result, together with the considerations made in Sect. 3.1, simplifies the conclusion of Theorem 3.1 in the two-dimensional case.
This paper is organized as follows: Sect. 2 presents the notation and recalls crucial properties of the boundary measures of holomorphic maps. Also, therein we prove the afore-mentioned lemma allowing for the decomposition of measures' n-tuple. Then, we introduce objects describing vital geometric properties of convex tube domains in C n . In Sect. 3, we formulate the main result of this paper, Theorem 3.1, and demonstrate its applications to special classes of tube domains, providing also a number of illustrating examples. Section 4 contains the proof of Theorem 3.1 together with additional remarks. In Sect. 5, we apply results from Sect. 3 to obtain formulas for extremal holomorphic mappings in some classes of Reinhardt domains in C n . Section 5 concludes this paper.

Preliminaries
We begin by introducing basic concepts and notation setting grounds for the rest of this study. Let the symbols D, T, C * denote the unit disc in C, the unit circle in C and the punctured plane, i.e. the set C \ {0}, respectively. By δ λ 0 , we mean the Dirac delta at a point λ 0 ∈ T, while χ A is the characteristic function χ A : T → {0, 1} of a set A ⊂ T. Let also e 1 , . . . , e n be the canonical basis of R n or C n . The Poincaré distance in D is denoted by ρ. By x, y , we understand the standard inner product of vectors x, y ∈ R n , by · the euclidean norm in R n and by B n the unit euclidean ball in R n . For a set A ⊂ R n , the symbol A ⊥ denotes the set {v ∈ R n : ∀a ∈ A : v, a = 0}.
We use the symbol ·, ·· also for measures and functions. For example, if μ is a tuple (μ 1 , . . . , μ n ) of real, i.e. complex with real values, Borel measures on T and v = (v 1 , . . . , v n ) is a real vector or a bounded Borel-measurable mapping from T to R n , then dμ, v or v, dμ is the measure n j=1 v j dμ j , etc. The fact that a real measure ν is positive (resp. negative, null) is shortly written as ν ≥ 0 (resp. ν ≤ 0, ν = 0). The variation of a complex measure ν is denoted by |ν|. Unless stated otherwise, any measure considered in this paper is understood as a real Borel measure on T. Henceforth, we shall use the following families of mappings: We have and (see e.g. [5,Lemma 8.4.6]), In particular, for h ∈ H 1 + it follows thatλh (λ) = c|λ − d| 2 , λ ∈ T. Hence, h has at most one zero on T (counting without multiplicities). Moreover, the choice of the field, R or C, has no effect on the linear dependence, or independence, of functions h 1 , . . . , h m ∈ H 1 . Both of these properties are equivalent in view of the fact thatλh j (λ) ∈ R for all λ ∈ T and j = 1, . . . , m.
Let us now recall the most important results for the boundary measures of holomorphic maps. A real Borel measure μ on T is called the boundary measure of a holomorphic function ϕ : D → C, if the Schwarz formula is satisfied or, equivalently, after taking the real parts of the both sides of the above equation, if If it exists, the measure μ is uniquely determined by ϕ. For a mapping ϕ = (ϕ 1 , . . . , ϕ n ) ∈ O(D, C n ), we introduce the boundary measure of ϕ as the n-tuple (μ 1 , . . . , μ n ) of the boundary measures of ϕ 1 , . . . , ϕ n , provided that they exist. Then, with the coordinate-wise integration, formulas analogous to (1) and (2) connect ϕ and μ.
Next, we define the family : ϕ admits the boundary measure}.
Here we treat complex measures as linear functionals on C(T), i.e. the space of all complexvalued continuous functions on T equipped with the supremum norm. The aforementioned weak-* convergence means that Also, the following fact is of great importance: if μ = g dL T + μ s is the Lebesgue-Radon-Nikodym decomposition of μ with respect to L T , i.e. g ∈ L 1 (T, L T ) and μ s is a real Borel measure on T singular to L T , then Re ϕ * (λ) = g(λ) for L T -a.e. λ ∈ T (see e.g. [8, p. 11]). In particular, Re ϕ * ∈ L 1 (T, L T ) and if ϕ s is a holomorphic function with the boundary measure μ s , then Re ϕ * s (λ) = 0 for L T -a.e. λ ∈ T. Henceforth, given a n-tuple μ = (μ 1 , . . . , μ n ) of real Borel measures on T, by its Lebesgue-Radon-Nikodym decomposition with respect to L T we understand a unique decomposition where g = (g 1 , . . . , g n ) : T → R n is a Borel-measurable map, g 1 , . . . , g n ∈ L 1 (T, L T ) and μ s = (μ s,1 , . . . , μ s,n ) is a n-tuple of real Borel measures on T, each being singular to L T . In other words, for every j, is the Lebesgue-Radon-Nikodym decomposition of μ j . We refer to g dL T and μ s shortly as the absolutely continuous part and the singular part of μ, respectively, assuming they are meant with respect to L T . The following essential lemma allows for expressing the singular part of μ 'spherically': Lemma 2.1 Let μ be a n-tuple of real Borel measures on T. Then there exist a unique finite positive Borel measure ν on T singular to L T , a unique, up to a set of ν measure zero, Borel-measurable map : T → ∂B n and a unique, up to a set of L T measure zero, Borel-measurable map g : T → R n with components in L 1 (T, L T ) such that In particular, g dL T and dν are the absolutely continuous and the singular parts of μ in its Lebesgue-Radon-Nikodym decomposition, respectively.
One can immediately deduce Lemma 2.1 by applying Lemma 2.2 to the singular part of μ.

Lemma 2.2 If (X, A)
is a measurable space and μ = (μ 1 , . . . , μ n ) is a n-tuple of real measures μ j : A → R, then there exist a unique finite positive measure ν : A → [0, ∞) and a unique, up to a set of ν measure zero, A-measurable map : X → ∂B n such that μ = dν.
Since μ 1 , . . . , μ n are absolutely continuous with respect to ν, the classical Radon-Nikodym theorem ensures the existence of an A-measurable map F = (F 1 , . . . , F n ) : X → R n such that F 1 , . . . , F n ∈ L 1 (X, ν) and Then |μ j | = |F j | d ν. Therefore, which gives the desired decomposition. It remains to show the uniqueness. To this end, assume that ν , satisfy the same conditions as ν, . Clearly, it holds that dν = dν . We put ω := ν +ν and let G, G : X → [0, ∞) be A-measurable functions, integrable with respect to ω and such that ν = G dω and ν = G dω. We have Thus, the maps G and G are equal ω-a.e. on X . This yields In consequence, ν = ν and ν-almost everywhere on X it holds that = , because dν = dν .

Example 2.3
In this example, we employ Lemma 2.1 in order to obtain the corresponding decomposition of the following n-tuple of measures: where g = (g 1 , . . . , g n ), g 1 , . . . , g n ∈ L 1 (T, L T ), α 1 , . . . , α n ∈ R and λ 1 , . . . , λ n ∈ T. As the measure ν is required to be singular with respect to L T , the first part of the desired decomposition is clearly equal to g dL T . The second part arises from Lemma 2.2 applied to (α 1 δ λ 1 , . . . , α n δ λ n ). To find it, we follow the proof of the lemma with X = T and A being the σ -field of Borel subsets of T. For j ∈ {1, . . . , n} let We have ν = |α 1 |δ λ 1 + · · · + |α n |δ λ n , so we may set Let us underline that the mapping F is well defined ν-almost everywhere, because if l∈A j |α l | = 0 for some j, then χ {λ j } is ν-a.e. equal to 0 and so is the jth coordinate of the right-hand side of the above definition. Since ν = F(λ) d ν(λ), the measure ν is supported on the set {λ 1 , . . . , λ n } and This yields where # A j denotes the number of elements of the set A j . A map : T → ∂B n has to be taken so that the equality F(λ) = (λ) F(λ) holds for ν-a.e. λ ∈ T, or equivalently, for ν-a.e. λ ∈ T. It means that The right-hand side is ν-almost everywhere well defined, and it does not matter what values takes outside the set {λ j : j ∈ {1, . . . , n}, l∈A j |α l | = 0}. The desired decomposition consists of the map g, the measure ν given by (4) and a map satisfying (5).
The problem simplifies in the case when λ 1 , . . . , λ n are pairwise disjoint. Indeed, then ν = |α 1 |δ λ 1 + · · · + |α n |δ λ n and = α 1 Let us now introduce special sets describing certain geometric aspects of the base of a given convex tube domain D ⊂ C n . Define and, for a vector v ∈ R n , It is clear that all these sets are convex, P D (v) ⊂ ∂Re D and if v ∈ S D , w ∈ W D and t ≥ 0, then tv ∈ S D and tw ∈ W D , that is, the sets S D and W D are infinite cones. A number of their elementary geometric properties are presented by the next observation.

Observation 2.4
Let D ⊂ C n be a convex tube domain and let v ∈ R n . Then: , then the vectors p − q and v are orthogonal, (iv) if the domain Re D is strictly convex (in the geometric sense, i.e. it is convex and ∂Re D does not contain any non-trivial segments), then the set P D (v) contains at most one element, Combining this with the fact that it is non-positive on the open set Re D, we conclude that it is negative on Re D.
The proofs are immediate.

Description of complex geodesics in an arbitrary convex tube domain and its applications in special classes of domains
One of the goals of this section is to present Theorem 3.1, which is the main result of this paper. It is formulated in terms of geometric properties of a domain, namely the sets P D (v), W D and S D . Next, we use the theorem to formulate a more detailed characteristic of complex geodesics for certain classes of tube domains. The proof of Theorem 3.1 is later derived in Sect. 4.

Theorem 3.1 Let D ⊂ C n be a convex tube domain containing no complex affine lines and let ϕ ∈ M n be a holomorphic map with the boundary measure μ. Consider the decomposition
. . , g n ) : T → R n and : T → ∂B n are Borel-measurable maps, g 1 , . . . , g n ∈ L 1 (T, L T ) and ν is a finite positive Borel measure on T singular to L T . Then

ϕ(D) ⊂ D and ϕ is a complex geodesic for D
iff there exists a map h ∈ H n , h ≡ 0, such that the following conditions hold:

Moreover, if ϕ(D) ⊂ D, ϕ is a complex geodesic for D and h ∈ H n , h ≡ 0 is a map satisfying the conditions (i)-(iv), then additionally the following statements are true:
Note that from (vi) and (vii) it follows that the measure ν is supported on the set {λ ∈ T : Remark 3.2 Let us notice that none of the above conditions, except (iv), contains at the same time the absolutely continuous and the singular part of μ. This makes it relatively not difficult to construct a measure which gives a complex geodesic for D. The part g dL T has to satisfy (i), while for the part dν we require that the conditions (ii) and (iii) hold. To create a measure μ defining a complex geodesic for D, it suffices to choose a map h ∈ H n , h ≡ 0 such that and next: • take a Borel map g with integrable components satisfying (i) (note that it may happen that it is impossible, even if (6) holds-see Example 3.7), • take a measure ν singular to L T and satisfying (vi), • take a Borel map : T → ∂B n satisfying (v).
Then, if μ = g dL T + dν and additionally 1 2π μ(T) ∈ Re D, which simply means that Re ϕ(0) ∈ Re D, then μ is the boundary measure of a complex geodesic for D. Remark 3.3 If D ⊂ C n is a convex tube domain with the base bounded, then W D = R n and S D = {0}. So, Theorem 3.1 (iii) yields that (λ) = 0 for ν-a.e. λ ∈ T. Hence, ν is the null measure, because the image of lies in ∂B n . Then also the condition (ii) is automatically fulfilled. Thus, a holomorphic map ϕ with the boundary measure μ is a complex geodesic for D iff μ = g dL T for some g, h satisfying (i) and (iv).
In general case, when the base of D is unbounded, Theorem 3.1 (i) determines the absolutely continuous part of μ by the same token. However, to find the singular part, we will need to appeal to the entire scope of Theorem 3.1.

Convex tube domains with W D = [0, ∞) n
In this subsection, we study the family D n consisting of all convex tube domains D ⊂ C n satisfying the equality A convex tube domain D belongs to D n if and only if e 1 , . . . , e n ∈ W D and The base of such a domain D contains no real affine lines. Also, it follows that Corollary 3.4 provides a characterization, more detailed than Theorem 3.1, of all complex geodesics for a domain D ∈ D n . It is next employed in Sect. 5, where we establish formulas for extremal mappings in certain classes of Reinhardt domains in C n .

Corollary 3.4
Let D ∈ D n , n ≥ 2, and let ϕ ∈ M n be a holomorphic map with the boundary measure μ. Consider the decomposition . . , g n ∈ L 1 (T, L T ) and ν is a finite positive Borel measure on T singular to L T . Then ϕ(D) ⊂ D and ϕ is a complex geodesic for D iff there exists a map h ∈ H n , h ≡ 0 such that the following conditions hold: Let us note that the condition (iii) simply means that the singular part of μ, i.e. the measure dν, is just a n-tuple of negative measures.

Proof of Corollary 3.4 Assume that ϕ(D) ⊂ D and that ϕ is a complex geodesic for D.
Taking h is as in Theorem 3.1, we immediately obtain the statements (i) -(iv) from the conclusion and it remains only to show the condition (v). The expression λ h (λ), (λ) , vanishing ν-almost everywhere in view of Theorem 3.1 (v), is equal to the sum of ν-almost everywhere non-positive termsλh 1 (λ) 1 (λ), . . . ,λh n (λ) n (λ). Thus, all these terms are νa.e. equal to zero. If j is such that h j ≡ 0, then the function h j ∈ H 1 + has at most one root on T (counting without multiplicities). Hence, up to a set of ν measure zero, On the other hand, if h is such that the conditions (i)-(v) are satisfied, then the statements (i), (iii) and (iv) from Theorem 3.1 clearly hold. To prove that ϕ is a complex geodesic for D, it suffices to ensure that (ii) is also fulfilled. From the assumption (v), we conclude that if j is such that h j ≡ 0, then This implies that λ h (λ), (λ) dν(λ) is the null measure, what involves the condition (ii). The proof is complete.
The base of D is drawn on Fig. 1.
If the functions h 1 , h 2 are linearly dependent, then arguing similarly as in Example 3.6, we conclude that for some γ > 0, λ 0 ∈ T and α 1 , α 2 ≤ 0 such that α 1 + α 2 < 0. And again, any holomorphic map with the boundary measure given by (14) is a complex geodesic for D. We see that in this example every complex geodesic admitting a map h with linearly independent components extends holomorphically on a neighbourhood of the closed unit disc D. However, even for some 'similar' domains, this claim no longer remains valid. For example, let Set h(λ) := ((λ + 1) 2 , λ) (it belongs to the family H 2 + ) and take g so that g(λ) ∈ P D (λh (λ)) for L T -a.e. λ ∈ T, i.e.
Both components of g lie in L 1 (T, L T ). Corollary 3.4 yields that if α 1 ≤ 0, then the holomorphic map given by the boundary measure μ := g dL T + (α 1 δ −1 , 0) is a complex geodesic for D . But this map does not extend continuously on D.
Analysing these examples, one can also notice the possibility that for some h there is no map g with components in L 1 (T, L T ) such that g(λ) ∈ P D (λh (λ)) for L T -a.e. λ ∈ T, even if the latter sets are non-empty for L T -a.e. λ ∈ T (cf. Remark 3.2).
Remark 3.8 Although in the examples presented above we focused on domains whose base are subsets of (−∞, 0) n , the family D n is essentially broader. For example, the base of D := {(x 1 , x 2 ) ∈ R 2 : x 2 < −e x 1 } + iR 2 is not contained in any set of the form a + (−∞, 0) 2 , a ∈ R 2 , but W D = {(0, 0)} ∪ [0, ∞) × (0, ∞) and thus D ∈ D 2 . Applying Corollary 3.4 in the same way as previously, we can find the boundary measures of all complex geodesics for D.

Domains in C 2
Let D ⊂ C 2 be a convex tube domain containing no complex affine lines. In view of Observation 2.4, the set W D is a closed, convex, infinite cone with the vertex at the origin and non-empty interior. Thus, W D is precisely one of the following: the whole R 2 , a half-plane or a convex infinite angle, i.e. the set {(r cos θ, r sin θ) : r ≥ 0, θ ∈ [θ 1 , θ 2 ]} for some θ 1 < θ 2 < θ 1 + π. If W D is the whole R 2 , then Re D is bounded. Convex tubes with bounded base are analysed in Remark 3.3. If W D is an angle, then D is affinely equivalent to a convex tube domain D ⊂ C 2 having W D = [0, ∞) 2 , i.e. belonging to the family D 2 . These domains are intensively studied in Sect. 3.1. In this subsection, we discuss the only remaining possibility, that is the situation when W D is a half-plane. Changing coordinates if necessary, we may focus on the case when W D = R × (−∞, 0]. Then the equality holds and D is of the form Here f − and f + denote the one-sided derivatives of f . Depending on a, b and f , the set W D \ W D will be the empty set, a horizontal half-line starting at the origin or the horizontal line R×{0}. Corollary 3.9 unifies all of these cases, since, as it is illustrated by the proceeding proof, W D is conclusive, not W D itself. (v) if h 2 ≡ 0, then ν = αδ λ 0 for some α ∈ [0, ∞) and λ 0 ∈ T such that αh 2 (λ 0 ) = 0.
The condition (iii) means that dν = (0, ν). In particular, the first component of μ is absolutely continuous with respect to L T and ν is equal to the singular part of the second component of μ.
Proof of Corollary 3.9 Assume that ϕ(D) ⊂ D and ϕ is a complex geodesic for D and take h as in Theorem 3.1. The statements (ii), (iii) and (iv) follow from Theorem 3.1. Moreover, (i) is a consequence of the fact thatλh (λ) ∈ W D for every λ ∈ T. If h 2 ≡ 0, then h 2 has at most one root on T (counting without multiplicities), so the set {λ ∈ T :λh (λ) ∈ ∂ W D } contains at most one element. Thus, Theorem 3.1 (vi) yields the part (v) of the conclusion.
Conversely, Theorem 3.1 guarantees that if h is taken so that the conditions (i) -(v) are fulfilled, then ϕ(D) ⊂ D and ϕ is a complex geodesic for D.
It is a convex tube domain of the type considered in Corollary 3.9, because otherwise.
If h 1 , h 2 are linearly dependent, then applying Corollary 3.9 (v) and following the argument of the previous examples we conclude that μ = γ, γ 2 dL T + 0, αδ λ 0 for some α < 0, λ 0 ∈ T and γ > 0. Any holomorphic map having the boundary measure of the above form is a complex geodesic for D.

Proof of Theorem 3.1 and further remarks
This section is devoted to the proof of the main result of this paper, Theorem 3.1. Let us begin by recalling Theorem 1.2 from Zając [16], which provides an equivalent condition for a holomorphic map to be a complex geodesic.
We have ν z = χ T\S dν z + χ S dν z and, by (17), and If the condition (16) holds, that is, ν z ≤ 0 for every z ∈ D, then (i) is a consequence of Zając [16,Lemma 3.7] and (ii) follows from the equality (19). Conversely, if both (i) and (ii) are fulfilled, then (18) and (19) yield that for each z ∈ D the measures χ T\S dν z and χ S dν z are negative. In consequence, ν z ≤ 0.
Proof Choose S ⊂ T so that all of the equalities in (17) are fulfilled. Assume that ϕ(D) ⊂ D. The first conclusion is clear. If v ∈ W D , then for a constant C ∈ R there is x, v < C for all x ∈ Re D. In particular, Re ϕ(λ), v < C for λ ∈ D, what gives a similar inequality for measures: Taking the weak-* limit when r tends to 1, we get Hence what together with (17) gives If w ∈ W D , then there exists a sequence (v m ) m ⊂ W D tending to w. The measure μ s , w is the weak-* limit of the sequence μ s , v m of negative measures, so it is also negative. Now assume that both (i) and (ii) are satisfied. To prove the conclusion, it suffices to check that whether p ∈ R n \ Re D and v ∈ R n are such that x − p, v ≤ 0 for every x ∈ Re D, then Re ϕ(λ) − p, v ≤ 0 for every λ ∈ D. Fix p, v and λ. It is clear that v ∈ W D and Re ϕ * (ζ ) − p, v ≤ 0 for L T -a.e. ζ ∈ T. We have The proof is complete.
Proof of Theorem 3.1 Since the singular part μ s of μ is equal to dν and Re ϕ * (λ) = g(λ) for L T -a.e. λ ∈ T, the statements Lemma 4.2 (ii) and Lemma 4.3 (ii) can be written equivalently as respectively. Now it is clear that if for a map h ∈ H n , h ≡ 0 the conditions (i)-(iv) from This 'almost every' may a priori depend on w, but one can omit this problem in the following way. Take a dense, countable subset {w j : j = 1, 2, . . .} ⊂ W D and for each j choose a Borel set A j ⊂ T so that ν(T \ A j ) = 0 and (λ), w j ≤ 0 for every λ ∈ A j . Denote A := ∩ ∞ j=1 A j . It is clear that ν(T \ A) = 0 and (22) holds for all w ∈ W D and λ ∈ A. Thus, (λ) ∈ S D for ν-a.e. λ ∈ T.
This is exactly the condition (iii). Nonetheless, we still need to prove the last part of the theorem, namely that if h ∈ H n , h ≡ 0 satisfy the conditions (i)-(iv), then it fulfils also (v), (vi) and (vii).
From (i), it follows that for L T -a.e. λ ∈ T there isλh (λ) ∈ W D . Hence, (vii) is a consequence of the continuity of h.

Example 4.5
In the previously analysed examples, the singular part of the boundary measure of a complex geodesic was expressed by Dirac deltas, provided that the components of the corresponding map h were linearly independent. This example shows that there exist domains in which the singular part may be almost arbitrary, even for the above h. For instance, consider a tube domain in C 3 with the base being a half-cone, namely One can verify that Define h ∈ H 2 be the formula h(λ) := 1 2 λ 2 + 1, −iλ 2 + i, −2λ so that λh (λ) = (Re λ, Im λ, −1), λ ∈ T. Note that for any λ ∈ T it holds thatλh (λ) ∈ ∂ W D iff Im λ ≥ 0. Let ν be an arbitrary finite positive Borel measure on T singular to L T and supported on the half-circle {λ ∈ T : Im λ ≥ 0}. Define μ := g dL T + dν = dν and let ϕ ∈ O(D, D) be given by the boundary measure μ. One can check that the conditions (i), (ii) and (iii) from Theorem 3.1 are fulfilled. Now if we choose ν so that 1 2π μ(T) ∈ Re D, then in view of Theorem 3.1 the map ϕ is a complex geodesic for D. To do so, we can, for example, put ν := ω + δ 1 + δ i for a finite positive Borel measure ω singular to L T and supported on the set {λ ∈ T : Im λ ≥ 0}.

Remark 4.6
Let D ⊂ C n be a convex tube domain containing no complex affine lines. Then a map ϕ ∈ O(D, D) is a complex geodesic for D iff there exists a number m ∈ {1, 2, 3} and a real m × n matrix V with linearly independent rows such that the domain D := {V · z : z ∈ D} ⊂ C m is a convex tube containing no complex affine lines and V · ϕ is a complex geodesic for D . This claim follows from Zając [16,Lemma 4.3]: if ϕ is a complex geodesic for D and h(λ) =āλ 2 + bλ + a (a ∈ C n , b ∈ R n ) is as in Theorem 3.1, then V may be chosen so that its rows form a basis of the space X h := span R {Re a, Im a, b}. Moreover, if we affinely change the coordinates so that X h = R m × {0} n−m , then the map (ϕ 1 , . . . , ϕ m ) has to be a complex geodesic for D , while the components ϕ m+1 , . . . , ϕ n may be arbitrarily chosen, provided that ϕ(D) ⊂ D.
Finally, we obtain that every complex geodesic for D is derived, in a corresponding sense, from a complex geodesic for a convex tube domain lying in C, C 2 or C 3 and containing no complex affine lines.

Applications of Theorem 3.1 in Reinhardt domains in C n
In this section, we employ the results of Sect. 3.1 to establish formulas, or strictly speaking, a necessary condition, for extremal mappings with respect to the Lempert function and the Kobayashi-Royden pseudometric in certain classes of complete Reinhardt domains in C n . Recall that a non-empty open set G ⊂ C n is called a complete Reinhardt domain if (λ 1 z 1 , . . . , λ n z n ) ∈ G for all (z 1 , . . . , z n ) ∈ G and λ 1 , . . . , λ n ∈ D. We associate such a domain G ⊂ C n with its logarithmic image log G := {(log |z 1 |, . . . , log |z n |) ∈ R n : (z 1 , . . . , z n ) ∈ G ∩ (C * ) n } and introduce the tube domain Then Re D G = log G and the map exp : D G (z 1 , . . . , z n ) → (e z 1 , . . . , e z n ) ∈ G ∩ (C * ) n is a holomorphic covering. If the domain G is bounded and pseudoconvex, then D G belongs to the family D n . We will appeal to an argument from Edigarian and Zwonek [3] to obtain a relationship between extremal mappings for G and complex geodesics for D G , which will allow us to apply Corollary 3.4 and postulate formulas for extremal mappings in G.
Let D ⊂ C n be a domain. The Lempert function D : D × D → [0, ∞) for D is given by We say that a holomorphic map f : We shall often use the following basic fact: given σ 1 , σ 2 ∈ D, σ 1 = σ 2 and f ∈ O(D, D), the equality ρ(σ 1 , For a convex tube domain D ⊂ C n containing no complex affine lines, let G(D) denote the family of all Borel-measurable maps g = (g 1 , . . . , g n ) : T → R n such that g 1 , . . . , g n ∈ L 1 (T, L T ) and there exists h ∈ H n satisfying g(λ) ∈ P D (λh (λ)) for L T -a.e. λ ∈ T.
It follows from Theorem 3.1 that if g ∈ G(D) and ϕ ∈ M n is a map with the boundary measure g dL T , then either ϕ is a complex geodesic for D (when ϕ(0) ∈ D) or its image lies in ∂ D (in the opposite case). Note that if ϕ(D) ⊂ ∂ D and in addition the domain Re D is strictly convex in the geometric sense, then ϕ is just a constant map.
In the following two propositions, we present formulas for G -extremal and κ G -extremal maps in certain classes of bounded pseudoconvex complete Reinhardt domains.
Proposition 5.2 Let G ⊂ C 2 be a bounded pseudoconvex complete Reinhardt domain and let R 1 , G) is a G -extremal or a κ G -extremal map, then at least one of the following conditions is true: where ϕ = (ϕ 1 , ϕ 2 ) ∈ M 2 is a map with the boundary measure g dL T .
Proof of Proposition 5. 1 We present the proof only for the case when f is a G -extremal map, because the proof for a κ G -extremal map is analogous. Let σ 1 , σ 2 ∈ D be such that ρ(σ 1 , σ 2 ) = G ( f (σ 1 ), f (σ 2 )) and σ 1 = σ 2 . It is clear that k > 0. The domain π A (G) satisfies the same assumptions as G, namely it is a bounded pseudoconvex complete Reinhardt domain with Re D π A (G) being strictly convex. Moreover, if z = (z 1 , . . . , z n ), w = (w 1 , . . . , w n ) ∈ G are such that z j = w j = 0 for every j / ∈ A, then G (z, w) = π A (G) (π A (z), π A (w)). In particular, what means that π A • f is a π A (G) -extremal map. Thus, it suffices to prove the conclusion for the domain π A (G) and the mapping π A • f . But the latter map has no identically vanishing components, so we can as well assume that f 1 , . . . , f n ≡ 0 and A = {1, . . . , n}.
The claim just proved yields that for L T -a.e. λ ∈ T it holds that ϕ * (λ) ∈ ∂ D G and hence f * (λ) ∈ ∂G. Let us point out, as we will employ this fact several times, that from the considerations made so far it follows that the latter statement remains valid for every G -extremal map.
These considerations, together with Proposition 5.2, lead to the conclusion that if f = ( f 1 , f 2 ) ∈ O(D, G a, p,q ) is a G a, p,q -extremal or a κ G a, p,q -extremal map, then one of following conditions is satisfied: for some ψ ∈ O(D, S), β ∈ R and B 1 , B 2 ∈ Aut (D) ∪ {1} with B 1 B 2 ≡ 1.