Functional Analysis and Its Applications, Vol. 52, No. 1, pp. 21–34, 2018 Translated from Funktsional nyi Analiz i Ego Prilozheniya, Vol. 52, No. 1, pp. 26–42, 2018 c by B. N. Khabibullin and A. P. Rozit Original Russian Text Copyright 

On the Distribution of Zero Sets of Holomorphic Functions ∗ B. N. Khabibullin and A. P. Rozit Received November 13, 2016

Abstract. Let M be a subharmonic function with Riesz measure νM in a domain D in the ndimensional complex Euclidean space Cn , and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ⊂ D, and satisfies |f |  exp M on D. Then restrictions on the growth of νM near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework. Key words: holomorphic function, zero set, subharmonic function, Riesz measure, Jensen measure.

1. Introduction 1.1. Earlier results. The multidimensional results obtained prior to mid-1990s on the zero sets of holomorphic functions defined in a domain D with restrictions on the growth of their modulus near the boundary ∂D can be found in the book [1], the papers [2]–[4], and surveys [5, 6.5] and [6, §6]. For example, it is well known that the volume/area of the zero set of a bounded nonzero holomorphic function is bounded (part of the classical Khenkin–Skoda theorem), while the zero set of a nonzero holomorphic function of finite order of growth has the same order of growth (part of the Dautov–Skoda theorem). The case of entire functions of several variables is sufficiently completely presented in [7]–[12]. Some subharmonic multidimensional results of this type have been obtained relatively recently by Favorov and Radchenko in [13]. In the case of functions holomorphic and subharmonic in domains on the complex plane C, the history of the problems in question is described in sufficient detail in [14]. In the present paper, we consider only an “easy part” of the problem of description of zero (sub)sets of holomorphic functions with a given majorant; namely, we consider necessary conditions in the form of restrictions on the growth of the “area/volume” of the zero set near the boundary of the domain. But this “easy part” of the problem is given in the most general form, i.e., for arbitrary domains and for a very wide range of restrictions on the growth of the moduli of holomorphic functions and various integral constraints on their zero sets. A basic typical result of this kind obtained earlier is given by the following theorem. Theorem A ([15, Main Theorem], [16, Corollary 1]). Let K be a compact set with nonempty interior in a domain D ⊂ Cn , let M ≡ −∞ be a subharmonic function in D with Riesz meain D \ K satisfying the condition sure νM , and let v be a positive bounded subharmonic function  lim supDz  →z v(z  ) = 0 for any point z ∈ ∂D such that D\K v dνM < +∞. If f is a nonzero holomorphic function in D with zero set Z = {z ∈ D : f (z) = 0} for which log |f (z)|  M (z) for all z ∈ D, then the integral Z\K v dσ2n−2 , where σ2n−2 is the (2n − 2)-dimensional surface measure, or the Hausdorff (2n − 2)-measure, is finite. We develop Theorem A for uniformly bounded subclasses of test functions v (cf. the results obtained by B. Korenblum and K. Seip about uniform Bergman classes [17] and [18, §5]) and for significantly more general majorants M , i.e., arbitrary differences of subharmonic functions in the form of which we present a very wide class of functions, the simplest of which are subharmonic functions themselves and arbitrary functions twice differentiable on D. ∗

The work is supported by the Russian Foundation for Basic Research (grant no. 16-01-00024). 0016–2663/18/5201–0021

c 2018 Springer Science+Business Media, Inc.

21

Along with the material in the papers listed above, we often use information given in [19]–[22] without mentioning the source. The proofs do not use the results in the yet unpublished paper [14], which is closely related to the present paper in the topic and methods. The authors thank the referee for several useful remarks and corrections. 1.2. Basic notation, definitions, and conventions. The reader can skip this Subsection 1.2 and return to it only if necessary. 1.2.1. Sets, topology, order. N is the set of positive integers, N0 := {0} ∪ N, R = R1 and C = C1 are respectively the sets of all real and complex numbers, R+∞ := R∪{+∞}, R±∞ := {−∞}∪R+∞ , + R+ := {x ∈ R : x  0}, and R+ +∞ := R ∪ {+∞}. We supplement the order relation  on R with the inequalities −∞  x  +∞ for all x ∈ R±∞ . For n, m ∈ N, the affine spaces Cn and Rm over C and R, respectively, are equipped with the standard Euclidean modulus norm | · |. By Rm ∞ := (Rm )∞ , Cn∞ := (Cn )∞ , and C∞ := (C1 )∞ we denote the one-point Aleksandrov compactifications; |∞| := +∞. If necessary, we identify Cn and Cn∞ with R2n and R2n ∞ , respectively (over R). Further, m if possible, we introduce the notation and give definition only for Rm and Rm ∞ . For a subset S ⊂ R∞ , we let clos S , int S , ∂S , and S := Rm ∞ \ S denote its closure, interior, boundary, and complement m is an open connected subset in Rm . For S ⊂ S ⊂ Rm , we write . A (sub)domain in R in Rm 0 ∞ ∞ ∞ ∞ + S0  S if clos S0 is a compact subset in S in the topology induced from Rm ∞ on S . For r ∈ R+∞ m  m  and x ∈ R , we set B(x, r) := {x ∈ R : |x − x| < r} (i.e., B(x, r) is an open ball of radius r centered at x), B(r) := B(0, r) and B(x, +∞) = Rm ; B(∞, r) := {x ∈ Rm : |x| > 1/r} and B(∞, +∞) := Rm ∞ \ {0}. The same symbol 0 is used, depending on the context, to denote the number zero, the origin, a zero vector, a zero function, a zero measure, etc. For a subset X of an ordered vector space of numbers, functions, measures, etc. with order relation , we write X + := {x ∈ X : x  0}; i.e., X + is the set of all positive elements in X ; x+ := max{0, x}. The positiveness is everywhere understood as  0 according to the context. The class of Borel subsets of a set S ⊂ Rm ∞ is denoted by B(S). 1.2.2. Functions. The value f (x) ∈ Y of an arbitrary function f : X → Y is not necessarily defined for all x ∈ X . The restriction of a function f to X0 ⊂ X is denoted by f |X0 . A function f is said to be extended numerical if its image is a subset of R±∞ . For extended numerical functions f , we set dom f := f −1 (R). The order relation on the space of extended numerical functions is induced by the order relation on R±∞ as a pointwise relation. At the same time, for extended numerical functions f : X → Y and g : X1 → Y for X0 ⊂ X ∩ X1 , we write “f  g on X0 ” and “f = g on X0 ” if f |X0  g|X0 and f |X0 = g|X0 , respectively. For an open subset O ⊂ Rm ∞ , by har(O) we denote the vector space over R of functions harmonic (affine for m = 1) in O ; sbh(O) is the convex cone over R+ of functions subharmonic (convex for m = 1) in O functions. We denote the function identically equal to −∞ on O by the symbol −∞ ∈ sbh(O); sbh∗ (O) := sbh(O) \ {−∞}. For O ⊂ Cn∞ , we let Hol(O) denote the vector space over C of functions holomorphic on O . For ∗ m  2, a function u is holomorphic or (sub)harmonic in an open neighborhood O of the point ∞ if, for the inversion transformation centered at zero (and at the point ∞) ⎧ ⎪ for x = ∞, ⎨0  2 (1.1)  : x → x := x/|x| for x = 0, ∞, ⎪ ⎩ ∞ for x = 0, (1.1)

the Kelvin transform u of the function u defined by the relations ∗∗ u (x ) = |x|m−2 u(x), x ∈ O ⊂  Rm ∞ , is respectively a holomorphic or (sub)harmonic function but already in a neighborhood O :=  m {x : x ∈ O} of the point 0. The inversion x0 centered at x0 ∈ R is defined as a superposition of the translation x → x − x0 and the successive inversion 0 := . For a proper subset S ⊂ Rm ∞ , the class sbh(S) consists of the restrictions to S of functions subharmonic in some (in general, its own ∗

For m = 1, the case of ∞ ∈ O ⊂ R∞ is not discussed in this paper. A reference mark over a symbol of (in)equality, inclusion, or more general binary relation, etc. means that this relation is somehow related to this reference. ∗∗

22

+ + for each function) open set O ⊂ Rm ∞ containing S ; sbh (S) := (sbh(S)) . The spaces har(S) and + har (S) are defined similarly. For u ∈ sbh(S), we set (−∞)u (S) := {z ∈ S : u(z) = −∞}, which is the (−∞)-set of the function u in S [20, 3.5]; we often simply write (−∞)u without indicating S . By consta1 ,a2 ,... ∈ R we denote constants that, in general, depend on a1 , a2 , . . . and, unless otherwise specified, only on them; const+ ...  0. 1.2.3. Measures. In what follows, M (S) is the class of Borel real measures on subsets in S ∈ B(Rm ∞ ) which range in R±∞ and are also called charges; Mc (S) is a subclass of measures in M (S) with compact support supp ν  S , M + (S) := (M (S))+ . For a charge μ ∈ M (S), we let μ+ , μ− := (−μ)+ and |μ| := μ+ + μ− , respectively, denote its upper, lower, and total variations. A measure μ ∈ M (S) is concentrated on a subset S0 ⊂ S measurable with respect to the measure μ if μ(S  ) = μ(S  ∩ S0 ) for any subset S  ⊂ S measurable with respect to the measure μ. The set of all subsets in S on which the measure μ ∈ M (S) is concentrated is denoted by Conc μ. Obviously, supp μ ∈ Conc μ. For x ∈ Rm ∞ and 0 < r ∈ R+ , we set μ(x, r) := μ(B(x, r)). The Riesz measure of a function u ∈ sbh(O), O ⊂ Rm ∞ , is most often denoted by

νu :=

1 sm−1

Δu ∈ M + (O),

sm−1 :=

2π m/2 max{1, (m − 2)} ; Γ(m/2)

(1.2)

sm−1 is the area of the (m − 1)-dimensional unit sphere ∂B(1) in Rm , Δ is the Laplace operator acting in the sense of the theory of distributions or generalized functions, and Γ is the gamma function. Such measures νu are Radon measures, i.e., define a positive continuous and bounded linear functional on the space C0 (O) of continuous finite functions on O . In particular, νu (S) < +∞ for each subset S  O measurable with respect to the measure νu . For u = −∞ ∈ sbh(O), by definition, ν−∞ (S) := +∞ for all S ⊂ O . We let λm ∈ M + (S) denote restrictions of the Lebesgue measure to Borel proper subsets / S ; δx ∈ M + (S) S ⊂ Rm ∞ , where, as ∞ ∈ S , we preliminary use the inversion x0 centered at x0 ∈ is the Dirac measure at a point x ∈ S , i.e., supp δx = {x} and δx ({x}) = 1. In the notation of the Lebesgue measure, the index m is often omitted. For p ∈ R+ , we let σp denote the Hausdorff p-dimensional (outer) measure or Hausdorff p-measure in open proper subsets O ⊂ Rm ∞ (for / O ). In the present paper, the Hausdorff ∞ ∈ O , we again use the inversion centered at x0 ∈ p-measure is used only for integer p ∈ N0 :

 p rj : S ⊂ B(xj , rj ), 0  rj < r , σp (S) := bp lim inf r→0

j∈N

j∈N

(1.3)

sp−1 ; b0 := 1, bp := p (1.2)

bp for p ∈ N is a normalizing factor equal to the volume of the unit ball B(1) in Rp . For such a normalization with p = 0, for any subset S ⊂ Rm ∞ , its Hausdorff 0-measure σ0 (S) is equal to the cardinality of the set S , i.e., to the number of points in S , and for p = m, we have σm = λm , (1.2)

(1.3)

sm−1 = σm−1 (∂B(1)), bp = σp (B(1)). In the space Cn∞ identified with R2n ∞ , in the present paper, we use only the Hausdorff (2n − 2)-measure σ2n−2 , which is often called a (2n − 2)-dimensional surface measure. At the same time, in (1.2) and in (1.3), (1.2)

s2n−1 =

2π n max{1, 2n − 2} , (n − 1)!

(1.3)

b2n−2 :=

π n−1 max{1, 2(n − 2)} . (n − 1)!

(1.4)

1.2.4. Zeros of holomorphic functions ([7, §11]–[12], [21, Ch. 1]). Let D be a subdomain in Cn∞ , 0 = f ∈ Hol(D). The divisor of zeros of a function f is the function Zerof : D → N0 equal to the multiplicity of the zero of the function f at each point z ∈ D. For f = 0 ∈ Hol(D), by definition, Zero0 ≡ +∞ on D. Further, 0 = f ∈ Hol(D) everywhere in this section. The divisor of zeros Zerof is upper semicontinuous in D. The support supp Zerof is the principal analytic set of pure dimension n − 1 over C and dimension 2n − 2 over R for which always σ2n−2 (supp Zerof \ reg supp Zerof ) = 0, 23

 and reg supp Zerof is the set of regular points. Let reg supp Zerof = j Zj be the representation in the form of a union of at most countably many connected components Zj , j = 1, 2, . . . . Then the family {Zj } is locally finite in D; i.e., each subset S  D meets at most finitely many components Zj . The divisor of zeros Zerof is constant on each component Zj ; i.e., the value Zerof (Zj ) is uniquely determined for each j ∈ N. To each divisor of zeros Zerof we assign the counting measure of zeros nZerof ∈ M + (D) defined as a Radon measure by the formulas  (1.3)  ϕ Zerof dσ2n−2 over all ϕ ∈ C0 (D), or, equivalently, as a Borel nZerof (ϕ) := ϕ dnZerof := (1.3) measure on D according to the rule nZerof (B) = j Zerof (Zj )σ2n−2 (B ∩ Zj ) for all B ∈ B(D). Poincar´ e–Lelong formula [23]. Let D = ∅ be a proper subdomain in Cn∞ , n ∈ N, 0 = f ∈ Hol(D). The following relations hold for the Riesz measure νlog |f | of the function log |f | ∈ sbh∗ (D): (1.2)

νlog |f | =

1 s2n−1

(1.4)

Δ log |f | =

2π n

(n − 1)! Δ log |f |=nZerof . max{1, 2n − 2}

(1.5)

The function Z : D → R+ is a subdivisor of zeros for f ∈ Hol(D) if Z  Zerof on D. Obviously, for f ∈ Hol(D), its divisor of zeros is the subdivisor of zeros for f . The integrals with respect to a positive measure whose integrands contain a subdivisor are everywhere below treated as upper integrals. 1.2.5. Domain D and its compactification. Everywhere below in this paper, n D = ∅ is a proper subdomain in Rm ∞ = D or in C∞ = D,

and also ∞ ∈ / clos D for m = 1; i.e., D ⊂ R is an open interval of finite length. We shall use the one-point Aleksandrov compactification of such a domain D, which we denote by D∂ := D ∪ {∂D}, where the added point ∂D is also called the boundary of the domain D. In particular, we have n n already encountered (Rm )∂ = Rm ∞ and (C )∂ = C∞ . The base of open (closed) neighborhoods of this point ∂D ∈ D∂ can be all possible sets (∂D) ∪ (D \ S), where S runs through the set of all compact sets S in D (respectively, open subsets S  D). Further, the phrase “near ∂D” means “in some punctured neighborhood of the point ∂D ∈ D∂ ” or “on a set of the form D \ S , where S  D.” For constant constm,n,D,... , we do not indicate the dependence on the dimensions m, n and the domain D and do not discuss it. 2. Corollaries of the Main Theorem 2.1. Subharmonic test functions. For an extended numerical function v defined near ∗ ∂D, the expression lim∂D v = y (v tends to y ∈ Y in ∂D ∈ D∂ ) means that limDx→∂D v(x) = y in D∂ . In the usual interpretation of the boundary ∂D as the closed subset of the compact space Rm ∞, this is equivalent to the relations limDx →x v(x ) = y for all points x ∈ ∂D. For a measure μ and a function v near ∂D, we use the notation

∂D v dμ for v dμ with any Borel subset S  D, (2.1) D\S

when only the finiteness (convergence) of the  ∞ integral is important; this is an analog of the widely without indicating the lower limit of integration. used representation of improper integrals An analog of finite positive test functions in L. Schwartz’s theory of distributions, or generalized functions, in terms of which the order relation is defined on distributions or measures/charges is given in the framework of the present paper by the following definition. ∗

Here one can and possibly must use the notion of the set of germs of functions defined near ∂D inside D, i.e., of functions in a punctured neighborhood of the point ∂D ∈ D∂ with the corresponding equivalence relation “coincide near ∂D” and factorization with respect to this equivalence relation. 24

Definition 1. A subharmonic positive function v  0 near ∂D ∈ D∂ as lim∂D v = 0 is called a test function for D near ∂D ∈ D∂ , and the class of all such functions is denoted by sbh+ 0 (∂D). For b ∈ R+ and a subset S  D closed in Rm , i.e., for S = clos S  D, the subclasses of test functions ∞ 

+ (2.2b) sbh+ 0 (D \ S;  b) := v ∈ sbh (D \ S) : sup v  b, lim v = 0 ,

sbh+ 0 (D \ S; < +∞) :=

D\S

∂D

sbh+ 0 (D \ S;  b),

(2.2S)

b∈R+

which degenerate for S = ∅ into a one-point set {0}, generate sbh+ sbh+ 0 (∂D) = 0 (D \ S; < +∞).

(2.2∂)

SD

2.2. Basic results for the subharmonic majorant M . The results presented in this Section 2.2 are corollaries of the main theorem in §3 in the case of D ⊂ Cn∞ , n ∈ N. (2.1)  ∂D w dνM < Theorem 1 (individual). Let M ∈ sbh∗ (D) with Riesz measure νM , and let +∞ for an extended numerical function w defined near ∂D. Suppose that 0 = f ∈ Hol(D) and |f |  exp M on D ⊂ Cn∞ . Then, for any function v on any Borel set in Conc νM

(2.2∂)

∈ sbh+ 0 (∂D) satisfying the constraint v  w  ∂D 1.2.4 near ∂D in notation (2.1), the relation vZ dσ2n−2 < +∞ holds

1.2.4

for any subdivisor Z  Zerof . Everywhere below in Section 2.2, we assume that ∅ = int S ⊂ S = clos S  D. For w : D \ S → R+ +∞ ,

b := sup lim sup w(x ) < +∞, x∈∂S D\Sx →x

(2.3)

we define the greatest minorant with respect to the cone sbh0 (D \ S) := {v ∈ sbh(D \ S) : lim∂D v = 0} as a function gm w := sup{v ∈ sbh0 (D \ S) : v  w on D \ S}. At the same time, by gm∗ w we denote the upper semicontinuous regularization of the function gm w : D \ S → R+ +∞ . Theorem 2 (individual). Let M ∈ sbh(D) with Riesz measure νM , M ≡ −∞, and let w be an upper semicontinuous function in (2.3) satisfying one of the two conditions: (i) lim w = 0 Let also

∂D

 D\S

w dνM

or

(ii) D is a regular domain* (for the Dirichlet problem ). (2.4)  < +∞. If 0 = f ∈ Hol(D) and |f |  exp M on D, then Z\S (gm∗ w)Z dσ2n−2 < 1.2.4

+∞ for any subdivisor Z  Zerof . When the zero sets are studied in “rigid” weighted classes of holomorphic functions with more or less explicit majorants M , it is also necessary to have uniform upper bounds for integrals of functions of test character in a certain class over the zero set of a holomorphic function. This approach was realized most obviously for uniform Bergman and similar spaces in the unit disk ([17, Ch. 4], [18, SS5, 6]) and in the unit ball [3]. The following result gives similar estimates in the general situation. Theorem 3 (uniform). Let z0 ∈ int S , let M (z0 ) = −∞ for M ∈ sbh∗ (D) with Riesz + charge νM , and let b ∈ R+ . Then there exist constants C := const+ z0 ,S,b > 0 and C M := constz0 ,S,M such that the inequality

1.2.4 v Zerof dσ2n−2  v dνM − C log |f (z0 )| + CC M (2.5) D\S

D\S

*For simple and visual sufficient conditions for regularity of a domain of geometric character, see, e.g., [19, Theorem 2.11], and for m = 2, i.e., in C, see [20, 4.2]. 25

holds for any nonzero function f ∈ Hol(D) with the constraint |f |  exp M on D and for any test function v

(2.2b)

∈ sbh+ 0 (D \ S;  b). 3. Main Theorem

3.1. δ-Subharmonic functions and the main theorem. Following [24]–[25], for each m ∈ N we define the functions h1 (t) = t, h2 (t) = log |t|, and hm (t) = −|t|2−m for m  3. Along with the function −∞, we also consider the function +∞ identically equal to +∞. A function M : D → R±∞ in D ⊂ Rm ∞ is said to be trivial δ-subharmonic if M = −∞ or M = +∞ on D and nontrivial δ-subharmonic with Riesz charge νM ∈ M (D) if the following three conditions–conventions are satisfied. 1. There exist u1 , u2 ∈ sbh(D) \ {−∞} with Riesz measures νu1 , νu2 ∈ M + (D) for which / (−∞)u1 ∪ (−∞)u2 . For νM := νu1 − νu2 ∈ M (D) the M (x) := u1 (x) − u2 (x) ∈ R for x ∈ + − − νM is uniquely determined. Hahn–Jordan decomposition νM := νM 2. The determining set dom M ⊂ D is the set of points x ∈ D for each of which one of the following conditions holds for some rx > 0:     rx |νM |(x, t)   dt < +∞ or hm (|x − x|) d|νM |(x ) > −∞ . (3.1) tm−1 0 B(x,rx ) In the case of m = 1, we always have dom M = D; i.e., (3.1) holds for all x ∈ D. For m > 1, we complete the definition of the function M on all x ∈ dom M by averaging over the spheres ∂B(x, r),

1 (1.2) M (x + x ) dσm−1 (x ) ∈ R for x ∈ dom M , M (x) = lim r→0 sm−1 r m−1 ∂B(0,r) Such an extension of the definition always agrees with the preliminary values of the function M in the preceding item 1 on Rm ∞ \ ((−∞)u1 ∪ (−∞)u2 ) ⊂ dom M . In particular ([19]–[20]), the determining set dom u for u ∈ sbh∗ (D) is D \ (−∞)u . 3. M (x) = +∞ for ∗ x ∈ D \ dom M . Following [24, Theorem 11], to avoid explicit mentioning of subharmonic functions, we can replace item 1 by the following assertion. 1  . M is a function locally integrable with respect to the Lebesgue measure λm on D which has the following property: for any subdomain D   D, there exists a constant C  ∈ R+ such that, for any compactly supported twice continuously differentiable function f : D  → R with support   (1.2) supp f ⊂ D  , we have  D M Δf dλm   C  maxz∈D |f (z)|. At the same time, νM := s−1 m−1 ΔM in the sense of the theory of distributions. Under this approach, we accordingly replace item 2 by the following assertion. 2  . For the determining set dom M ⊂ D which, as above, is defined by the condition that integrals of the form (3.1) are finite, we set

1 (1.3) M (x + x ) dλm (x ) for all x ∈ dom M , M (x) = lim r→0 bm r m B(0,r) where the integral with the factor is the averaging over the ball B(x, r) and D \ dom M is the set of capacity zero [19, Ch. 5]. In particular, λm (D \ dom M ) = 0. We preserve the assertion in item 3 or assume that the function M is not defined in D \dom M . The class of functions δ-subharmonic in D is denoted by δ-sbh(D), and the subclass of nontrivial functions, by δ-sbh∗ (D) := δ-sbh(D) \ {±∞}. Main theorem. Let M ∈ δ-sbh∗ (D) be a function with Riesz charge νM and determining set dom M ⊂ D, and let ∅ = int S ⊂ S = clos S  D ⊂ Rm ∞ = D. Then, for any point x0 ∈ ∗

In [25, 2], the authors assumed that M (x) = 0 for x ∈ / dom M , but M (x) = +∞ is more preferable for our purposes, because the function M here plays the role of a majorant. 26

 ⊂ Rm with the Green function g  ( · , x0 ) int S ∩ dom M , any number b > 0, any regular domain D ∞ D   with a pole at x0 which satisfies the conditions S  D ⊂ D and Rm ∞ \ clos D = ∅, any function (2.2b)

u ∈ sbh∗ (D) satisfying the inequality u  M on D, and any test function v ∈ sbh+ 0 (D \ S;  b), the following inequality holds:

− v dνu  v dνM + v dνM + CC M , (3.2) Cu(x0 ) + D\S

 D\S

D\S

where b > 0, (3.3) inf z∈∂S gD (x, x0 )   − := D\{x g  ( · , x0 ) dνM + D\S gD ( · , x0 ) dνM + M + (x0 ),   0} D

C := const+

 x0 ,S,D,b

:=

and the value +∞ is possible for C M   D, this is a certain constant C M := const+ but for D

 x0 ,S,D,M

< +∞.

3.2. Jensen measures and potentials. m Definition 2 ([4], [11], [12], [14]–[16], [26], [27]). A measure μ ∈ M +  (R∞ ) is called a Jensen + measure inside a domain D at a point x0 ∈ D if μ ∈ Mc (D) and u(x0 )  u dμ for all u ∈ sbh(D). The class of all such Jensen measures is denoted by Jx0 (D).

Obviously, each measure μ ∈ Jx0 (D) is a probability measure; i.e., μ(D) = 1. Further, without mentioning, we use the fact that μ(E \ {x0 }) = 0 for any set of capacity zero E ⊂ D, in particular, for E = (−∞)u , u ∈ sbh∗ (D), and for the measure μ ∈ Jx0 (D) [26, Corollary 1.8].  = ∅ be a subdomain in D with nonpolar boundary, and let Example 3.1. Let m  2, let D m  = ∅. The harmonic measure ω  (x0 , · ) for (or with respect to) D  at a point x0 ∈ D  R∞ \ clos D D  (see [20, 4.3], [19, 3.6, 5.7.4]) under the condition that D  D is an example of the Jensen measure in Jx0 (D). The degenerate case is the Dirac measure δx0 .  = (a, b)  R at a point x0 ∈ (a, b) is the For m = 1, the harmonic measure for the interval D x0 −a b−x0  = (a, +∞) ⊂ D ⊂ R measure ω(a,b) (x0 , · ) = b−a δa + b−a δb . For domains-intervals of the form D  = (−∞, a) ⊂ D ⊂ R with a ∈ R, there exists no harmonic measure at any point x0 ∈ D. and D Definition 3 ([4], [11], [12], [14]–[16], [27]). A function V ∈ sbh+ (Rm ∞ \ {x0 }) for x0 ∈ D is called a Jensen potential inside D with a pole at a point x0 if the following two conditions are satisfied: (1) There exists a domain DV  D containing the point x0 for which V (x) ≡ 0 for all = 0 (finiteness in D). x ∈ Rm ∞ \ DV , i.e., V |Rm ∞ \DV (2) A seminormalization at the point x0 is prescribed; namely, for m = 1 and x0 ∈ R,

(3.4a)

for m = 2,

(3.4b)

lim sup |x − x0 |m−2 V (x)  1

for m  3 and x0 = ∞,

(3.4c)

lim sup V (x)  1

for m  3 and x0 = ∞.

(3.4d)

lim sup V (x) < +∞ x→x0

lim sup x→x0

V (x) 1 − log |x − x0 |

x→x0

x→∞

We denote the class of all such Jensen potentials by P Jx0 (D). Definitions 1 and 3 imply the following obvious assertion. Proposition 3.1. Each Jensen potential in P Jx0 (D) is a test function for D near ∂D ∈ D∂ , (2.2b)

(2.2S)

+ and for S  D and int S = ∅, the inclusions P Jx0 (D) ⊂ sbh+ 0 (D\S;  sup∂S V ) ⊂ sbh0 (D\S; < +∞) hold for any point x0 ∈ int S .

27

Example 3.2. Under the conditions of Example 3.1, the Green function gD ( · , x0 ) for (or with  continued to Rm  with a pole at a point x0 ∈ D respect to) D ∞ according to the rule (see [19, 3.7, 5.7], [20, 4.4])    lim supDx   →x gD  (x , x0 ) for x ∈ ∂ D, (3.5) gD (x, x0 ) :=  0 for x ∈ Rm ∞ \ clos D   D. The degenerate version is a is an example of a Jensen potential in P Jx0 (D) provided that D m function identically zero on R∞ \ {x0 }. Everywhere below, to avoid technically cumbersome consideration of different cases, we do not consider the Jensen potentials and Green functions with a pole at the point ∞. Definition 4. The logarithmic potential of kind 0 of a measure μ ∈ Mc+ (Rm ∞ ) with a pole at \ {x } as the function [27, Definition 3] x0 ∈ Rm is defined for all y ∈ Rm 0 ∞

(hm (|y − x|) − hm (|y − x0 |)) dμ(x), (3.6) Vμ (y) := D

where the integrand is supplemented with the value 0 for y = ∞. In what follows, we only discuss the dimension m  2 in detail, because the one-dimensional situation m = 1 is rather specific and, in our opinion, requires a separate study. At the same time, the case of m = 1 also belongs in our general scheme, and the main theorem and its corollaries in Section 2.2 also hold for m = 1. We point out the main interconnections between the Jensen measures and potentials for m  2. The first is the following assertion about duality. Proposition 3.2 [27, Proposition 1.4, duality theorem]. The mapping (3.6)

P : Jx0 (D) → P Jz0 (D),

P(μ) := Vμ ,

μ ∈ Jx0 (D),

is a bijection, P(tμ1 + (1 − t)μ2 ) = tP(μ1 ) + (1 − t)P(μ2 ) for all t ∈ [0, 1], and     1 V (x) −1 δx0 , ΔV  + 1 − lim sup V ∈ P Jx0 (D). P (V ) = sm−1 x→x0 −hm (|x − x0 |) D\{x0 }

(3.7)

(3.8)

 this is the classical relation  ⊂ D and for x0 ∈ D, In particular, for a regular domain D P(ωD (x0 , · )) = gD ( · , x0 ),

 ⊂ D. x0 ∈ D

The second is a generalization of the classical Poisson–Jensen formula [19, Theorem 3.14]. Proposition 3.3 [27, Proposition 1.2]. Let μ ∈ Jx0 (D). Then the extended Poisson–Jensen formula

Vμ dνu = u dμ (3.9) u(x0 ) + D\{x0 }

D

holds for u ∈ sbh(D) with Riesz measure νu and for u(x0 ) = −∞. In particular, for a bounded (3.6)

domain D in Rm and for μ = ωD (x0 , · ) and, as a consequence, with Vμ = gD ( · , x0 ), this is a generalization of the Poisson–Jensen formula [19, Theorem 5.27]. 3.2.1. Continuation of test functions to D \ {x0 }. Proposition 3.4. For ∅ = int S ⊂ S = clos S  D, for any points x0 ∈ int S , a number  ⊂ Rm such that S  D  ⊂ D and Rm \ clos D  = ∅, and a number b ∈ R+ , a regular domain D ∞ ∞ c := const+ 

 x0 ,S,D,b

28

=

1 inf g  (x, x0 ) > 0, b x∈∂S D

(3.10)

(2.2)

any test function v ∈ sbh+ 0 (D\S;  b) with Riesz measure μv allows one to construct the following function by using the continued Green function gD ( · , x0 ): ⎧ on S \ {x0 }, ⎪ ⎪gD ( · , x0 ) ⎪ ⎨max{g ( · , x ),   \ S, c · v} on D 0  D V := V ∈ sbh+ (Rm (3.11) ∞ \ {x0 }).  ⎪ c  · v on D \ D, ⎪ ⎪ ⎩ 0 on Rm ∞ \ D, This function has the properties V |int S\{x0 } ∈ har(int S \ {x0 }),

lim V =0,

lim

x→x0

∂D

V (x) −hm (|x − x0 |)

(3.4b)–(3.4d)

=

1.

(3.12)

(2.2)

Proof. We can assume that the function v ∈ sbh+ 0 (D \ S;  b) is continued by zero to D, and we denote the continuation again by v. Obviously, the continued function v is positive subharmonic everywhere on Rm ∞ \S and v = 0 on D. It follows from the constraint supz∈D\S v  b in Definition (2.2b) that supx∈∂S lim supD\Sx →x v(x )  b. Then lim sup

 (Rm ∞ \S)x →x

v(x )  b

for all x ∈ ∂(Rm ∞ \ S).

(3.13)

 It follows from Consider the Green function gD ( · , x0 ) continued as in (3.5) for the domain D. the properties of the Green function ([19, 3.7], [20, Theorem 4.4.3]) that it is harmonic and  \ {x0 }, and therefore, a := inf x∈∂(Rm \S) g  (x, x0 ) > 0. For the function strictly positive on D D ∞ v0 := (b/a)gD ( · , x0 ) ∈ sbh+ (Rm ∞ \ {x0 }), according to the properties of the Green function ([20, Theorem 4.4.9], [19, 1.5.1]), we have lim

Dx→x0

b v0 (x) = , −hm (|x − x0 |) a

 v0 = 0 on D \ D,

 \ {x0 }). v0 |D\{x ∈ har(D  0}

(3.14)

Moreover, in view of (3.13)–(3.14) and by the construction of the function v0 , we have (3.14)

(3.13)

 b  v0 |∂(Rm ∞ \S)

sup

lim sup

m  x∈∂(Rm ∞ \S) (R∞ \S)x →x

v(x ),

v0 |D = 0  v|D .

(3.15)

Now we need the following assertion. Gluing theorem (see [22, Corollary 2.4.4]). Let O and O0 be open sets in Rm ∞ , and let O ⊂ O0 . Let v ∈ sbh(O) and v0 ∈ sbh(O0 ). If lim sup v(x )  v0 (x) Ox →x

then the function

for all points x ∈ O0 ∩ ∂O ,

 max{v, v0 } v := v0

on O, on O0 \ O

(3.16)

(3.17)

is subharmonic on O ; i.e.,  v ∈ sbh(O0 ). m We apply the gluing theorem for O0 := Rm ∞ \ {x0 } and O := R∞ \ S to the function v0 and the function v continued to O . It follows from the first relation in (3.15) that these functions satisfy condition (3.16). We use (3.14)–(3.15) to describe the function v ∈ sbh+ (Rm ∞ \ {x0 }) constructed in (3.17) in terms of the corresponding restrictions in more detail, ⎧ on S \ {z0 }, ⎪ ⎪ v0 ⎪ ⎨max{v, v } on D  \ S, 0 v ∈ sbh+ (Rm (3.18) 0 v= ∞ \ {x0 }).  ⎪ v on D \ D, ⎪ ⎪ ⎩ 0 on Rm ∞ \ D, 29

At the same time, the following normalization holds: lim

 Dx→x 0

v(x) (3.14) b = ; −hm (|x − x0 |) a

(3.14)

v|int S\{x0 } ∈ har(int S \ {x0 }).

(3.19)

(3.18)

In view of (3.19), the function V := (a/b) v ∈ sbh+ (Rm ∞ \ {x0 }) satisfies the normalization condition, i.e., the last relation in (3.12). We set  c :=

1 a = inf g  (x, x0 ) b b x∈∂S D

as in (3.10). Then, by multiplying (3.18) by  c , we obtain precisely (3.11). Finally, all above-listed properties in (3.12) are direct consequences of the construction (3.18)–(3.19) and well-known properties of the Green function gD ( · , z0 ) participating in this construction, starting from the construction of the function v0 = (b/a)gD ( · , x0 ). 3.2.2. Continued function V , i.e., the limit of Jensen potentials. Proposition 3.5. Let V be the function in (3.11) constructed in Section 3.2.1. Then Vn := (V − 1/n)+ := max{0, V − 1/n} ∈ sbh+ (Rm ∞ \ {x0 }),

n ∈ N,

(3.20)

are the Jensen potentials Vn ∈ P Jx0 (D) for which limn→+∞ Vn = V pointwise and Vn  Vn+1 on D \ {x0 } for all n ∈ N. Moreover, for a number r0 > 0 such that B∗ (x0 , r0 ) := B(x0 , r0 ) \ {x0 }  S and for a number n0 ∈ N, we have Vn |B∗ (x0 ,r0 ) ∈ har(B∗ (x0 , r0 )) lim

(D\{x0 })x→x0

for all n  n0 ; (3.21)

Vn (x) (3.4) = 1. −hm (|x − x0 |)

Proof. For any n ∈ N, for each function Vn in (3.20), its subharmonicity and positiveness in Rm ∞ \ {x0 } is a consequence of its Definition (3.11), the normalization relation in (3.21) follows from the last relation in (3.12), the harmonicity in B∗ (x0 , r0 ) in (3.21) follows from the first relation in (3.12), and the finiteness in D of the Jensen potentials, which is required in Definition 3, follows from the construction (3.20) and the property lim∂D V = 0 in (3.12). By Definition 3, the functions Vn ∈ P Jx0 (D) are the Jensen potentials inside D with a pole at x0 ∈ D. By the construction (3.20), the sequence (Vn )n∈N is increasing and pointwise tends to V . 3.3. Proof of the main theorem. In this proof, we use the sequence of Jensen potentials (Vn )n∈N with the properties (3.21), which was constructed in Section 3.2.1 and increases to the function V in (3.11). By Proposition 3.2 with the mapping P in (3.7), each Jensen potential Vn is associated with the Jensen measure μn inside the domain D at a point x0 ∈ D \ dom M , μn := P −1 (Vn )

(3.8), (3.21)

=

1 sm−1

ΔVn ∈ P Jx0 (D),

(3.21)

supp μn ⊂ D \ B(x0 , r0 ),

(3.22)

where r0 > 0 and B(x0 , r0 ) ⊂ int S . Let M = u1 − u2 , where u1 and u2 are functions in sbh(D) + − , νM ∈ M + (D). In view of x0 ∈ dom M , by the definition of with respective Riesz measures νM δ-subharmonic function, we have u1 (x0 ) = −∞ and u2 (x0 ) = −∞. At the same time, if u(x0 ) = −∞ or the conditions

− − v dνM < +∞, gD ( · , x0 ) dνM < +∞ (3.23) D\S

 D\S

are not satisfied, then inequality (3.2) is trivial. Therefore, we can assume that u(x0 ) = −∞ and relations (3.23) are satisfied simultaneously. According to the extended Poisson–Jensen formula (3.9) 30

in Proposition 3.3 applied to subharmonic functions u, u1 , and u2 we obtain

(3.9), (3.22) Vn dνu = u dμn , u(x0 ) + D\{x0 } D\B(x0 ,r0 )

+ (3.9), (3.22) Vn dνM = u1 dμn , u1 (x0 ) + D\{x0 } D\B(x0 ,r0 )

− (3.9), (3.22) u2 (x0 ) + Vn dνM = u2 dμn . D\{x0 }

(3.24u) (3.24u1 ) (3.24u2 )

D\B(x0 ,r0 )

From the condition u  M = u1 − u2 on D for the right-hand sides of (3.24), we obtain



u dμn  M dμn = u1 dμn − u2 dμn . D\B(x0 ,r0 )

D\B(x0 ,r0 )

D\B(x0 ,r0 )

This and three relations (3.24) imply that

Vn dνu + u(x0 ) + D\{x0 }

D\{x0 }

− Vn dνM  M (x0 ) +

 M (x0 ) +

D\B(x0 ,r0 )

D\{x0 }

D\{x0 }

+ Vn dνM + , V dνM

(3.25)

because the sequence of functions Vn increases and tends to V on D pointwise. For the integral on the right-hand side, the value +∞ is admissible. If this integral is indeed equal to +∞, then, since   + (3.11) + = D\D  c v dνM . the integrals (3.1) are finite for x = x0 ∈ dom M , we obtain +∞ = D\D V dνM Since the first integral in (3.23) is finite, this implies that the first integral on the right-hand side in formulas (3.2) is also +∞, and we have nothing to prove. Therefore, we further assume that the integral on the right-hand side in formula (3.25) is finite. Applying the theorem of monotone convergence of integrals to the left-hand side of (3.25), we obtain

− +   . (3.26) V dνu + V dνM  M (x0 ) + V dνM u(x0 ) + D\{x0 }

Here, by (3.11),

D\{x0 }

D\{x0 }

⎧ ⎪ on S \ {x0 }, ⎨gD ( · , x0 )   \ S, V = max{gD ( · , x0 ),  cv} on D ⎪ ⎩  cv  on D \ D.

Applying these relations to (3.26), we obtain

gD ( · , x0 ) dνu + cv dνu  u(z0 ) + S\{x0 } D\S

 M (x0 ) + gD ( · , x0 ) dνM + S\{x0 }

 D\S

max{gD ( · , x0 ),  cv} dνM +

 D\D

 cv dνM . (3.27)

The second positive integral term on the left-hand side in (3.27) can be removed. Dividing inequality (3.27) by  c > 0 in (3.10) implies the inequality

v dνu Cu(x0 ) + D\S

+  v dνM + C gD ( · , x0 ) dνM + max{CgD ( · , x0 ), v} dνM + CM (x0 )  D\D

S\{x0 }

 D\S

31

(3.10)

for C := 1/ c = b/ inf x∈∂S gD (x, x0 ) > 0 of the form (3.3). From this, using the inequality max{CgD ( · , x0 ), v}  CgD ( · , x0 ) + v on D \ S , which is obvious for positive functions, we obtain

v dνu Cu(x0 ) + D\S

 v dνM + D\S

 D\S

− v dνM

+C

gD ( · , z0 ) dνM  D\{x 0}

+C

− gD ( · , z0 ) dνM  D\S

+ CM (x0 ),

  D, both the second integral on the and this proves the desired inequality (3.2). Finally, for D  \ S and the integrals right-hand side in (3.2) owing to the boundedness of the function v on D participating in the definition of C M are finite for x0 ∈ dom M in view of (3.1) with x = x0 . The proof of the main theorem is complete. 3.4. Subharmonic majorant M . Everywhere below, D ⊂ Cn∞ and z0 = x0 ∈ int S  D. Proof of Theorem 3. Obviously, we can assume that f (z0 ) = 0 and the integral on the righthand side in formula (2.5) is finite. According to the main theorem, for u := log |f | ∈ sbh∗ (D), in − = 0 and, on the right-hand side of inequality the special case of M ∈ sbh∗ (D), it is obvious that νM − are zero. We (3.2) just as in the expression for C M , the integrals with respect to the measure νM   can always choose a regular domain D  D described in [19, 1.4.3, Definition] for which S  D. (2.2)

Thus, for any test function v ∈ sbh+ 0 (D \ S;  b), inequality (3.2) implies that

v dνu  v dνM − Cu(z0 ) + C gD ( · , z0 ) dνM + CM + (z0 ). D\S

D\S

 D

(3.28)

 is completely specified only by the mutual location of the domain The choice of the domain D S  D and the domain D; i.e., when choosing the constant C in (3.3), we can replace the influence   by the dependence on S and D. Likewise, the constant C M (3.2) :=  g  ( · , z0 ) dνM + of the domain D D D

M + (z0 ) depends only on z0 , S , D, and M , as desired. According to the Poincar´e–Lelong formula in Section 1.2.4, the left-hand side of (3.28) can be replaced by the left-hand side of inequality (2.5). The proof of Theorem 3 is complete. Proof of Theorem 2. It follows from Theorem 3 that it suffices to prove the following lemma. Lemma 3.1. For the upper semicontinuous function w in (2.3), for any of the two conditions (2.4)(i) or (2.4)(ii), the upper semicontinuous regularization gm∗ w of its greatest minorant gm w with respect to sbh0 (D \ S) is a test function in sbh+ 0 (D \ S; < +∞) satisfying the constraint ∗ gm w  w on D \ S .

Proof of Lemma 3.1. Even without the condition (2.4), by construction, we have gm∗ w ∈ sbh+ (D\S) and gm∗ w  w on D\S , because the function w is upper semicontinuous. Then, under (2.2S)

condition (2.4)(i), we have 0  lim∂D gm∗ w  lim∂D w = 0 and gm∗ w ∈ sbh+ 0 (D \ S; < +∞), as desired. In the case of (2.4)(ii), we assume that z0 ∈ int S and B(z0 , r0 ) ⊂ S for some r0 > 0. We  is played by the use the Green function gD ( · , z0 ) in (3.5), continued to D, where the role of D domain D. It follows from the regularity of the domain D that gD ( · , z0 ) = 0 on ∂D, and since (2.3)

B(z0 , r0 ) ⊂ S  D, it follows that there exists a C ∈ R+ such that CgD ( · , z0 )  b on ∂S . At the same time, CgD ( · , z0 )|D\{z0 } ∈ har(D \ {z0 }) and CgD ( · , z0 ) ∈ sbh(Cn∞ \ {z0 }). In particular, the continued Green function gD ( · , z0 ) is upper semicontinuous on Cn∞ \ {z0 }. For any function v ∈ sbh0 (D \ S) satisfying the constraint v  w on D \ S , it follows from the maximum principle that CgD ( · , x0 )v on D \ S , because the left-hand side is harmonic on D \ {z0 }. This and the fact that the function CgD ( · , z0 ) is upper semicontinuous on Cn∞ \{z0 } imply that gm∗ w  CgD ( · , z0 ) 32

on D \ S , and this implies that lim∂D gm∗ w = 0 and gm∗ w Lemma 3.1 is complete.

(2.2b)

∈ sbh+ 0 (D \ S;  b). The proof of

(2.2∂)

Proof of Theorem 1. Let v ∈ sbh+ 0 (∂D), and let v  w on a Borel set B in Conc νM . Then, by Definition 1 of test functions, we can choose a point z0 ∈ D for which M (z0 ) = −∞ and f (z0 ) = 0 and a closed subset S  D with int S z0 so that the function v is subharmonic on (2.2b)

+ D \ int  S and v ∈ sbh0 (D \ S;  b) for b = sup∂S v < +∞, while BS := B ∩ (D \ S) ∈ B(D \ S) and BS wdνM < +∞. According to the main theorem with u := log |f |, inequality (3.2) with a loss of part of information (just as in the proof of Theorem 3) can be written for a constant C0 ∈ R+ (3.28)     in the form D\S v dνu  D\S v dνM + C0 = BS v dνM + C0  BS w dνM + C0 < +∞. By the Poincar´e–Lelong formula in Section 1.2.4, the left-hand side of notation (2.1) can here be replaced (2.1)  ∂D vZ dσ2n−2 < +∞ for any subdivisor Z  Zerof . The proof of Theorem 1 is by the integral complete.

Concluding remarks. 1. Various methods for constructing specific test functions in Definition 1 and their subclasses (radial for the disk, plane, and annulus, and those which can be expressed in terms of Green functions, a function of the distance to a subset on ∂D, a hyperbolic or conformal radius of the domain, etc.) are given in [14] in the case of a single variable. In [28], the initial constructions are outlined for test functions (radial for the ball and those which can be expressed in terms of the Green function) of several variables. 2. The “plurisubharmonic” transition to pseudoconvex domains in Cn for the plurisubharmonic majorant M with a significant extension of the class of test functions for n > 1 is somewhat more complicated, but is still promising on the basis of the complex theory of potentials with the use of the technique of analytic disks ([22], [4], and [12]) and the theory of holomorphic flows and disk envelopes [29]. References [1] Ph. Griffiths and J. King, “Nevanlinna theory and holomorphic mappings between algebraic varieties,” Acta Math., 130 (1973), 145–220. [2] Sh. A. Dautov and G. M. Khenkin, “Zeros of holomorphic functions of finite order and weighted ¯ estimates for solutions of the ∂-equations,” Mat. Sb., 107(149):2(10) (1978), 163–174; English transl.: USSR Sb. Math., 35:4 (1979), 449–459. [3] J. Bruna and X. Massaneda, “Zero sets of holomorphic functions in the unit ball with slow growth,” J. Anal. Math., 66 (1995), 217–252. [4] B. N. Khabibullin, “Dual representation of superlinear functionals and its application in function theory. II,” Izv. Ross. Akad. Nauk Ser. Mat., 65:5 (2001), 167–190; English transl.: Russian Acad. Sci. Izv. Math., 65:5 (2001), 1017–1039. [5] G. M. Khenkin, “The method of integral representations in complex analysis,” in: Complex analysis — several variables — 1, Itogi Nauki i Tekhniki. Sovrem. Probl. Mat. Fund. Napr., vol. 7, VINITI, Moscow, 1985, 23–124. [6] S. V. Shvedenko, “Hardy classes and related spaces of analytic functions in the unit disc, polydisc, and ball,” in: Itogi Nauki i Tekhniki. Mat. Analiz, vol. 23, VINITI, Moscow, 1985, 3–124. [7] L. I. Ronkin, Elements of the Theory of Analytic Functions of Several Variables [in Russian], Naukova dumka, Kiev, 1977. [8] L. I. Ronkin, “Entire functions,” in: Complex analysis—several variables—3, Itogi Nauki i Tekhniki. Sovrem. Probl. Mat. Fund. Napr., vol. 9, VINITI, Moscow, 1986, 5–36. [9] P. Lelong and L. Gruman, Entire Functions of Several Comples Variables, Springer-Verlag, Berlin–Heidelberg, 1986. [10] L. I. Ronkin, Functions of Completely Regular Growth, Mathematics and Its Applications (Soviet Series), Kluver Academic Publishers Group, Dordrecht–Boston–London, 1992. 33

[11] B. N. Khabibullin, “Completeness of systems of entire functions in spaces of holomorphic functions,” Mat. Zametki, 66:4 (1999), 603–616; English transl.: Math. Notes, 66:4 (1999), 495–506. [12] B. N. Khabibullin, Completeness of systems of exponentials and sets of uniqueness, 4th ed., completed, RITs BashGU, Ufa, 2012. [13] S. Yu. Favorov and L. D. Radchenko, “The Riesz measure of functions subharmonic in the exterior of a compact set,” Matematichni Studii, 40:2 (2013), 149–158. [14] B. N. Khabibullin and N. R. Tamindarova, “Distribution of zeros and masses for holomorphic and subharmonic functions. I, II. Hadamard- and Blaschke-type conditions,” Mat. Sb, 2018 (to appear); http://arxiv.org/abs/1512.04610v2. [15] B. Khabibullin and N. Tamindarova, “Distribution of zeros for holomorphic functions: Hadamard- and Blaschke-type conditions,” in: Abstracts of International Workshop on “Nonharmonic Analysis and Differential Operators” (May 25-27, 2016), Institute of Mathematics and Mechanics of Azerbaijan National Academy of Sciences, Azerbaijan, Baku, 2016, 63. [16] B. Khabibullin and N. Tamindarova, “Uniqueness theorems for subharmonic and holomorphic functions of several variables on a domain,” Azerb. J. Math., 7:1 (2017), 70–79. [17] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. [18] E. G. Kudasheva and B. N. Khabibullin, “The distribution of zeros of holomorphic functions of moderate growth in the unit disc and the representation of meromorphic functions there,” Mat. Sb, 200:9 (2009), 95–126; English transl.: Russian Acad. Sci. Sb. Math., 200:9 (2009), 1353–1382. [19] W. Hayman and P. Kennedy, Subharmonic Functions, Academic Press, London–New York, 1976. [20] Th. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995. [21] E. M. Chirka, Complex Analytic Sets, Kluwer Academic Publishers, Dordrecht, 1985. [22] M. Klimek, Pluripotential Theory, Clarendon Press, Oxford University Press, New York, 1991. [23] P. Lelong, “Propri´et´es m´etriques des vari´et´es analytiques complexes d´efinies par une ´equation,” Ann. Sci. Ecole Norm. Sup., 67 (1950), 393–419. [24] M. G. Arsove, “Functions representable as differences of subharmonic functions,” Trans. Amer. Math. Soc., 75 (1953), 327–365. [25] A. F. Grishin, Nguen Van Quynh, and I. V. Poedintseva, “Representation theorems of δ-subharmonic functions [in Russian],” Vestnik Kharkov Nat. Univ. Ser. Mat., Prikl. Math. i Mekh., 1133:70 (2014), 56–75; http://vestnik-math.univer.kharkov.ua/ Vestnik-KhNU-1133-2014-grish.pdf. [26] B. J. Cole and T. J. Ransford, “Subharmonicity without upper semicontinuity,” J. Func. Anal., 147 (1997), 420–442. [27] B. N. Khabibullin, “Criteria for (sub)harmonicity and continuation of (sub)harmonic functions,” Sibirsk. Mat. Zh., 44:4 (2003), 905–925; English transl.: Siberian. Math. J., 44:4 (2003), 713–728. [28] B. N. Khabibullin and N. R. Tamindarova, “Subharmonic test functions and the distribution of zero sets of holomorphic functions,” Lobachevskii J. Math., 38:1 (2017), 38–43; http://arxiv.org/abs/1606.06714v1. [29] E. A. Poletsky, “Disk envelopes of functions II,” J. Funct. Anal., 163:1 (1999), 111–132. Bashkir State University, Ufa, Russia e-mail: [email protected] Bashkir State University, Ufa, Russia e-mail: [email protected]

Translated by M. A. Shishkova

34