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We study the limits of pluricomplex Green functions with four poles tending to the origin in a hyperconvex domain, and the (related) limits of the ideals of holomorphic functions vanishing on those points. Taking subsequences, we always assume that the directions defined by pairs of points stabilize as they tend to $0$. We prove that in a generic case, the limit of the Green functions is always the same, while the limits of ideals are distinct (in contrast to the three point case). We also study some exceptional cases, where only the limits of ideals are determined. In order to do this, we establish a useful result linking the length of the upper or lower limits of a family of ideals, and its convergence., Comment: 16 pages

In the paper we show the equality between the Lempert function and the Green function with two poles with equal weights in the bidisc thus giving the positive answer to a conjecture of Coman in the simplest unknown case. Actually, a slightly more general equality is proven which in some sense is natural when studied from the point of view of the Nevanlinna-Pick problem in the bidisc.

The spectral unit ball $\Omega_n$ is the set of all $n\times n$ matrices with spectral radius less than $1$. Let $\pi(M) \in \mathbb C^n$ stand for the coefficients of its characteristic polynomial of $M$ (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisk is $\mathbb G_n:=\pi(\Omega_n)$. When investigating Nevanlinna-Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisk (for instance to obtain continuity results for the Lempert function): if $\psi \in \mathcal O(\mathbb D, \Omega_n)$, then $\pi \circ \psi \in \mathcal O(\mathbb D, \mathbb G_n)$. Given a map $\varphi \in \mathcal O(\mathbb D, \mathbb G_n)$, we are looking for necessary and sufficient conditions for this map to "lift through given matrices", i.e. find $\psi$ as above so that $\pi \circ \psi = \varphi$ and $\psi (\alpha_j) = M_j$, $1\le j \le N$. A natural necessary condition is $\varphi(\alpha_j)=\pi(M_j)$, $1\le j \le N$. When the matrices $M_j$ are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of $\varphi$ at the points $\alpha_j$. Those conditions are necessary and sufficient for a local lift. We give a scheme which shows that the necessary conditions are also sufficient for a global lift in small dimensions (up to $5$), and a counter-example to show that the scheme fails in dimension $6$ (and above)., Comment: 22 pages; many small mistakes and typos fixed, thanks to the referee. To appear in Complex Analysis and Operator Theory

We continue the study of convergence of multipole pluricomplex Green functions for a bounded hyperconvex domain of $\mathbb C^n$, in the case where poles collide. We consider the case where all poles do not converge to the same point in the domain, and some of them might go to the boundary of the domain. We prove that weak convergence will imply convergence in capacity; that it implies convergence uniformly on compacta away from the poles when no poles tend to the boundary; and that the study can be reduced, in a sense, to the case where poles tend to a single point. Furthermore, we prove that the limits of Green functions can be obtained as limits of functions of the type $\max_{1\le i\le 3n} \frac{1}{p} \log |f_i|$, where the $f_i$ are holomorphic functions., Comment: To appear in Indiana University Mathematics Journal

We prove the non-hyperbolicity of the Kobayashi distance for $\mathcal{C}^{1,1}$-smooth convex domains in $\mathbb{C}^{2}$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. Moreover, examples of smooth, non pseudoconvex, Gromov hyperbolic domains are given; we prove that the symmetrized polydisc and the tetrablock are not Gromov hyperbolic and write down some results about Gromov hyperbolicity of product spaces., Comment: 17 pages ; typos corrected, some proofs rewritten for clarity following the referee's comments. To appear in Forum Mathematicum

We construct an example to show that no condition of slow decrease of the modulus of a function is sufficient to make it cyclic in the Hardy space of the bidisc. This is similar to what is well known in the case of the Hardy space of the disc, but in contrast to the case of the Bergman space of the disc., Comment: 4 p

It is known (implicit in [HMNT]) that when $\Lambda$ is an interpolating sequence for the Nevanlinna or the Smirnov class then there exist functions $f_\lambda$ in these spaces, with uniform control of their growth and attaining values 1 on $\lambda$ and 0 in all other $\lambda'\neq\lambda$. We provide an example showing that, contrary to what happens in other algebras of holomorphic functions, the existence of such functions does not imply that $\Lambda$ is an interpolating sequence., Comment: 9 pages, 1 figure

Let us have a family of ideals of holomorphic functions vanishing at N distinct points of a complex manifold, all tending to a single point. As is known, convergence of the ideals does not guarantee the convergence of the pluricomplex Green functions to the Green function of the limit ideal; moreover, the existence of the limit of the Green functions was unclear. Assuming that all the powers of the ideals converge to some ideals, we prove that the Green functions converge, locally uniformly away from the limit pole, to a function which is essentially the upper envelope of the scaled Green functions of the limits of the powers. As examples, we consider ideals generated by hyperplane sections of a holomorphic curve near its singular point. In particular, our result explains recently obtained asymptotics for 3-point models., Comment: 15 pages; typesetting errors fixed

We investigate the limits of the ideals of holomorphic functions vanishing on three points in $\C^2$ when all three points tend to the origin, and what happens to the associated pluricomplex Green functions. This is a continuation of the work of Magnusson, Rashkovskii, Sigurdsson and Thomas, where those questions were settled in a generic case., Comment: 15 pages

Let $X$ be a complex space and $H$ a positive homogeneous plurisubharmonic function $H$ on $X\times\C^m$. Consider the Hartogs-type domain $\Omega_{H}(X):=\{(z,w)\in X\times \C^m:H(z,w)<1 \}$. Let $S$ be an analytic subset of $X$. We give necessary and sufficient conditions for hyperbolicity and tautness modulo $S\times \C^m$ of $\Omega_{H}(X)$, with the obvious corollaries for the special case of Hartogs domains., Comment: 10 pp

It is shown that a domain in $\C^n$ which is locally convex and has $\mathcal C^1$-smooth Levi-flat boundary is locally linearly equivalent to a Cartesian product of a planar domain and $\C^{n-1}.$ This result does not extend to the case where the Levi form has locally constant rank for some $k

A domain in $\C^n$ with Levi-flat boundary near a given point is characterized in terms of the boundary behavior of the Kobayashi or Bergman metrics, or of the Bergman kernel. Some results are given in the case of intermediate values of the rank of the Levi form., Comment: v2: We have added remarks comparing our results to the previous work of S. Fu, the relevance of which we had not realized

The Lempert function for a set of poles in a domain of $\mathbb C^n$ at a point $z$ is obtained by taking a certain infimum over all analytic disks going through the poles and the point $z$, and majorizes the corresponding multi-pole pluricomplex Green function. Coman proved that both coincide in the case of sets of two poles in the unit ball. We give an example of a set of three poles in the unit ball where this equality fails., Comment: v3: proof of the upper estimate for the Green function added; accepted in Pacific Journal of Mathematics

Let $S_\epsilon$ be a set of $N$ points in a bounded hyperconvex domain in $C^n$, all tending to 0 as$\epsilon$ tends to 0. To each set $S_\epsilon$ we associate its vanishing ideal $I_\epsilon$ and the pluricomplex Green function $G_\epsilon$ with poles on the set. Suppose that, as $\epsilon$ tends to 0, the vanishing ideals converge to $I$ (local uniform convergence, or equivalently convergence in the Douady space), and that $G_\epsilon$ converges to $G$, locally uniformly away from the origin; then the length (i.e. codimension) of $I$ is equal to $N$ and $G \ge G_I$. If the Hilbert-Samuel multiplicity of $I$ is strictly larger than $N$, then $G_\epsilon$ cannot converge to $G_I$. Conversely, if the Hilbert-Samuel multiplicity of $I$ is equal to $N$, (we say that $I$ is a complete intersection ideal), then $G_\epsilon$ does converge to $G_I$. We work out the case of three poles; when the directions defined by any two of the three points converge to limits which don't all coincide, there is convergence, but $G > G_I$., Comment: 41 p., version 2. A section linking our notion of convergence to the topology of the Douady space has been added. Some typos have been corrected

We prove that an open set $D$ in $\C^n$ is pseudoconvex if and only if for any $z\in D$ the largest balanced domain centered at $z$ and contained in $D$ is pseudoconvex, and consider analogues of that characterization in the linearly convex case., Comment: v2: Proposition 14 is improved; v3: Example 15 and the proof of Proposition 14 are changed

We prove that a $C^{1,1}$-smooth bounded domain $D$ in $\C^n$ is linearly convex if and only if the convex hull of any two discs in $D$ with common center lies in $D.$, Comment: to appear in Math. Scand.; v3: Appendix is added

The Green function of the spectral ball is constant over the isospectral varieties, is never less than the pullback of its counterpart on the symmetrized polydisk, and is equal to it in the generic case where the pole is a cyclic (non-derogatory) matrix. When the pole is derogatory, the inequality is always strict, and the difference between the two functions depends on the order of nilpotence of the strictly upper triangular blocks that appear in the Jordan decomposition of the pole. In particular, the Green function of the spectral ball is not symmetric in its arguments. Additionally, some estimates are given for invariant functions in the symmetrized polydisc, e.g. (infinitesimal versions of) the Carath\'eodory distance and the Green function, that show that they are distinct in dimension greater or equal to $3$., Comment: 12 pages

We find the precise growth of some invariant metrics near a point on the boundary of a domain where the Levi form has at least one negative eigenvalue. We also introduce a new invariant pseudometric which is convenient in this context, and give some of its general properties., Comment: v5: a lower estimate for the Kobayashi metric is added

The Nevanlinna-Pick problem and the simplest case of the Carath\'eodory-Fej\'er problem on the spectral ball $\Om_3$ are reduced to interpolation problems on the symmetrized three-disc $\G_3.$, Comment: v2: minor changes

We discuss some extremal bases for $\CC$-convex domains., Comment: Few typos have been corrected in v3

We find all matrices $A$ from the spectral unit ball $\Omega_n$ such that the Lempert function $l_{\Omega_n}(A,\cdot)$ is continuous., Comment: small corrections; one reference is added

We give some further criteria for continuity or discontinuity of the Lempert funtion of the spectral ball $\Omega_n$, with respect to one or both of its arguments, in terms of cyclicity the matrices involved., Comment: 12 pages; the converse to theorem 1.3, mistakenly claimed in v1 & 2, excised from v3, is now proved in some cases

An upper estimate for the Lempert function of any $C^{1+\epsilon}$-smooth bounded domain in $\Bbb C^n$ is found in terms of the boundary distance., Comment: misprints are corrected; to appear in Math. Z

Pseudoconvexity of a domain in $\Bbb C^n$ is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point.

Necessary and sufficient conditions for Lipschitzness of the Lempert and Green functions are found in terms of their boundary behaviors., Comment: minor changes

If F is an automorphism of the spectral unit ball, we show that, in a neighborhood of any cyclic (i.e. non-derogatory) matrix of the ball, the map F can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result., Comment: 4 pages

Given $A\in\Omega_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a "generalized" tangent vector at $A$ to an entire curve in $\Omega_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A.$ In the case of $\Omega_3,$ the zero set of this metric is completely described., Comment: minor changes; to appear in Ann. Polon. Math

Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given., Comment: After submitting this paper to a journal in January the authors received a paper by G. Bharali (see arXiv:0704.1966), where some of the problems are similar to that in our paper

Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space $Hol (\D,\Omega)$ of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ., Comment: 24 pages; many typos corrected thanks to the referee of Arkiv for Matematik

We propose a definition of sampling set for the Nevanlinna class in the disk, i.e. a subset of the disk such that the analogue of the norm of a function in the Nevanlinna class can be recovered only from its values on the subset. We show it is equivalent with the notion of determination set for the same class, that is, subsets such that any function in the class which is bounded on the subset must be bounded everywhere (by the same bound, in fact); and more restrictive than the condition of being a determination set for the class of differences of positive harmonic functions (which had been studied in particular by Hayman, Lyons, and Gardiner). We give sufficient conditions for sampling to hold, as well as necessary conditions (different but closely related), and show that in the case of certain (natural) regular sets, the necessary and sufficient conditions coincide and characterize the sampling property in a numerically precise way. In particular, we observe a remarkable agreement with the results of Joaquim Ortega-Cerd\`a and Kristian Seip about "champagne subdomains", the complement in the unit disk of a union of hyperbolic disks centered on a maximal hyperbolically separated subsequence, the radii of which decrease uniformly as they approach the unit circle (arXiv:math.CV/0305075). Our methods involve a careful analysis of the decrease of the modulus of a Blaschke product., Comment: 28 pages, 2 figures

The Lempert function for several poles $a_0, ..., a_N$ in a domain $\Omega$ of $\mathbb C^n$ is defined at the point $z \in \Omega$ as the infimum of $\sum^N_{j=0} \log|\zeta_j|$ over all the choices of points $\zeta_j$ in the unit disk so that one can find a holomorphic mapping from the disk to the domain $\Omega$ sending 0 to $z$. This is always larger than the pluricomplex Green function for the same set of poles, and in general different. Here we look at the asymptotic behavior of the Lempert function for three poles in the bidisk (the origin and one on each axis) as they all tend to the origin. The limit of the Lempert functions (if it exists) exhibits the following behavior: along all complex lines going through the origin, it decreases like $(3/2) \log |z|$, except along three exceptional directions, where it decreases like $2 \log |z|$. The (possible) limit of the corresponding Green functions is not known, and this gives an upper bound for it., Comment: 16 pages; references added to related work of the author

A sequence of points $z_k$ in the unit disk is said to be thin for a given decrease function $\rho$, if there is a nontrivial bounded holomorphic function such that the infinite series $\sum_k \rho(1-|z_k|)|f(z_k)|$ converges. All sequences will be assumed hyperbolically separated. We give necessary and sufficient conditions for the problem of thinness of a sequence to be non-trivial (one way or the other), and for two different decrease functions to give rise to the same thin sequences. Along the way, some concrete conditions (necessary or sufficient) for a sequence to be thin are obtained., Comment: 15 pages, LaTeX; some typos corrected. To appear in the issue of Complex Variables dedicated to the memory of Matts Essen

We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain logarithmic Lipschitz upper envelope of it (relevant because of the Harnack inequality) admits a superharmonic majorant. We discuss the logarithmic Lipschitz regularity of this superharmonic majorant, and show that in general it cannot be better than that of the Poisson kernel. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant., Comment: 13 pages ; this expands and improves on the previous version, due to the two authors named last

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other., Comment: 25 pages

We provide results of uniqueness for holomorphic functions in the Nevanlinna class bridging those previously obtained by Hayman and Lyubarskii-Seip. Namely, we propose certain classes of hyperbolically separated sequences in the disk, in terms of the rate of non-tangential accumulation to the boundary (the endpoints of this spectrum of classes being respectively the sequences with a non-tangential cluster set of positive measure, and the sequences violating the Blaschke condition); and for each of those classes, we give a critical condition of radial decrease on the modulus which will force a Nevanlinna class function to vanish identically., Comment: 13 pp ; rewritten introduction, proofs revised for clarity, a number of minor corrections

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension of such algebras. The conditions are expressed in terms of the inner part of a function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra. Versions of these results for the case of the disk algebra are given., Comment: 22 pages ; a number of minor mistakes have been corrected, and some points clarified. Conditionally accepted by Journal d'Analyse Mathematique