We develop and use some key concepts of potential theory, such as balayage and duality between measures and their potentials, to study the distribution of masses of subharmonic functions while restrictions to their growth near the boundary of their domain of definition. They are formulated and proved in terms of the Riesz measure for a subharmonic function. In this article, applications to holomorphic functions concern the distribution of their zero sets under restrictions to growth of this functions. Auxiliary results of our article may also be of independent interest. This is, first of all, 1) theorems on gluing of subharmonic functions and on gluing a subharmonic function with Green's function; 2) an internal description of the duality between balayage of measures and their potentials; 3) the duality between the Jensen and Arens-Singer measures on the one hand and the Jensen and Arens-Singer potentials of on the other; 4) a generalization of classical Poisson-Jensen formula connecting subharmonic functions, their Riesz measures, harmonic measures, and Green's functions.
We investigate some properties of balayage, or, sweeping (out), of measures with respect to subclasses of subharmonic functions. The following issues are considered: relationships between balayage of measures with respect to classes of harmonic or subharmonic functions and balayage of measures with respect to significantly smaller classes of specific classes of functions; integration of measures and balayage of measures; sensitivity of balayage of measures to polar sets, etc.
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 investigate some properties of balayage of charges and measures for subclasses of subharmonic functions and their relationship to the geometry of domain or open set in finite-dimensional Euclidean space where this balayage is considered.
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…
Our main results are certain developments of the classical Poisson--Jensen formula for subharmonic functions. The basis of the classical Poisson--Jensen formula is the natural duality between harmonic measures and Green's functions. Our generalizations use some duality between the balayage of measures and and their potentials.
In the second part of this work was developed a technique of balayage of finite genus $q=0,1,2,\dots$ for measures (charges) and ($δ$-) subharmonic functions of finite order to an arbitrary closed system of rays $S$ with vertex at origin on the complex plane $\mathbb C$. In this third part of our work, we use only the case $q=1$ when $S$ is a pair of oppositely directed rays, i.e., $S$ is a straight line as the point set, and balayage is made from both sides of this line. We consider measures and subharmonic functions of finite type of order $1$. This bilateral balayage of genus $1$ will be ap…
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…
We establish the theory of balayage for the Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, on $\mathbb R^n$, $n\geqslant3$, alternative to that suggested in the book by Landkof. A need for that is caused by the fact that the balayage in that book is defined by means of the integral representation, which, however, so far is not completely justified. Our alternative approach is mainly based on Cartan's ideas concerning inner balayage, formulated by him for the Newtonian kernel. Applying the theory of inner Riesz balayage thereby developed, we obtain a number of criteria for the existence of an inner e…
Balayage of measures with respect to classes of all subharmonic or harmonic functions on an open set of a plane or finite-dimensional Euclidean space is one of the main objects of potential theory and its applications to the complex analysis. For a class $H$ of functions on $O$, a measure $ω$ on $O$ is a balayage of a measure $δ$ on $O$ with respect to this class $H$ if $\int_O h\, d δ\leq \int_O h\, dω$ for each $h\in H$. In our previous works we used this concept to study envelopes relative to classes of subharmonic and harmonic functions and apply them to describe zero sets of holomorphic f…
For the Riesz kernel $κ_α(x,y):=|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$, we introduce the inner pseudo-balayage $\hatω^A$ of a (Radon) measure $ω$ on $\mathbb R^n$ to a set $A\subset\mathbb R^n$ as the (unique) measure minimizing the Gauss functional \[\intκ_α(x,y)\,d(μ\otimesμ)(x,y)-2\intκ_α(x,y)\,d(ω\otimesμ)(x,y)\] over the class $\mathcal E^+(A)$ of all positive measures $μ$ of finite energy, concentrated on $A$. For quite general signed $ω$ (not necessarily of finite energy) and $A$ (not necessarily closed), such $\hatω^A$ does exist, and it maintains the basic features of …
Previously, we developed the technique of balayage of measures or charges and ($δ$-)subharmonic functions of finite order onto an closed system of rays $S$ with a vertex at zero on the complex plane $\mathbb C$. In this article, we use only two kinds of balayage of measure and charge, as well as of subharmonic functions of finite type under the order $1$ and their differences. First, it is a classical balayage of the genus $q=0$ on a system of four closed rays: positive and negative, and real and imaginary semi-axis $\mathbb R^+$, $-\mathbb R^+$, $i\mathbb R^+$, $-i\mathbb R$. Second, it is tw…
Let $ν$ be a charge distribution on the complex plane $\mathbb C$, i.e. the real Radon measure on $\mathbb C$ with total variation $|ν|$. The charge distribution $ν$ is of finite upper density under order of $1$ if $$ \limsup_{0<r\to+\infty} \frac{1}{t}|ν|\Bigl(\bigl\{z\in \mathbb C \bigm| |z|\leq r\bigr\}\Bigr)<+\infty. $$ The charge distribution $ν$ satisfies the Lindelöf condition of genus $1$ if $$ \limsup_{1\leq r\to +\infty}\biggl|\int_{1\leq |z|\leq r} \frac{1}{z}\operatorname{d}\!ν(z)\biggr|<+\infty. $$ These concepts play a key role in the study of entire functions of exponential type…
The paper deals with the theory of inner (outer) capacities on locally compact spaces with respect to general function kernels, the main emphasis being placed on the establishment of alternative characterizations of inner (outer) capacities and inner (outer) capacitary measures for arbitrary sets. The analysis is substantially based on the close interaction between the theory of capacities and that of balayage. As a by-product, we provide a rigorous justification of Fuglede's theories of inner and outer capacitary measures and capacitability (Acta Math., 1960). The results obtained are largely…
Let $X$ be a class of extended numerical functions on a domain $D$ of $d$-dimensional Euclidean space $\mathbb R^d$, $H\subset X$. Given $u,M\in X$, we write $u\prec_H M$ if there is a function $h\in H$ such that $u+h\leq M$ on $D$. We consider this special preorder $\prec_H$ for a pair of subharmonic unctions $u, M$ on $D$ in cases where $H$ is the space of all harmonic functions on $D$ or $H$ is the convex cone of all subharmonic functions $h \not\equiv -\infty$ on $D$. Main results are dual equivalent forms for this preorder $\prec_H$ in terms of balayage processes for Riesz measures of sub…
Let $M$ be a subharmonic function on a domain $D$ in the complex plane $\mathbb C$ with the Riesz measure $ν_M$. Let $f$ be a non-zero holomorphic function on $D$ such that $\log |f|\leq M$ on $D$ and the function $f$ vanish on a sequence ${\tt Z}=\{{\tt z}_k\}_{k=1,2, \dots}\subset D$ {\large(}$u\not\equiv -\infty$ be a subharmonic function on $D$ with the Riesz measure or the mass distribution $ν_u$, and $u\leq M$ on $D$ resp.{\large)}. Then restrictions on the growth of the Riesz measure $ν_M$ of the function $M$ near the boundary of the domain $D$ entail certain restrictions on the distrib…
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 …
Given two compact sets, $E$ and $F$, on the unit circle, we study the class of subharmonic functions on the unit disk which can grow at the direction of $E$ and $F$ (sets of singularities) at different rate. The main result concerns the Blaschke-type condition for the Riesz measure of such functions. The optimal character of such condition is demonstrated.
Focusing first on the inner $α$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $μ^A$ the inner $α$-Riesz balayage of a Radon measure $μ$ to $A\subset\mathbb R^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass, establish the vague continuity of the map $y\mapsto\varepsilon_y^A$ outside the inner $α$-irregular points for $A$, and obtain necessary and sufficient conditions for $\varepsilon_y^A$ to be of finite energy (more generally, for $\varepsilon_y^A$ to be absolutely continuous with respect to inner cap…
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.
First introduced by J. Deny, the classical principle of positivity of mass states that if $κ_αμ\leqslantκ_αν$ everywhere on $\mathbb{R}^n$, then $μ(\mathbb{R}^n)\leqslantν(\mathbb{R}^n)$. Here $μ,ν$ are positive Radon measures on $\mathbb{R}^n$, $n\geqslant2$, and $κ_αμ$ is the potential of $μ$ with respect to the Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,2]$, $α<n$. We strengthen Deny's principle by showing that $μ(\mathbb{R}^n)\leqslantν(\mathbb{R}^n)$ still holds even if $κ_αμ\leqslantκ_αν$ is fulfilled only on a proper subset $A$ of $\mathbb{R}^n$ that is not inner $α$-thin at infinity; …
Let $M$ be a subharmonic function with Riesz measure $ν_M$ in a domain $D$ in the $n$-dimensional complex Euclidean space $\mathbb C^n$, and let $f$ be a nonzero function that is holomorphic in $D$, vanishes on a set ${\sf Z}\subset D$, and satisfies $|f| \leq \exp M$ on $D$. Then restrictions on the growth of $ν_M$ near the boundary of $D$ imply certain restrictions on the area/volume of $\sf Z$. We give a quantitative study of this phenomenon in the subharmonic framework.
Let $D$ be a domain in the complex plane $\mathbb C$. It follows from first part of our work that if a non-zero holomorphic function $f$ on $D$ vanishes on a sequence ${\sf Z}\subset D$ and satisfies $|f|\leq M$ on $D$, where $M$ is a subharmonic function on $D$, then the the distribution of ${\sf Z}$ is subordinated to the Riesz measure $ν_M$ of $M$ in a certain sense. Here we show that this result is "almost reversible".
Let $M$\/ be a subharmonic function with Riesz measure $μ_M$ on the unit disk $\mathbb D$ in the complex plane $\mathbb C$. Let $f$ be a nonzero holomorphic function on $\mathbb D$ such that $f$ vanishes on ${\sf Z}\subset \mathbb D$, and satisfies $|f| \leq \exp M$ on $\mathbb D$. Then restrictions on the growth of $μ_M$ near the boundary of $D$ imply certain restrictions on the distribution of $\sf Z$. We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using $ρ$-trigonometrically convex functions.
