arXiv subjects
Publications and source records attributed to Evgeny A. Poletsky.
In this paper we prove the basic facts for pluricomplex Green functions on manifolds. The main goal is to establish properties of complex manifolds that make them analogous to relatively compact or hyperconvex domains in Stein manifolds. The final version to appear in JGA.
Using pluricomplex Green functions we introduce a compactification of a complex manifold $M$ invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form $V$ on $M$ such that all negative plurisubharmonic functions on $M$ are in $L^1(M,V)$. Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point $w\in M$ with the normalized pluricomplex Green function with pole at $w$ we get an imbedding of $M$ into a compact set and the closure of $M$ in this set is the pluripotential compactification.
Let $M$ be a complex manifold and $PSH^{cb}(M)$ be the space of bounded continuous plurisubharmonic functions on $M$. In this paper we study when functions from $PSH^{cb}(M)$ separate points. Our main results show that this property is equivalent to each of the following properties of $M$: (1) the core of $M$ is empty. (2) for every $w_0\in M$ there is a continuous plurisubharmonic function $u$ with the logarithmic singularity at $w_0$. Moreover, the core of $M$ is the disjoint union of 1-pseudoconcave in the sense of Rothstein sets $E_j$ with the following Liouville property: every function from $PSH^{cb}(M)$ is constant on each of $E_j$.
Let $(W,Π)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\bd\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $Π\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ are called {\it $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into $\rS_{1,w_0}({\bar{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an equivalence relation and the equivalence class of $f\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ will be denoted by $[f]$ and the set of all equivalence classes by $η_1(W,M,w_0)$. There is a natural mapping $ι_1:\,η_1(W,M,w_0)\toπ_1(W,w_0)$ generated by assigning to $f\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ its restriction to $\mathbb T$. We introduce on $η_1(W,M,w_0)$ a binary operation $\star$ which induces on $η_1(W,M,w_0)$ a structure of a semigroup with unity. Moreover, $ι_1([f_1]\star[f_2])=ι_1([f_1])\cdotι_1([f_2])$, where $\cdot$ is the standard operation on $π_1(W,w_0)$. Then we establish standard properties of $η_1(W,M,w_0)$ and provide some examples. In particular, we completely describe $η_1(W,M,w_0)$ when $W$ is a finitely connected domain in $M=\mathbb C$ and $Π$ is an identity. In particular, we show for a general domain $W\subset\mahbb C$ that $[f_1]=[f_2]$ if and only if $ι_1([f_1])=ι_1([f_2])$.
Let $W$ be a domain in a connected complex manifold $M$ and $w_0\in W$. Let ${\mathcal A}_{w_0}(W,M)$ be the space of all continuous mappings of a closed unit disk $\overline D$ into $M$ that are holomorphic on the interior of $\overline D$, $f(\partial\mathbb D)\subset W$ and $f(1)=w_0$. On the homotopic equivalence classes $η_1(W,M,w_0)$ of ${\mathcal A}_{w_0}(W,M)$ we introduce a binary operation $\star$ so that $η_1(W,M,w_0)$ becomes a semigroup and the natural mappings $ι_1:\,η_1(W,M,w_0)\toπ_1(W,w_0)$ and $δ_1:\,η_1(W,M,w_0)\toπ_2(M,W,w_0)$ are homomorphisms. \par We show that if $W$ is a complement of an analytic variety in $M$ and if $S=δ_1(η_1(W,M,w_0))$, then $S\cap S^{-1}=\{e\}$ and any element $a\inπ_2(M,W,w_0)$ can be represented as $a=bc^{-1}=d^{-1}g$, where $b,c,d,g\in S$. \par Let ${\mathcal R}_{w_0}(W,M)$ be the space of all continuous mappings of $\overline D$ into $M$ such that $f(\partial{\mathbb D})\subset W$ and $f(1)=w_0$. We describe its open dense subset ${\mathcal R}^{\pm}_{w_0}(W,M)$ such that any connected component of ${\mathcal R}^{\pm}_{w_0}(W,M)$ contains at most one connected component of ${\mathcal A}_{w_0}(W,M)$.
In this paper we completely characterize those weighted Hardy spaces that are Poletsky--Stessin Hardy spaces $H^p_u$. We also provide a reduction of $H^\infty$ problems to $H^p_u$ problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.
In this paper we show that that on a strongly pseudoconvex domain $D$ the projective limit of all Poletsky--Stessin Hardy spaces $H^p_u(D)$, introduced in \cite{PS}, is isomorphic to the space $H^\infty(D)$ of bounded holomorphic functions on $D$ endowed with a special topology. To prove this we show that Carathéodory balls lie in approach regions, establish a sharp inequality for the Monge--Ampére mass of the envelope of plurisubharmonic exhaustion functions and use these facts to demonstrate that the intersection of all Poletsky--Stessin Hardy spaces $H^p_u(D)$ is $H^\infty(D)$.
The classical results about the boundary values of holomorphic or harmonic functions on a domain $D$ state that under additional integrability assumptions these functions have limits along specific sets approaching boundary. The proofs of these results are based on properties of smooth boundaries used to define the approach regions and on estimates of representing kernels along these regions. This paper attempts to look at the situation when no assumptions about the boundary smoothness are made and, consequently, no natural definitions of approach regions could be given.
We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain $W$ in a Stein manifold to a larger domain $X$ under suitable conditions on $W$ and $X$. We introduce a related equivalence relation on the space of analytic discs in $X$ with boundary in $W$. The quotient, if it is Hausdorff, is a complex manifold with a local biholomorphism to $X$. We use our disc formula to generalise Kiselman's minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.
In this paper we provide sufficient conditions for the graphs of holomorphic mappings on compact sets in complex manifolds to have Stein neighborhoods. We show that under these conditions the mappings have properties analogous to properties of holomorphic functions on compact sets in $\mathbb C^n$.
We define $F$-polynomials as linear combinations of dilations by some frequencies of an entire function $F$. In this paper we use Pade interpolation of holomorphic functions in the unit disk by $F$-polynomials to obtain explicitly approximating $F$-polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set $K\subset\mathbb C$ then optimal choices for the frequencies of interpolating polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfinite diameter of $K$. In case of the Laplace transforms of measures on $K$, we show that the coefficients of interpolating polynomials stay bounded provided that the frequencies are Fekete points. Finally, we give a sufficient condition for measures on the unit circle which ensures that the sums of the absolute values of the coefficients of interpolating polynomials stay bounded.
Edwards' Theorem establishes duality between a convex cone in the space of continuous functions on a compact space and the set of representing or Jensen measures for this cone. In this paper we prove non-compact versions of this theorem.
Let $α\in(0,1)\setminus{\Bbb Q}$ and $K=\{(e^z,e^{αz}):\,|z|\leq1\}\subset{\Bbb C}^2$. If $P$ is a polynomial of degree $n$ in ${\Bbb C}^2$, normalized by $\|P\|_K=1$, we obtain sharp estimates for $\|P\|_{Δ^2}$ in terms of $n$, where $Δ^2$ is the closed unit bidisk. For most $α$, we show that $\sup_P\|P\|_{Δ^2}\leq\exp(Cn^2\log n)$. However, for $α$ in a subset ${\mathcal S}$ of the Liouville numbers, $\sup_P\|P\|_{Δ^2}$ has bigger order of growth. We give a precise characterization of the set ${\mathcal S}$ and study its properties.
In this paper we solve the Dirichlet problems for different classes of plurisubharmonic functions on compact sets in $\mathbb C^n$ including continuous, pluriharmonic and maximal functions.
Suppose that $h$ and $g$ belong to the algebra $\B$ generated by the rational functions and an entire function $f$ of finite order on ${\Bbb C}^n$ and that $h/g$ has algebraic polar variety. We show that either $h/g\in\B$ or $f=q_1e^p+q_2$, where $p$ is a polynomial and $q_1,q_2$ are rational functions. In the latter case, $h/g$ belongs to the algebra generated by the rational functions, $e^p$ and $e^{-p}$.
In thius paper we introduce the Hardy and Bergman spaces on hyperconvex domains relative to a acontinuous exhaustion function. We prove their basic properties and study their composition operators induced by holomorphic mappings between domains.
In this paper we study the consequences of overinterpolation, i.e., the situation when a function can be interpolated by polynomial, or rational, or algebraic functions in more points that normally expected. We show that in many cases such a function has specific forms.
In this paper we obtain estimates for certain transcendence measures of an entire function $f$. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial $P(z,w)$ in ${\Bbb C}^2$ along the graph of $f$. These inequalities provide, in turn, estimates for the number of zeros of the function $P(z,f(z))$ in the disk of radius $r$, in terms of the degree of $P$ and of $r$. Our estimates hold for arbitrary entire functions $f$ of finite order, and for a subsequence $\{n_j\}$ of degrees of polynomials. But for special classes of functions, including the Riemann $ζ$-function, they hold for all degrees and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set of values $f(E)$, in terms of the size of the set $E$.