Balayage of Measures on the Complex Plane with respect to Harmonic Polynomials and Logarithmic Kernels
aa r X i v : . [ m a t h . C V ] A ug Balayage of Measures on the Complex Plane with respect toHarmonic Polynomials and Logarithmic Kernels
B. N. Khabibullin * and E. B. Menshikova ** Bashkir State University, Bashkortostan, Russian Federation
Received August 03, 2020
Keywords and phrases: balayage of measure, polynomial, logarithmic potential, subharmonicfunction, Riesz measure, Weierstrass – Hadamard representation, polar set
1. BALAYAGE OF MEASURESAs usual, N := { , , . . . } is set of all natural numbers. We denote singleton sets by a symbolwithout curly brackets. So, N := 0 ∪ N := { } ∪ N . R is the real line, or the real axis of the complexplane C , with the standard Euclidean norm-module | · | and order ≤ , R + := { r ∈ R : 0 ≤ r } is the positive closed semiaxis, and R + \ is the positive open semiaxis. The extended real line R := R ∪ ±∞ is the order completion of R by the union with + ∞ := sup R and −∞ := inf R , and inf ∅ := + ∞ , sup ∅ := −∞ for the empty set ∅ etc. Besides, R + := R + ∪ + ∞ , x · ( ±∞ ) := ±∞ =: ( − x ) · ( ∓∞ ) for x ∈ R + \ , but · ( ±∞ ) := 0 unless otherwise specified.Denote by C ∞ := C ∪ {∞} the Alexandroff one-point compactification of the complex plane C with the standard Euclidean norm-module | · | , but |∞| := + ∞ . For S ⊂ C ∞ we let ∁ S := C ∞ \ S , clos S , int S := ∁ (clos ∁ S ) , and ∂S := clos S \ int S denote its complement, closure, interior, and boundary always in C ∞ , and S is equipped with the topology induced from C ∞ . If clos S ′ ⊂ S , thenwe write S ′ ⋐ S .Let X, Y are sets. We denote by Y X the set of all functions f : X → Y .Let Bor ( S ) be the class of all Borel subsets in S ⊂ C ∞ . We denote by Meas ( S ) the class ofall Borel signed measures, or, charges on a set S ∈ Bor ( C ∞ ) , and Meas cmp ( S ) is the class ofcharges µ ∈ Meas ( S ) with a compact support supp µ ⋐ S ; Meas + ( S ) := { µ ∈ Meas ( S ) : µ ≥ } isthe subclass of all positive measures ; Meas +cmp ( S ) := Meas cmp ( S ) ∩ Meas + ( S ) ; Meas ( S ) := { µ ∈ Meas + ( S ) : µ ( S ) = 1 } is the class of probability measures on Bor ( S ) . Besides, Meas := Meas ( C ) , Meas cmp := Meas cmp ( C ) , Meas + := Meas ( C ) , etc. For S ∈ Bor ( C ) and a charge ν ∈ Meas ( S ) , we let ν + ∈ Meas + ( S ) , ν − := ( − ν ) + ∈ Meas + ( S ) , | ν | := ν + + ν − ∈ Meas + ( S ) respectively denote upper,lower, and total variations of ν . * E-mail: [email protected] ** E-mail: [email protected]
B. N. KHABIBULLIN, E. B. MENSHIKOVA
Definition 1 (see [18], [16], [6], [11], cf. [14, Definition 2]) . For δ, ω ∈ Meas + and H ⊂ R C , letus assume that the integrals R h d δ and R h d ω are well defined with values in R for each function h ∈ H . We say that the measure ω is a balayage, or, sweeping (out), of the measure δ with respectto H , or, briefly, ω is a H -balayage of δ , if Z h d δ ≤ Z h d ω ∈ R for each h ∈ H . (1) Obviously, (1) implies the equality Z h d δ = Z h d ω ∈ R if ± h ∈ H and Z h d δ = −∞ . (2)For S ⊂ C , by har ( S ) and sbh ( S ) we denote the classes of all harmonic and subharmonic functionson open neighborhoods of S , respectively; sbh ∗ ( S ) := { u ∈ sbh ( S ) : u
6≡ −∞} . Widely used variantsof classes H ⊂ R C from Definition 1 are har := har ( C ) , sbh := sbh ( C ) , sbh ∗ := sbh ∗ ( C ) [6], [16]. Foran open set O ⊂ C , the Riesz measure of u ∈ sbh ∗ ( O ) is a positive Borel measure ∆ u := 12 π △ u ∈ Meas + ( O ) , (3)where △ is the Laplace operator acting in the sense of the theory of distributions or generalizedfunctions.For p ∈ R and Z := ( − N ) ∪ N , we set ⌊ p ⌋ :=: floor ( p ) := sup { k ∈ Z : k ≤ p } ∈ Z ∪ ±∞ (4f)(the integer part of p if p ∈ R ), and ⌈ p ⌉ :=: ceil ( p ) := inf { k ∈ Z : k ≥ p } ∈ Z ∪ ±∞ . (4c)Let p ∈ R + , let Mon p := n z z ∈ C cz k : p ≥ k ∈ N , c ∈ C o (4f) = Mon ⌊ p ⌋ ⊂ C C (5)be the class of complex monomials of degree at most p , and let Pol p := ⌊ p ⌋ X k =0 Mon k (6)be the class of all complex polynomials P ∈ z ∈ C C [ z ] of degree deg P ≤ p .In this article, we often consider the following two classes as a class H ⊂ R C from Definition 1.The first class mon p := Re Mon p = Im Mon p = mon ⌊ p ⌋ ⊂ har , mon ∞ := mon + ∞ , (7)consists of harmonic real homogeneous polynomials of degree ≤ p . The real linear span of of thisclass mon p coincides with the real space of all harmonic polynomials of degree at most p .The second class is the union lnmon p := ln C ∪ mon p ⊂ sbh ∗ , where ln C := n z z ∈ C ln | z − w | : w ∈ C o (8) LOBACHEVSKII JOURNAL OF MATHEMATICS
ALAYAGE OF MEASURES ON THE COMPLEX PLANE . . . 3 is the class of logarithmic functions generated by the logarithmic kernel ln | · − · | : ( z, w ) ln | w − z | if z = w, −∞ if z = w, ( z, w ) ∈ C . We denote by D ( z, t ) := (cid:8) z ′ ∈ C : | z ′ − z | < t (cid:9) , D ( z, t ) := (cid:8) z ′ ∈ C : | z ′ − z | ≤ t (cid:9) , ∂D ( z, t ) := D ( z, t ) \ D ( z, t ) an open disk, a closed disk, a circle of radius t ∈ R + centered at z ∈ C , respectively; D ( t ) := D (0 , t ) , D ( t ) := D (0 , t ) , ∂D ( t ) := ∂D (0 , t ) , and D := D (1) , D := D (1) , ∂ D = ∂ D := ∂D (1) are the open unit disk, the closed unit disk, the unit circle, respectively.For ν ∈ Meas , z ∈ C , and ≤ r < R ∈ R + , we set ν ( z, r ) := ν (cid:0) D ( z, r ) (cid:1) , ν rad ( r ) := ν (0 , r ) , (9 ν ) N ν ( z, r, R ) := Z Rr ν ( z, t ) t d t, N rad ν ( r, R ) := N ν (0 , r, R ) , (9N)and also, for a function t t ∈ [ r, R ] f ( t ) ∈ R , provided ν ( C ) = ν rad (+ ∞ ) ∈ R , N ⋆ν (cid:0) r, R ; t f ( t ) (cid:1) := Z Rr ν rad (+ ∞ ) − ν rad ( t ) t f ( t ) d t. (9 ⋆ )We list some elementary properties of balayage with respect to certain classes from (7) and (8)and the classes har and sbh . Proposition 1.
Let p ∈ R + , δ ∈ Meas + , ω ∈ Meas + , and < δ ( C ) = δ rad (+ ∞ ) < + ∞ , < ω ( C ) = ω rad (+ ∞ ) < + ∞ . (10)[b1] A measure ω ∈ Meas + is a mon p -balayage of δ ∈ Meas + for p < if and only if ω ( C ) = δ ( C ) . [b2] The following tree statements are equivalent: (i) ω is a mon p -balayage of δ ; (ii) R C z k d δ ( z ) = k ∈ N R C z k d ω ( z ) for each k ≤ p ; (iii) equality (2) is fulfilled for each polynomial h (6) ∈ Pol p . [b3] The integral R h d ω ∈ R ∪ −∞ are well defined for each function h (8) ∈ lnmon p under thecondition N ⋆ν (cid:0) , + ∞ ; t t ⌊ p ⌋ (cid:1) (9 ⋆ ) < + ∞ .Suppose that, in addition to (10) , δ ∈ Meas +cmp and ω ∈ Meas +cmp . [b4] If ω is a mon ∞ -balayage of δ , then ω is a har -balayage of δ . [b5] If ω is a lnmon ∞ -balayage of δ , then ω is a sbh -balayage of δ . We omit simple proofs of statements [b1]–[b5] of Proposition 1 and note only that statements[b4] and [b5] are implicitly proved in [10, § LOBACHEVSKII JOURNAL OF MATHEMATICS
B. N. KHABIBULLIN, E. B. MENSHIKOVA
2. LOGARITHMIC POTENTIALS, BALAYAGE, AND POLAR SETS
Definition 2. [21], [1], [2], [11, Definition 2], [15, 3.1, 3.2]
Let ν ∈ Meas be a charge such that N ⋆ | ν | (1 , + ∞ ; t < + ∞ , i.e., Z + ∞ | ν | rad (+ ∞ ) − | ν | rad ( t ) t d t < + ∞ , Its logarithmic potential pt ν ( z ) := Z ln | w − z | d ν ( w ) , (11) is uniquely determined on [1], [15, 3.1] Dom pt ν (9N) := n z ∈ C : min (cid:8) N ν − ( z, , , N ν + ( z, , (cid:9) < + ∞ o (12) by values in R , and the set E := ( ∁ Dom pt ω ) \∞ is polar G δ -set with zero outer capacity Cap ∗ ( E ) =0 . Evidently, pt ν ∈ har (cid:0) C \ supp | ν | (cid:1) . If ν ∈ Meas +cmp , then pt ν ∈ sbh ∗ , and ∆ pt ν (3) = ν ∈ Meas + . Theorem 1.
Let E ∈ Bor ( C ) be a polar set, and let ω ∈ Meas + be a ln C -balayage of δ ∈ Meas + provided N ⋆δ + ω (1 , + ∞ ; t < + ∞ . p If R pt ν d δ ∈ R for each pt ν
6≡ −∞ with ν ∈ Meas +cmp , then ω ( E ) = 0 . p If δ ∈ Meas +cmp , then ω ( E \ supp δ ) = 0 .Proof. We can consider only E ⋐ C . There is a potential pt ν with ν ∈ Meas +cmp such that this set E is included in the minus-infinity G δ -set E ν := ( −∞ ) pt ν := (cid:8) z ∈ C : pt ν ( z ) = −∞ (cid:9) ⋐ C . p
1. The condition R pt ν d δ > −∞ entails −∞ < Z pt ν d δ (1) ≤ Z pt ν d ω = Z E ν + Z C \ E ν ! pt ν d ω = ( −∞ ) · ω ( E ν ) + Z C \ E ν pt ν d ω. This is only possible when ω ( E ν ) = 0 . p
2. For any k ∈ N there exists an finite cover of supp δ ⋐ C by disks D ( x j , /k ) ⋐ C such thatthe open sets O k := [ j D ( x j , /k ) ⋐ C , supp δ ⋐ O k +1 ⊂ k ∈ N O k , supp δ = \ k ∈ N O k , have complements ∁ O k without isolated points. Then every open subset O k ⋐ C is regular forthe Dirichlet problem. It suffices to prove that the equality ω ( E ν \ O k ) = 0 holds for each k ∈ N .Consider the functions u k = pt ν on C \ O k , the harmonic extension of pt ν from ∂O k into O k on O k , k ∈ N . LOBACHEVSKII JOURNAL OF MATHEMATICS
ALAYAGE OF MEASURES ON THE COMPLEX PLANE . . . 5
We have u k ∈ sbh ∗ , and u k is bounded from below in supp δ ⋐ O k . Hence − ∞ < Z u k d δ (1) ≤ Z u k d ω = Z C \ ( E ν \ O k ) + Z E ν \ O k ! u k d ω ≤ const ν Z C log(2 + | z | ) d ω ( z ) + ( −∞ ) · ω ( E ν \ O k ) ≤ const ν,ω + ( −∞ ) · ω ( E ν \ O k ) , where here and further we denote by const a ,a ,... constants that depend only on the parameters-indexes a , a , . . . ; const + a ,a ,... ∈ R + . This is only possible when ω ( E ν \ O k ) = 0 . Remark 1.
Theorem 1 is not true for mon p -balayage (see [19, Example]).3. DUALITY FOR mon p -BALAYAGE AND lnmon p -BALAYAGEFor numbers r ∈ R + \ and q ∈ N , we consider the classical subharmonic Weierstrass –Hadamard kernel of genus q [2, 3], [13, (3.2)], [12, §
2, Example 3( E q )], [4], [7], [8], [9], [20] k q ( w, z ) := log | w − z | if ( w, z ) ∈ r D × C , (13 ) k q ( w, z ) := log (cid:12)(cid:12)(cid:12) − zw (cid:12)(cid:12)(cid:12) + q X k =1 Re z k kw k if ( w, z ) ∈ ( C \ r D ) × C , (13 ∞ )where P k =1 · · · := 0 , r is indicated only when necessary, and the Riesz measure ∆ k q ( w, · ) (3) ∈ Meas is the
Dirac measure at w with supp ∆ k q ( w, · ) = w for each w ∈ C . Proposition 2. If δ, ω ∈ Meas + are measures such that N ⋆δ + ω (1 , + ∞ ; t t ⌊ p ⌋ ) (9 ⋆ ) < + ∞ for p ∈ R + ,and ω is a mon p -balayage of δ , then Z C k ⌊ p ⌋ ( w, z ) d( ω − δ )( z ) = pt ω − δ ( w ) for each w (12) ∈ Dom pt ω − δ . (14) If ω is a lnmon p -balayage of δ , then Z C k ⌊ p ⌋ ( w, z ) d δ ( z ) ≤ Z C k ⌊ p ⌋ ( w, z ) d ω ( z ) for each w ∈ C . (15) Proof. If w (13 ) ∈ r D , then (14) is obvious. If w (13 ∞ ) ∈ C \ r D , then, by statements [b2] and [b3] ofProposition 1, the integration of k ⌊ p ⌋ from (13) with respect to the charge ω − δ ∈ Meas gives (14).If ω is a lnmon p -balayage of δ , then, using definition (13) of the Weierstrass – Hadamard kernel ofgenus ⌊ p ⌋ , we obtain (15) by Definition 1 with (1) for h (8) ∈ ln C and Proposition 1 with [b1]–[b2].For r ∈ R + and u : ∂D ( z, r ) → R we define M u ( z, r ) := sup ∂D ( z,r ) u, M rad u ( r ) := M u (0 , r ) . (16)If u ∈ sbh , then M u ( z, r ) = sup ≤ t ≤ r M u ( z, t ) , and M rad u ( r ) = sup ≤ t ≤ r M rad u ( t ) .An elementary consequence the classical Borel – Carath´eodory inequality for disks D (2 r ) is LOBACHEVSKII JOURNAL OF MATHEMATICS
B. N. KHABIBULLIN, E. B. MENSHIKOVA
Proposition 3. If f is an entire function on C , then M | f | ( r ) ≤ M Re f (2 r ) + 3 (cid:12)(cid:12) f (0) (cid:12)(cid:12) for each r ∈ R + . Theorem 2. If ω ∈ Meas +cmp is a mon p -balayage of δ ∈ Meas +cmp , then pt δ ∈ sbh ∗ ∩ har ( C \ supp δ ) , pt ω ∈ sbh ∗ ∩ har ( C \ supp ω ) , (17p) pt ω ( w ) = pt δ ( w ) + O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) as w → ∞ . (17O) If ω is a lnmon p -balayage of δ , then, in addition to (17) , we have pt ω ≥ pt δ on C . (18) Conversely, suppose that there are a set S ⋐ C , and functions d ∈ sbh ∩ har ( C \ S ) , v ∈ sbh ∩ har ( C \ S ) (cf. (17p)) (19p) such that v ( w ) = d ( w ) + O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) as w → ∞ (cf. (17O)) . (19O) Then the Riesz measure ω := ∆ v (3) := 12 π △ v (19) ∈ Meas + (clos S ) ⊂ Meas +cmp (20) of v is a mon p -balayage of the Riesz measure δ := ∆ d (3) := 12 π △ d (19) ∈ Meas + (clos S ) ⊂ Meas +cmp (21) of d . If, in addition to (19) , we have v ≥ d on C (cf. (18)), then ω is a lnmon p -balayage of δ .Proof. The first property (17p) for potentials pt δ and pt ω with compact supports supp δ, supp ω ⋐ C is obvious. Let’s prove property (17O). By Proposition 2 we have (14). If | w | (13) > R := max { r , s } , where s := sup (cid:8) | z | : z ∈ supp ( δ + ω ) (cid:9) , (22)then w (12) ∈ Dom pt ω − δ , and we can use the Taylor series expansion for the integrand expression k ⌊ p ⌋ of integral from the left-hand side of (14) in the form (cid:12)(cid:12) pt ω − δ ( w ) (cid:12)(cid:12) (14) = (cid:12)(cid:12)(cid:12)Z C k ⌊ p ⌋ ( w, z ) d( ω − δ )( z ) (cid:12)(cid:12)(cid:12) (13 ∞ ) = (cid:12)(cid:12)(cid:12)(cid:12)Z (cid:18) ∞ X k = ⌊ p ⌋ +1 − Re z k kw k (cid:19) d( ω − δ )( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∞ X k = ⌊ p ⌋ +1 (cid:12)(cid:12)(cid:12) − Re z k kw k (cid:12)(cid:12)(cid:12) d( ω + δ )( z ) (22) ≤ Z D ( s ) | z | ⌊ p ⌋ +1 | w | ⌊ p ⌋ +1 ∞ X m =0 (cid:12)(cid:12)(cid:12) zw (cid:12)(cid:12)(cid:12) m d( ω + δ )( z ) (22) ≤ | w | ⌊ p ⌋ +1 Z D ( s ) | z | ⌊ p ⌋ +1 d( ω + δ )( z ) = O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) as w → ∞ . The latter means (17O). Finally, if ω is a ln C -balayage of δ , then by inequalities (1) for h ∈ ln C inDefinition 1 we obtain pt δ ≤ pt ω on C . LOBACHEVSKII JOURNAL OF MATHEMATICS
ALAYAGE OF MEASURES ON THE COMPLEX PLANE . . . 7
Conversely, (19p) implies (20) and (21). First, we prove that v is the logarithmic potential pt ω under the assumption that the function d is the logarithmic potential d ( w ) ≡ w ∈ C pt δ ( w ) = O (cid:0) ln | w | (cid:1) as w → ∞ . (23)It follows from (23) and (19O) that v ( w ) = O (cid:0) ln | w | (cid:1) as w → ∞ . By the Weierstrass – Hadamardrepresentation theorem, such subharmonic functions with the Riesz measure ω ∈ Meas + can berepresented in the form v := pt ω + C , where C is a constant. By [21, Theorem 3.1.2], we obtain pt ω ( w ) = ω ( C ) log | w | + O (cid:0) / | w | (cid:1) , pt δ ( w ) = δ ( C ) log | w | + O (cid:0) / | w | (cid:1) as w → ∞ . (24)Hence we have C ≡ w ∈ C v ( w ) − pt ω ( w ) (19O) = w → ∞ pt δ ( w ) + O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) − pt ω ( w ) (24) = w → ∞ (cid:0) δ ( C ) − ω ( C ) (cid:1) ln | w | + O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) + O (cid:16) | w | (cid:17) as w → ∞ . This is only possible if δ ( C ) − ω ( C ) = 0 and C = 0 . Thus, we have v = pt ω , ω ( C ) = δ ( C ) , pt ω − δ ( w ) (19O) = w → ∞ O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) . (25)In particular, from here we have pt ω − δ ( z ) = Z C ln (cid:12)(cid:12)(cid:12) − zw (cid:12)(cid:12)(cid:12) d( ω − δ )( z ) for each w (12) ∈ Dom pt ω − δ , (26)and, by Proposition 1[b1], ω is a mon p -balayage of δ in the case p < . Let’s prove that this measure ω ∈ Meas +cmp is a mon p -balayage of δ in the case p ≥ . By expanding in the Taylor series of thecorresponding analytic branch of the function z z ∈ D ln(1 − z ) , we obtain the following representation pt ω − δ ( w ) (26) = Z D ( s ) ⌊ p ⌋ X k =1 Re − z k kw k d( ω − δ )( z ) = Q ⌊ p ⌋ ( w ) + R ⌊ p ⌋ ( w ) (27p)if w (22) ∈ ∁ D ( R ) , where supp ( δ + ω ) (22) ⊂ D ( s ) , i.e., (cid:12)(cid:12)(cid:12) zw (cid:12)(cid:12)(cid:12) ≤ , (27 w ) Q ⌊ p ⌋ ( w ) ≡ | w | ≥ R Re ⌊ p ⌋ X k =1 q k w k with q k := − k Z D ( s ) z k d( ω − δ )( z ) (27 Q )is harmonic rational function on C ∞ \ with Q ⌊ p ⌋ ( ∞ ) = 0 , and, for | w | (27 w ) ≥ R , (cid:12)(cid:12) R ⌊ p ⌋ ( w ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z D ( s ) ∞ X ⌊ p ⌋ +1 Re − z k kw k d( ω − δ )( z ) (cid:12)(cid:12)(cid:12)(cid:12) (27 w ) ≤ s ⌊ p ⌋ +1 | w | ⌊ p ⌋ +1 ( ω + δ )( C ) . Hence, in view of (25) and (27), we have Q ⌊ p ⌋ ( w ) (27 Q ) = Re ⌊ p ⌋ X k =1 q k w k (27p) = O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) as w → ∞ . (28) LOBACHEVSKII JOURNAL OF MATHEMATICS
B. N. KHABIBULLIN, E. B. MENSHIKOVA
Therefore, for complex polynomial Q ( z ) ≡ z ∈ C ⌊ p ⌋ X k =1 q k z k , Q (0) = 0 , deg Q ≤ ⌊ p ⌋ , (29)we get Re Q ( z ) (28) = O (cid:0) | z | ⌊ p ⌋ +1 (cid:1) as z → . Hence, by Proposition 3, we have M | Q | ( r ) ≤ M Re Q (2 r ) + 3 (cid:12)(cid:12) Q (0) (cid:12)(cid:12) (29) = 2 M Re Q (2 r ) (29) = O (cid:0) | z | ⌊ p ⌋ +1 (cid:1) as z → .Thus, this polynomial Q of degree deg Q (29) ≤ ⌊ p ⌋ has a root of multiplicity at least ⌊ p ⌋ + 1 at .Therefore, Q = 0 , and q k = 0 for each k = 1 , . . . ⌊ p ⌋ . By definition (27 Q ) of q k and by Proposition1[b2], the measure ω is a mon p -balayage of δ . Evidently, the inequality pt ω = p ≥ pt δ on C meansthat ω is also a ln C -balayage of δ . Thus, the second final part of our Theorem is proved underassumption (23).If d (19p) ∈ sbh ∩ har ( C \ S ) is an arbitrary function with the Riesz measure δ from (21), then, bythe Weierstrass – Hadamard representation theorem, d admits representation d (21) = pt δ + h d , where h d is harmonic on C . Consider function ˜ d := pt δ instead of function d , and function ˜ v := v − h d instead of function v . For this pair of functions, we have δ = ∆ ˜ d and ω = ∆ ˜ v , and also ˜ v ( w ) = v ( w ) − h d ( w ) (19O) = w → ∞ d ( w ) − h d ( w ) + O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) = ˜ d ( w ) + O (cid:16) | w | ⌊ p ⌋ +1 (cid:17) as w → ∞ ,but assumption (23) for the function ˜ d instead of d is already fulfilled.4. BALAYAGE WITH RESPECT TO SUBHARMONIC FUNCTIONS OF FINITE ORDER Definition 3.
For a function f : [ r, + ∞ ) → R and f + := sup { , f } , values (see [17, § ord [ f ] := lim sup x → + ∞ ln(1 + f + ( x ))ln x ∈ R + ∪ + ∞ , (30o) type p [ f ] := lim sup x → + ∞ f ( x ) x p ∈ R ∪ + ∞ (30t) are the order of this function f , and the type of this function f under order p , respectively. Using Definition 3, for ν ∈ Meas , we define ord [ ν ] (30o) := ord (cid:2) | ν | rad (cid:3) , (31o) type p [ ν ] (30t) := type p (cid:2) | ν | rad (cid:3) if ord [ ν ] ≤ p ∈ R + . (31t) Proposition 4.
Let p ∈ R + , and let ν ∈ Meas + be a measure with type p [ ν ] (31t) < + ∞ under order p (31o) := ord [ ν ] ∈ R + , (32) and a finite measure ω ∈ Meas + satisfies the condition N ⋆ν (1 , + ∞ ; t t p ) (9 ⋆ ) < + ∞ when p ∈ R + \ N , N ⋆ν (1 , + ∞ ; t t p ln t ) (9 ⋆ ) < + ∞ when p ∈ N . (33) LOBACHEVSKII JOURNAL OF MATHEMATICS
ALAYAGE OF MEASURES ON THE COMPLEX PLANE . . . 9
Then there exist the following two equal repeated integrals −∞ ≤
Z (cid:18)Z k ⌊ p ⌋ ( w, z ) d ν ( w ) (cid:19) d ω ( z ) = Z (cid:18)Z k ⌊ p ⌋ ( w, z ) d ω ( z ) (cid:19) d ν ( w ) < + ∞ . (34) Remark 2.
Condition (33) can be written in the equivalent form as Z + ∞ ω rad (+ ∞ ) − ω rad ( t ) t t p ln ⌊ p ⌋−⌈ p ⌉ t d t < + ∞ , (35)since, obviously, ⌊ p ⌋ − ⌈ p ⌉ = if p ∈ Z , if p ∈ R \ Z . (36) Proof.
Standard classical estimates of the Weierstrass – Hadamard kernel k ⌊ p ⌋ of genus q from (13)give [9, 4.1.1] k ⌊ p ⌋ ( w, z ) ≤ K ⌊ p ⌋ ( w, z ) := const + p ln (cid:0) | z | (cid:1) if | w | < , ln (cid:0) | z | / | w | (cid:1) if | w | ≥ and ⌊ p ⌋ = 0 , (cid:0) | z | / | w | (cid:1) ⌊ p ⌋ min (cid:8) , | z | / | w | (cid:9) if | w | ≥ and ⌊ p ⌋ ∈ N , (37)The function K ⌊ p ⌋ is positive and Borel-measurable on C . Using these estimates and the condition(32) we estimate, as in [9, Lemma 4.4], the integral Z K ⌊ p ⌋ ( w, z ) d ν ( w ) ≤ const + ν,p · | z | p if p / ∈ N , (1 + | z | p ) log(2 + | z | ) if p ∈ N . (38)Hence, in view of (33) ⇔ (35), there exists the following repeated integral Z (cid:18)Z K ⌊ p ⌋ ( w, z ) d ν ( w ) (cid:19) d ω ( z ) < + ∞ . By the Fubini – Tonelli Theorem [5, Ch. V, §
8, 1, Scholium] there are two equal repeated integrals
Z (cid:18)Z K ⌊ p ⌋ ( w, z ) d ν ( w ) (cid:19) d ω ( z ) = Z (cid:18)Z K ⌊ p ⌋ ( w, z ) d ω ( z ) (cid:19) d ν ( w ) ∈ R . Hence, in view of k ⌊ p ⌋ (37) ≤ K ⌊ p ⌋ on C , by Fubini’s Theorem [9, Theorem 3.5] the integrals in (34)exist and coincide with possible value of −∞ .For r ∈ R + and u : ∂D ( z, r ) → R we define the integral average value of u on the circle ∂D ( z, r ) C u ( z, r ) := 12 π Z π u ( z + re is ) d s, C rad u ( r ) := C u (0 , r ) , (39C)and the integral average value of u : D ( z, r ) → R on the disk D ( z, r ) B u ( z, r ) := 2 r Z r C u ( z, t ) t d t, B rad u ( r ) := B u (0 , r ) . (39B) LOBACHEVSKII JOURNAL OF MATHEMATICS
Proposition 5.
Let p ∈ R + . If u ∈ sbh ∗ , type ⌊ p ⌋ +1 [ u ] (4c) = 0 , type p [ C rad u ] < + ∞ , (40) then type p [ ∆ u ] < + ∞ , there is a polynomial P ∈ Pol p such that u ( z ) = Z C k ⌊ p ⌋ ( w, z ) d ∆ u ( w ) | {z } I u ( z ) + Re P ( z ) for each z ∈ C , (41) and u ( z ) (36) = O (cid:16) | z | p ln ⌊ p ⌋−⌈ p ⌉ | z | (cid:17) as z → ∞ . (42) Proof.
By the Jensen – Privalov formula [3, Theorem 2.6.5.1] for subharmonic function u on D ( R ) , N rad ∆ u (1 , R ) = C rad u ( R ) − C rad u (1) for each R > . Hence, in view of type p [ C rad u ] (40) < + ∞ , we obtain type p (cid:2) N rad ∆ u (1 , · ) (cid:3) < + ∞ , and, as an easy consequence, type p [ ∆ u ] < + ∞ . This gives the Weierstrass –Hadamard representation [2], [9], [3] of the form (41), but so far only with an entire function P = 0 .The harmonic function Re P (41) := u − I u is the difference between the subharmonic function u with type ⌊ p ⌋ +1 [ u ] = 0 and the canonical Weierstrass – Hadamard integral I u satisfying I u ( z ) (36) = O (cid:16) | z | p ln ⌊ p ⌋−⌈ p ⌉ | z | (cid:17) as z → ∞ (43)since type p [ ∆ u ] < + ∞ (see (37)-(38)). In particular, type ⌊ p ⌋ +1 [ I u ] = 0 . Hence type ⌊ p ⌋ +1 [Re P ] = type ⌊ p ⌋ +1 [ u − I u ] = 0 [3, Theorems 2.9.3.2, 2.9.4.2]. Therefore type ⌊ p ⌋ +1 (cid:2) M | P | (cid:3) = 0 by Proposition3. This is possible only if the entire function P is a polynomial of degree deg P < ⌊ p ⌋ + 1 , i.e., P ∈ Pol p , and (42) follows from (41) and (43). Theorem 3.
Let p ∈ R + . If both ω ∈ Meas + and δ ∈ Meas + satisfy (33) ⇔ (35) for p ∈ R + , and ω is a mon p -balayage of δ , then for each function u of (40) the integrals R C u d ω ∈ R ∪ −∞ and R C u d δ ∈ R ∪ −∞ is well defined. If ω is a lnmon p -balayage of δ , then Z C u d δ ≤ Z C u d ω (44) for each u of (40) , i.e., ω is a balayage of δ with respect to the class of functions u satisfying (40) .Proof. In view of (33), it follows from Propositions 4 and 5 that the integration of two summandsin the right-hand side of (41) with respect to the measure ω is correct, and by representation (41).If ω is a lnmon p -balayage of δ , then Z C u d δ (41) = Z C Z C k ⌊ p ⌋ ( w, z ) d ∆ u ( w ) d δ ( z ) + Re Z C P ( z ) d δ ( z )= (cid:12)(cid:12)(cid:12) Propositions 5, 4 with (33), 1[b2] (cid:12)(cid:12)(cid:12) = Z C Z C k ⌊ p ⌋ ( w, z ) d δ ( z ) d ∆ u ( w ) + Re Z C P ( z ) d ω ( z ) (15) ≤ Z C k ⌊ p ⌋ ( w, z ) d ω ( z ) d ∆ u ( w ) + Re Z C P ( z ) d ω ( z ) , where the right-hand side is equal to R C u d ω by Propositions 4 with (33), and 5. LOBACHEVSKII JOURNAL OF MATHEMATICS
ALAYAGE OF MEASURES ON THE COMPLEX PLANE . . . 11
Remark 3.
The condition type p [ C rad u ] < + ∞ from (40) in both Proposition 5 and Theorem 3 canbe replaced by the condition type p [ B rad u ] (39B) < + ∞ . Acknowledgments.
The research is funded in the framework of executing the developmentprogram of Scientific Educational Mathematical Center of Volga Federal District by additionalagreement no. 075-02-2020-1421/1 to agreement no. 075-02-2020-1421 (first author), and also wassupported by a Grant of the Russian Foundation of Basic Research (Project no. 19-31-90007, secondauthor).
Compliance with ethical standards
Conflict of interest.
The authors declare that they have no conflict of interest.REFERENCES
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