Igor Chyzhykov
In certain classes of subharmonic functions u on C distinguished in terms of lower bounds for the Riesz measure of u, a sharp estimate is obtained for the rate of approximation by functions of the form log |f(z)|, where f is an entire function. The results complement and generalize those recently obtained by Yu. Lyubarskii and Eu. Malinnikova.
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Let $D$ be a bounded domain in $\mathbb C^n$. We study approximation of (not necessarily bounded from above) $m-$subharmonic function $D$ by continuous $m-$subharmonic ones defined on neighborhoods of $\overline{D}$. We also consider the existence of a $m-$subharmonic function on $D$ whose boundary values coincides with a given real valued continuous function on $\partial D$ except for a sufficiently small subset of $\partial D.$
Approximation of $m$-subharmonic functions in weighted energy classes with given boundary values
本文研究加权能量类中带给定边界值的 $m$-次调和函数的次延拓，并据此利用更大区域上的递增序列实现对该类函数的逼近。
We consider the pointwise approximation of a subharmonic function by the logarithm of the modulus of an entire function up to a bounded quantity. In the case of finite order an estimate from below of the planar Lebesgue measure of an exceptional set in such approximation is obtained.
Let u be a subharmonic function in D={|z|<1}. There exist an absolute constant C and an analytic function f in D such that \int_D |u(z)-log|f(z)|| dm(z)<C where m denotes the plane Lebesgue measure. We also consider uniform approximation.
Subextension and approximation of $m$-subharmonic functions with given boundary values
本文研究了$m$-亚调和函数在不改变Hessian测度的情况下子扩展问题，并通过递增序列近似具有给定边界值的$m$-亚调和函数。
It is known that a subharmonic function of finite order $ρ$ can be approximated by the logarithm of the modulus of an entire function at the point $z$ outside an exceptional set up to $C\log|z|$. In this article we prove that if such an approximation is made more precise, i. e. a constant $C$ decreases, then, beginning with $C=ρ/4$, the size of the exceptional set enlarges substantially. Similar results are proved for subharmonic functions of infinite order and functions subharmonic in the unit disk. These theorems improve and complement a result by Yulmukhametov.
Maximal subextension and approximation of $m-$subharmonic function
本文研究$m$-次调和函数在两类函数空间中的极大次延拓，并利用其结果探讨逼近性质。
Let $Ω\subset \mathbb C^n$ be a bounded domain, and let $f$ be a real-valued function defined on the whole topological boundary $\partial Ω$. The aim of this paper is to find a characterization of the functions $f$ which can be extended to the inside to a $m$-subharmonic function under suitable assumptions on $Ω$. We shall do so by using a function algebraic approach with focus on $m$-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of $m$-subharmonic functions.
We will prove that a function u(x,y) defined on a domain of RpxRq that is subharmonic in one variable and harmonic in the other is (jointly) subharmonic. This solves a long-standing open problem.
We prove that if $\{ φ_j\}_j$ is a sequence of subharmonic functions which are increasing to some subharmonic function $φ$ in $\mathbb{C}$, then the union of all the weighted Hilbert spaces $H(φ_j)$ is dense in the weighted Hilbert space $H(φ)$.
We prove that the upper envelope of a family of subharmonic functions defined on an open subset of $\mathbb{R}^{N}$, $(N\geq2)$, that is finite every where, is locally bounded above outside a closed nowhere dense set with no bounded components. Then we conclude as a consequence that a separately subharmonic function is subharmonic outside a closed nowhere dense set with no bounded components. It generalizes a result due to Cegrell and Sadullaev.
We prove some basic properties of quasinearly subharmonic functions and quasinearly subharmonic functions in the narrow sense.
We prove that the set of points where a subharmonic function fails to be continuous is polar.
The aim of this paper is to study Capacities and Hessians in a class of m-subharmonic functions
In our articles of recent years, the technique of gluing two subharmonic functions turned out to be very useful in studying the distribution of the roots or masses of holomorphic or subharmonic functions, respectively. Here we develop and improve this technique. Its applications will be given in our further works.
It is shown that harmonic functions on some subsets, subharmonic and coinciding everywhere outside of these sets, actually coincide everywhere.
Maximal subextension and stability on $m$-capacity of maximal subextension of $m$-subharmonic functions with given boundary values
本文研究给定边值条件下m-次调和函数的最大次延拓，并证明其最大次延拓在m-容量上的稳定性。
We show that the spaces of $A$-$m$-subharmonic and $B$-$m$-subharmonic functions differ in sufficently high dimensions. We also prove that the Monge-Ampère type operator $\mathcal M_m$ associated to the space of $m$-plurisubharmonic functions does not allow an integral comparison principle except in the classical cases $m=1$ and $m=n$. These answer in the negative two problems posed by A. Sadullaev.
We observe that a recent result by Gardiner and Sjödin, solving a problem of Král on subharmonic functions, can be easily generalized to yield a somewhat stronger result. This can be combined with a viscosity technique of ours, which we slightly improve, to obtain Radó- type theorems for plurisubharmonic functions. Finally, we study the Borel complexity of the critical set and the set where the gradient does not exist finitely of subharmonic functions and general real valued functions.
The purpose of this paper is to introduce into consideration an analogue of the concentration index in the class of subharmonic functions of infinite order. The one in the case of finite order is used in the interpolation theory.
We establish a general uniqueness theorem for subharmonic functions of several variables on a domain. A corollary from this uniqueness theorem for holomorphic functions is formulated in terms of the zero subset of holomorphic functions and restrictions on the growth of functions near the boundary of domain.
We expand the classical balayage of measures and subharmonic functions on a system of rays $S$ with a common origin on the complex plane $\mathbb C$. This allows for an arbitrary subharmonic function $v$ of finite order on $\mathbb C$ build $δ$-subharmonic function on $\mathbb C$, harmonic outside $S$ coinciding with $v$ on $S$ outside the polar set, of the same order of growth as $v$. We give applications to the relationship of the growth of an entire function in $S$ with the distribution of its zeros.
We study the density of polynomials in $H^2(Ω,e^{-φ})$, the space of square integrable holomorphic functions in a bounded domain $Ω$ in $\mathbb{C}$, where $φ$ is a subharmonic function. In particular, we prove that the density holds in Carathéodory domains for any subharmonic function $φ$ in a neighborhood of $\overlineΩ$. In non-Carathéodory domains, we prove that the density depends on the weight function, giving examples.
We obtain a complete description of the Riesz measures of almost periodic subharmonic functions with at most of linear growth on the complex plane; as a consequence we get a complete description of zero sets for the class of entire functions of exponential type with almost periodic modulus.
We investigate decay near boundary of the volume of sublevel sets in Cegrell classes of $m-$ subharmonic function on bounded domains in $\mathbb C^n.$ On the reverse direction, some sufficient conditions for membership in certain Cegrell's classes, in terms of the decay of the sublevel sets, are also discussed.
Answering a question of A.Zygmund in \cite{MR} G.MacLane and L.Rubel described boundedness of $L_2$-norm w.r.t. the argument of $\log |B|$, where $B$ is a Blaschke product. We generalize their results in several directions. We describe growth of $p$th means, $p\in(1, \infty)$, of subharmonic functions bounded from above in the unit disc. Necessary and sufficient conditions are formulated in terms of the complete measure (of a subharmonic function) in the sense of A.Grishin. We also prove sharp estimates of the growth of $p$th means of analytic and subharmonic functions of finite order in the u…
We supplement the result of the first part of the work with estimates of the integrals of the difference of subharmonic functions in measure with some deterioration of the absolute constants, but these estimates have the form of a criterion, which is formulated exclusively in terms of the measure and its potential. This criterion is also new for meromorphic functions on the complex plane.
We obtain two-bound estimates for the local growth of pluri-subharmonic functions in terms of Siciak and relative extremal functions. As applications, we give simple new proofs of "Bernstein doubling inequality" and the main result in [Alexander Brudnyi, Local inequalities for pluri-subharmonic functions, Annals Math. 149 (1999), No. 2, pp. 511--533]. We propose a conjecture similar to the comparison theorem in [H. Alexander and B. A. Taylor, Comparison of two capacities in $\mathbb{C}^n$, Math. Z. 186 (1984), 407--417], whose validity allows to obtain bounds for the local growth of pluri-subh…
We develop classical balayage (sweeping) measures and subharmonic functions on the ray system $S$ with a general origin on the complex plane $\mathbb C$. This allows for a subharmonic function $v$ on $\mathbb C$ to construct also a subharmonic function on $\mathbb C$ that is harmonic outside $S$ and coincides with $v$ on $S$. We give applications to the relationship between the growth of an entire function on $S$ and the distribution of its zeros, the conditions of complete regular growth of entire and subharmonic functions on the ray system, to the problems of incompleteness of exponential sy…
Let $G$ be a nonempty bounded domain in a finite-dimensional Euclidean space. The main results are general estimates from below at points from $G$ for an arbitrary subharmonic function $u\not\equiv -\infty$ on the closure of the domain $G$ through the maximum of the function $u$ on the boundary of the domain $G$. These results are new for planar domains $G$, and for intervals of $G$ on the numerical line have also not been previously noted. They show that the Harnack distance plays a key role in these estimates. Further applications to subharmonic, convex, holomorphic functions, as well as to …
We construct and apply the classic balayage (sweeping out) of measures and subharmonic functions on closed system of rays in the complex plane with vertex at the origin, including measures and subharmonic functions and infinite order. The need for such a procedure occurs in the study of the behavior of entire and subharmonic functions on systems of rays. The results apply to the complete regularity of the growth of subharmonic and entire functions on a system of rays, the study of the distribution of zeros of entire functions of exponential type of class A, touches upon the incompleteness of e…
In this article, we prove an extension of the mean value theorem and a comparison theorem for subharmonic functions. These theorems are used to answer the question whether we can conclude that two subharmonic functions which agree almost everywhere on a surface with respect to the surface measure must coincide everywhere on that surface. We prove that this question has a positive answer in the case of hypersurfaces, and we also provide a counterexample in the case of surfaces of higher co-dimension. We also apply these results to Ahlfors-David sets and we prove other versions of the main resul…
In this paper we study the existence of Lelong numbers of $m-$subharmonic currents of bidimension $(p,p)$ on an open subset of $\Bbb C^n$, when $m+p\geq n$. In the special case of $m-$subharmonic function $φ$, we give a relationship between the Lelong numbers of $dd^cφ$ and the mean values of $φ$ on spheres or balls. As an application we study the integrability exponent of $φ$. We express the integrability exponent of $φ$ in terms of volume of sub-level sets of $φ$ and we give a link between this exponent and its Lelong number.
Lower envelopes with respect to convex subcones of subharmonic and plurisubharmonic functions
本文研究了有限维欧氏空间域上次调和函数凸子锥的下包络存在性问题，为 $H-H$ 中的任意函数提出了其存在来自 $H$ 的下包络的判据。
Let $m,n\geq 1$ are integers and $D$ be a domain in the $$ $\mathbb C^n$ or in the $m$-dimensional real space $\mathbb R^m$. We build positive subharmonic functions on $D$ vanishing on the boundary $\partial D$ of $D$. We use such (test) functions to study the distribution of zero sets of holomorphic functions $f$ on $D\subset \mathbb C^n$ with restrictions on the growth of $f$ near the boundary $\partial D$.
General estimates from below of holomorphic and subharmonic functions play one of the key roles in the theory of growth of holomorphic and subharmonic functions and in general in the theory of potential. At the same time, the most diverse technical apparatus was often used for this. In one of our recent papers, we showed in fairly general cases that in fact such a wide set of tools is not required. The classic Harnack distance is almost always enough. In this article, we will illustrate how this works in the cases of the unit disc in the complex plane.
Integrability of $(ω,m)$-subharmonic functions on compact Hermitian manifolds
本文证明在紧 Hermitian 流形上，所有 $(\omega,m)$-次调和函数对任意 $p < \frac{n}{n-m}$ 均为 $L^p$ 可积，给出了可积性阈值。
We estimate the growth of the canonical integral of Hadamard-Weierstrass of measure of finite order on the complex plane by the type of counting function or average counting function of this measure
Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. In 2016, we obtained a result on the existence of an entire function $f\neq 0$ satisfying the estimate $\log|f|\leq {\sf B}_u$ on $\mathbb C$, where functions ${\sf B}_u$ are integral averages of $u$ for rapidly shrinking disks as it approaches infinity. We give another equivalent version of this result with $\log |f|\leq u$ outside a very small exceptional set if $u$ is of finite order.
We consider two balayage constructions on the complex plane $\mathbb C$ with real axis $\mathbb R$ for $0\leq b\in \mathbb R$. Let $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$ of order $$\operatorname{ord}[u]:=\limsup_{z\to \infty} \frac{\ln \max\{1,u(z)\}}{\ln |z|}\leq 1,$$ $U=u-v$ be the difference of subharmonic functions $u$ and $v\not\equiv -\infty$ on $\mathbb C$ with $\operatorname{ord}[v]\leq 1$, i.e., $δ$-subharmonic function on $\mathbb C$ of order $\operatorname{ord}[U]\leq 1$. Then there is a $δ$-subharmonic function $V\not\equiv \pm\infty$ on $\mathbb C$ of order…
Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. Then for any function $r\colon\mathbb C\to (0,1]$ satisfying the condition $$\inf_{z\in\mathbb C}\frac{\ln r(z)}{\ln(2+|z|)}>-\infty,$$ there is an entire function $f\not\equiv 0$ such that $$ \ln |f(z)|\leq \frac{1}{2π}\int_0^{2π}u(z+r(z)e^{iθ})\,{\mathrm d}θ\quad\text{for all $z\in\mathbb C$.}$$ A similar result is established for subharmonic functions of finite order with inequalities of the form $\ln|f(z)|\leq u(z)$ at all points $z\in\mathbb C\setminus E$, where the exceptional set $E$ is small in terms …
We use sup-convolution to find upper approximations of a bounded $m$-subharmonic function on a compact Kähler manifold with nonnegative holomorphic bisectional curvature. As an application, we show the Hölder continuity of solutions to $σ_m$ equation when the right hand side is in $L^p$, $p>\frac{n}{m}$. All these results generalize to more general complex Hessian equations.
We present an analytical proof that certain natural metric planar universal covers are Hadamard metric spaces. In particular if $ρ=φ\circ u$ where $u$ is locally Lipschitz and subharmonic in $Ω$, $φ$ is positive and increasing on an interval containing $u(Ω)$ with $\logφ$ convex, and if the metric space $(Ω,ρ(z)|dz|)$ is complete, then it has universal cover $(\tildeΩ,\tilde{d})$ which is a Hadamard space for which geodesics have Lipschitz continuous first derivatives.
Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ are two subharmonic functions in the complex plane $\mathbb C$ with the Riesz measures $ν_u$ and $μ_M$ such that $u(z)\leq O(|z|)$ and $M(z)\leq O(|z|)$ as $z\to \infty$. If the growth of a function $M$ in some sense exceeds the growth of a function $u$ on some straight line, then we can expect measure $μ_M$ to dominate measure $ν_u$ in some sense. We give quantitative forms of such dominance. The main results are illustrated by a new uniqueness theorem for entire functions of exponential type.
Contraction property of Fock type space of log-subharmonic functions in $\mathbb{R}^m$
本文证明了对数次调和函数的 Fock 型空间的收缩性质，通过建立超水平集测度的单调性，推广了 Carlen 关于全纯函数的结果。
The maximum of the modulus of a meromorphic function cannot be restricted from above by the Nevanlinna characteristic of this meromorphic function. But integrals from the logarithm of the module of a meromorphic function allow similar restrictions from above. This is illustrated by one of the important theorems of Rolf Nevanlinna in the classical monograph by A.A. Goldberg and I.V. Ostrovskii on meromorphic functions, as well as by the Edrei-Fuchs Lemma on small arcs and its versions for small intervals in articles by A.F. Grishin, M.L. Sodin, T.I. Malyutina. Similar results for integrals of d…
We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves developing a new technique to give an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, giving a sharp estimate for the growth in many cases.
In this paper we study the class $\mathcal{E}_{m}(Ω)$ of $m-$subharmonic functions introduced by Lu in \cite{L1}. We prove that the convergence in $m-$capacity implies the convergence of the associated Hessian measure for functions that belong to $\mathcal{E}_{m}(Ω)$. Then we extend those results to the class $\mathcal{E}_{m,χ}(Ω)$ that depends on a given increasing real function $χ$. A complete characterization of those classes using the Hessian measure is given as well as a subextension theorem relative to $\mathcal{E}_{m,χ}(Ω)$.
Let $u$ be a plurisubharmonic function. We prove the existence of a nonzero holomorphic function such that the logarithm of its modulus is not more than local averages of this function $u$. This is the abstract for scientific conference "Algebra, Analysis and Related Problems of Mathematical Modeling" (Vladikavkaz, June 26-27, 2015) dedicated to the 60th anniversary of Professor, Doctor of Physical and Mathematical Sciences Vladimir A. Koibaev
Hessian type equations for $m-ω-$subharmonic functions on bounded domains in $\mathbb C^n$
研究复\(n\)维空间中有界域上\(m - ω -\)次调和函数的黑塞型方程，借助文献结果证明此类方程有界解的存在性。
We study subharmonic functions whose Laplacian is supported on a null set and in connected components of of the complement to the support admit harmonic extensions to larger sets. We prove that if such a function has a piecewise holomorphic derivative, then it is locally piecewise harmonic. In generic cases it coincides locally with the maximum of finitely many harmonic functions. Moreover, we describe the support when the holomorphic derivative satisfies a global algebraic equation. The proofs follow classical patterns and our methods may also be of independent interest.
Contraction property of certain classes of log-$\mathcal{M}-$subharmonic functions in the unit ball in $\mathbb{R}^n$
本文证明单位球中绝对值对数超调和的光滑函数类的压缩性质，推广Kulikov结果至高维，并应用于复平面调和映射获得新结果。
Let $\mathbb C$ be the complex plane, $E$ be a measurable subset in a segment $[0, R]$ of the positive semiaxis $\mathbb R^+$, $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$. The main result of this article is an upper estimate of the integral of the module $|u|$ over a subset of $E$ through the maximum of the function $u$ on a circle of radius $R$ centered at zero and a linear Lebesgue measure of subset $E$. Our result develops one of the classical theorems of R. Nevanlinna in the case of $E=[0, R]$ and versions of so-called Small Arcs Lemma by Edrei-Fuchs for small intervals …
Let $f$ be a meromorphic function on the complex plane $\mathbb C$ with the maximum function of its modulus $M(r,f)$ on circles centered at zero of radius $r$. A number of classical, well-known and widely used results allow us to estimate from above the integrals of the positive part of the logarithm $\ln^+M(t,f)$ over subsets of $E\subset [0,r]$ via the Nevanlinna characteristic $T(r,f)$ and the linear Lebesgue measure of the set $E$. The paper gives similar estimates for the Lebesgue-Stieltjes integrals of $\ln^+M (t,f)$ over the increasing integration function of $m$. The main part of the p…
Let $U\not\equiv \pm\infty$ be the difference of subharmonic functions, i.e., a $δ$-subharmonic function, on a closed disc of radius $R$ centered at zero. In the preceding first part of our paper, we obtained general estimates for the integral of the positive part of the radial maximum growth characteristic ${\mathsf M}_U(t):=\sup\bigl\{U(z)\bigm| |z|=r\bigr\}$ over the increasing integration function $m$ on the segment $[0, r]$ via the Nevanlinna difference characteristic and the modulus of continuity of the function $m$. The second part of the work gives an explicit view for such estimates, …
Minimal m-subharmonic functions with nonmaximal Hessian measures
本文构造了闭单位球上Hölder连续的m次调和函数，它们在Cegrell类中最小但Hessian测度非极大，回答了第二作者的问题，并给出了最小性准则及乘积域上的Monge–Ampère例子族。
We prove that almost periodicity in the sense of distributions coincides with almost periodicity with respect to Stepanov's metric for the class of subharmonic functions in a horizontal strip. We also prove that Fourier coefficients of these functions are continuous functions in Im z. Further, if the logarithm of a subharmonic almost periodic function is a subharmonic function, then it is almost periodic.
In this course of lectures we give an account of the growth theory of subharmonic functions, which is directed towards its applications to entire functions of one and several complex variables.
Let $D$ be a domain in the complex plane, $M$ be an extended real function on $D$. If $f$ is a non-zero holomorphic function on $D$ with an upper constraint $|f|\leq \exp M$ on this domain $D$, then it is natural to expect that there must be some upper constraints on the distribution of zeros of this holomorphic function exclusively in terms of the function $M$ and the geometry of the domain $D$. We have investigated this question in detail in our previous works in the case when $M$ is a subharmonic function and the domain $D$ is arbitrary or with a non-polar boundary. The answer was given in …
We prove Richberg type theorem for $m$-subharmonic function. The main tool is the complex Hessian equation for which we obtain the existence of the unique smooth solution in strictly pseudoconvex domains.
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