Balayage of Measures with respect to Classes of Subharmonic and Harmonic Functions
BBalayage of Measures with respect to Classesof Subharmonic and Harmonic Functions
B. N. KhabibullinApril 15, 2020
Abstract
We investigate some properties of balayage, or, sweeping (out), of measures withrespect to subclasses of subharmonic functions. The following issues are considered:relationships between balayage of measures with respect to classes of harmonic orsubharmonic functions and balayage of measures with respect to significantly smallerclasses of specific classes of functions; integration of measures and balayage of mea-sures; sensitivity of balayage of measures to polar sets, etc.
The origins of the concept of balayage, or, “sweeping (out)” etc., of measures or functionsare the studies of Henri Poincar´e, de la Vall´ee Poussin, Henri Cartan, Marcel Brelot andmany others. A detailed historical review of potential theory is given in in [5]. In [18], weinvestigate various general concepts of balayage. In this article we deal with particularcases of such balayage with respect to special classes of subharmonic functions.The general concept of balayage can be defined as follows. Let R be a (pre-)orderedset with a (pre-)order relation ≤ . Let L be a set with a subset H ⊂ L . A function ω : L → R can be called the balayage of a function δ : L → R with respect to H , and wewrite δ (cid:22) H ω , if the function ω majorizes the function δ on H : δ ( h ) ≤ ω ( h ) for each h ∈ H ⊂ L. (1)In this article, R is the extended real line, L is the class of all upper semicontinuousfunctions on an open set O in a finite-dimensional Euclidean space, H is a subclass ofsubharmonic functions on O , and δ and ω is a pair of Radon positive measures on O withcompact supports in O . In this case, relationship (1) turns into inequalities of the form δ ( h ) := (cid:90) O h d δ ≤ (cid:90) O h d ω =: ω ( h ) for each h ∈ H ⊂ L. (2)1 a r X i v : . [ m a t h . C V ] A p r e investigate properties of balayage of measures with respect to classes of harmonic,subharmonic, and special subharmonic functions.We proceed to precise and detailed definitions and formulations. The reader can skip this Section 2 and return to it only if necessary.We denote by N := { , , . . . } , R , and R + := { x ∈ R : x ≥ } the sets of natural, of real, and of positive numbers, each endowed with its natural order ( ≤ , sup / inf), algebraic,geometric and topological structure. We denote singleton sets by a symbol without curlybrackets. So, N := { } ∪ N =: 0 ∪ N , and R + \ R + \{ } is the set of strictly positive numbers, etc.The extended real line R := −∞ (cid:116) R (cid:116) + ∞ is the order completion of R by the disjointunion (cid:116) with + ∞ := sup R and −∞ := inf R equipped with the order topology withtwo ends ±∞ , R + := R + (cid:116) + ∞ ; inf ∅ := + ∞ , sup ∅ := −∞ for the empty set ∅ etc.The same symbol 0 is also used, depending on the context, to denote zero vector, zerofunction, zero measure, etc.We denote by R d the Euclidean space of d ∈ N dimensions with the Euclidean norm | x | := (cid:112) x + · · · + x d of x = ( x , . . . , x d ) ∈ R d .We denote by R d ∞ := R d (cid:116)∞ the Alexandroff one-point compactification of R d obtainedby adding one extra point ∞ . For a subset S ⊂ R d ∞ or a subset S ⊂ R d we let (cid:123) S := R d ∞ \ S , clos S , int S := (cid:123) (clos (cid:123) S ), and ∂S := clos S \ int S denote its complement, closure,interior, and boundary always in R d ∞ , and S is equipped with the topology induced from R d ∞ . If S (cid:48) is a relative compact subset in S , i.e., clos S (cid:48) ⊂ S , then we write S (cid:48) (cid:98) S .We denote by B ( x, t ) := { y ∈ R d : | y − x | < t } , B ( x, t ) := { y ∈ R d : | y − x | ≤ t } , ∂B ( x, t ) := B ( x, t ) \ B ( x, t ) an open ball, closed ball, a circle of radius t ∈ R + centered at x ∈ R d , respectively. Besides, we denote by B := B (0 , B := B (0 ,
1) and ∂ B := ∂B (0 , open unit ball, the closed unit ball and the unit sphere in R d , respectively.Throughout this paper O (cid:54) = ∅ will denote an open subset in R d , and D (cid:54) = ∅ is a domain in R d , i.e., an open connected subset in R d .For S ⊂ R d ∞ , C ( S ) is the vector space over R of all continuous functions f : S → R withthe sup-norm, C ( S ) ⊂ C ( S ) is the subspace of functions f ∈ C ( S ) with compact support supp f (cid:98) S , and usc ( S ) is the convex cone over R + of all upper semicontinuous functions f : S → R ∪ −∞ = R \ + ∞ . For S ⊂ R d , har ( S ) and sbh ( S ) are the collections of allfunctions u which are harmonic and subharmonic on some open set O u ⊃ S , respectively.In addition, sbh ∗ ( S ) ⊂ sbh ( S ) consists only of functions u ∈ sbh ( S ) such that u (cid:54)≡ −∞ on each connected component of O u .The convex cone over R + of all Borel, or Radon, positive measures µ ≥ σ -algebra Bor ( S ) of all Borel subsets of S is denoted by Meas + ( S ); Meas +cmp ( S ) ⊂ Meas + ( S )is the subcone of measures µ ∈ Meas + ( S ) with compact support supp µ in S , Meas +1 ( S ) is2he convex set of probability measures on S , Meas ( S ) := Meas ( S ) ∩ Meas cmp ( S ). So, δ x ∈ Meas ( S ) is the Dirac measure at a point x ∈ S , i.e., supp δ x = { x } , δ x ( { x } ) = 1.We denote by µ (cid:12)(cid:12) S (cid:48) the restriction of µ to S (cid:48) ∈ Bor ( S ). The same notation is used forthe restrictions of functions and their classes to sets.Let (cid:52) be the Laplace operator acting in the sense of the theory of distributions, Γ bethe gamma function . For u ∈ sbh ∗ ( O ), the Riesz measure of u is a Borel (or Radon [24,A.3]) positive measure ∆ u := c d (cid:52) u ∈ Meas + ( O ) , c d := Γ( d/ π d/ max { , d − (cid:9) . (3) Let O be a topological space, and S ⊂ O . We denote by Conn O S the set of all connectedcomponents of S . We write S (cid:98) O , if the closure of S in O is a compact subset of O . Definition 1.
The union of a subset S ⊂ O with all connected component of C ∈ Conn O ( O \ S ) such that C (cid:98) O will be called the inward filling of S with respect to O and is denoted further asin-fill O S := S (cid:91)(cid:16)(cid:91)(cid:8) C ∈ Conn O ( O \ S ) : C (cid:98) O (cid:9)(cid:17) . Denote by O ∞ the Alexandroff one-point compactification of O with underlying set O (cid:116){∞} . Proposition 1 ([9, 6.3], [10]) . Let S be a compact set in an open set O ⊂ R d . Then (i) in-fill O S is a compact subset in O , and in-fill O (cid:0) in-fill O S (cid:1) = in-fill O S ; (ii) the set O ∞ \ in-fill O S is connected and locally connected; (iii) the inward filling of S with respect to O coincides with the complement in O ∞ ofconnected component of O ∞ \ S containing the point ∞ ; (iv) if O (cid:48) ⊂ R d ∞ is an open subset and O ⊂ O (cid:48) , then in-fill O S ⊂ in-fill O (cid:48) S ; (v) R d \ in-fill O S has only finitely many components, i. e., R d ∞ ( R d \ in-fill O S ) < ∞ . Proposition 2 ([9, Theorem 1.7]) . Let O be an open set in R d , let S be a compact subsetin O , and suppose that O ∞ \ S is connected. Then, for each u ∈ har ( S ) and each number b ∈ R + \ , there is h ∈ har ( O ) such that | u − h | < b on S . Proposition 3.
Let O be an open set in R d , and let S be a compact subset in O . If h ∈ har (cid:0) in-fill O S (cid:1) , then there are harmonic functions h j ∈ j ∈ N har ( O ) such that the sequence ( h j ) j ∈ N converges to this harmonic function h in C (in-fill O S ) . O S instead of S in Proposition 2. Proposition 4 ([9, Theorem 6.1], [11, Theorem 1], [10, Theorem 16]) . Let O be an openset in R d , let S be a closed subset in O , and suppose that O ∞ \ S is connected and locallyconnected. Then, for each u ∈ sbh ( S ) , there exists U ∈ sbh ( O ) such that u = U on S . The intersection of Proposition 1 (parts (i)–(ii)) and Proposition 4 is
Proposition 5.
Let O be an open set in R d , and let S be a compact subset in O . Then,for each u ∈ sbh (in-fill O S ) , there exists U ∈ sbh ( O ) such that u = U on in-fill O S . In this section 4 we traditional classical balayage that is particular case of (1) (see also[18]).
Definition 2 ([22], [3], [18, Definition 5.2]) . Let S ⊂ Bor ( R d ∞ ), δ ∈ Meas +cmp ( S ), ω ∈ Meas +cmp ( S ). Let H ⊂ usc ( S ) be a subclass of upper semicontinuous functions on S . Wewrite δ (cid:22) H ω and say that the measure ω is a balayage, or, sweeping (out), of the measure δ with respect to H , or, briefly, ω is H -balayage of δ , if (cid:90) h d δ (2) ≤ (cid:90) h d ω for each h ∈ H . (4)If δ (cid:22) H ω and at the same time ω (cid:22) H δ , then we write δ (cid:39) H ω . Proposition 6.
Let O ⊂ R d be an open set, ω ∈ Meas ( O ) be a H -balayage of δ ∈ Meas ( O ) , O (cid:48) ⊂ R d be an open set, and H (cid:48) ⊂ R O (cid:48) . (i) The binary relation (cid:22) H (respectively (cid:39) H ) on Meas +cmp ( S ) is a preorder, i.e., a re-flexive and transitive relation, (respectively, an equivalence) on Meas +cmp ( S ) . (ii) If H contains a strictly positive (respectively, negative) constant, then δ ( S ) ≤ ω ( S ) (respectively, δ ( S ) ≥ ω ( S ) ). (iii) If H (cid:48) ⊂ H , then ω is H (cid:48) -balayage of δ . (iv) If O (cid:48) ⊂ O and supp δ ∪ supp ω ⊂ O (cid:48) , then ω (cid:12)(cid:12) O (cid:48) is a balayage of δ (cid:12)(cid:12) O (cid:48) for H (cid:12)(cid:12) O (cid:48) . (v) If H = − H , then the order (cid:22) H is the equivalence (cid:39) H . So, if H = har ( S ) , then ω isa har ( S ) -balayage of δ if and only if δ (cid:39) har ( S ) ω , i.e., (cid:90) S h d δ = (cid:90) S h d ω for each h ∈ har ( S ) and δ ( S ) = ω ( S ) . (5)4vi) If δ (cid:22) sbh ( S ) ω , then δ (cid:22) har ( S ) ω . The converse is not true [20, XIB2], [23, Example].(vii) If ω ∈ Meas +cmp ( O ) is a (cid:0) sbh ( O ) ∩ C ∞ ( O ) (cid:1) -balayage of δ ∈ Meas +cmp ( O ) , where C ∞ ( O ) is the class of all infinitely differentiable functions on O , then δ (cid:22) sbh ( O ) ω , since foreach function u ∈ sbh ( O ) there exists a sequence of functions u j ∈ j ∈ N sbh ( O ) ∩ C ∞ ( O ) decreasing to it [7, Ch. 4, 10, Approximation Theorem].All statements of Proposition 6 are obvious. Example 1 ([8], [6], [25], [12]) . Let x ∈ O . If a measure ω ∈ Meas +cmp ( O ) is a balayageof the Dirac measure δ x with respect to sbh ( O ), then this measure ω is called a Jensenmeasure on O at x . The class of all Jensen measures on O at x ∈ O will be denoted by J x ( O ). Example 2 ([8], [8, 3], [16], [17, Definition 8]) . Let x ∈ O . If ω ∈ Meas +cmp ( O ) is har ( O )-balayage of the Dirac measure δ x , then the measure ω is called an Arens – Singermeasure on O at x ∈ O . The class of all Arens – Singer measures on O at x is denoted by AS x ( O ) ⊃ J x ( O ).For s ∈ R , we set k s ( t ) := (cid:40) ln t if s = 0 , − sgn( s ) t − s if s ∈ R \ t ∈ R + \ , (6k) K d − ( y, x ) := k d − (cid:0) | y − x | (cid:1) if y (cid:54) = x, −∞ if y = x and d ≥ , y = x and d = 1 , ( y, x ) ∈ R d × R d , (6K) k x : y (cid:55)−→ y ∈ R d K d − ( y, x ) ∈ sbh ( R d ) (cid:92) har ( R d \ x ) , x ∈ R d , (6 k x ) K ( X ) := { k x : x ∈ X } ⊂ sbh ∗ ( R d ) , X ⊂ R d . (6 K ) Theorem 1.
Let O ⊂ R d be an open set, and δ ∈ Meas +cmp ( O ) , ω ∈ Meas +cmp ( O ) . Themeasure ω is har ( O ) -balayage (respectively, sbh ( O ) -balayage) of the measure δ if and onlyif there exists a compact subset S (cid:98) O such that this measure ω is a balayage of δ withrespect to K ( O \ S ) (cid:91)(cid:0) − K ( O \ S ) (cid:1) , (7h) (cid:16) respectively, K ( O ) (cid:91)(cid:0) − K ( O \ S ) (cid:1)(cid:17) . (7s) Proof.
We set S O := in-fill O (supp δ ∪ supp ω ) . (8)5y Proposition 3, for each x / ∈ S O there are functions ± h xj ∈ j ∈ N har ( O ) such that thesequence ( ± h xj ) j ∈ N converges to ± k x ⊂ har ( S O ) in C ( S O ). Let (cid:16) δ (cid:22) har ( O ) ω (cid:17) ⇐⇒ (cid:16) δ (cid:39) har ( O ) ω (cid:17) (see Definition 2 and Proposition 6(i),(v)) . (9)If x / ∈ S O , then, (cid:90) ± K d − ( y, x ) d δ ( y ) (8) = (cid:90) S O ± K d − ( y, x ) d δ ( y ) = (cid:90) S O lim j →∞ ± h xj ( y ) d δ ( y ) = lim j →∞ (cid:90) S O ± h xj d δ (8) = lim j →∞ (cid:90) O ± h xj d δ (9) , (5) = lim j →∞ (cid:90) O ± h xj d ω (8) = lim j →∞ (cid:90) S O ± h xj d ω = (cid:90) S O lim j →∞ ± h xj d ω = (cid:90) S O ± K d − ( y, x ) d ω ( y ) (8) = (cid:90) ± K d − ( y, x ) d ω ( y ) . Thus, (9) implies that ω is a balayage of δ with respect to the class (7h) with S := S O .If δ (cid:22) sbh ( O ) ω , then, by Proposition 6(vi), δ (cid:22) har ( O ) ω , and δ (cid:22) K ( R d \ S O ) ∪ ( − K ( R d \ S O )) ω .Besides, in view of (6 k x ), we obtain (cid:90) K d − ( y, x ) d δ ( y ) (6 k x ) = (cid:90) O k x ( y ) d δ ( y ) (4) ≤ (cid:90) O k x ( y ) d δ ( y ) (6 k x ) = (cid:90) K d − ( y, x ) d ω ( y ) for each x ∈ R d .Thus, δ (cid:22) sbh ( O ) ω implies δ (cid:22) K ( R d ) ω and ω is a balayage of δ with respect to (7s) if S := S O .So, the necessary conditions of Theorem 1 are proved.In the opposite direction, let δ (7h) (cid:22) K ( O \ S ) ∪ ( − K ( O \ S )) ω, where S closed = clos S compact (cid:98) O. (10)Then, by Definition 2 and Proposition 6(v), according to equality (5), we have (cid:90) S K d − ( y, x ) d δ ( y ) = (cid:90) S K d − ( y, x ) d ω ( y ) for each x ∈ O \ S . (11)Let u ∈ har ( O ). Without loss of generality, we can assume thatsupp δ (cid:91) supp ω (cid:98) int S ⊂ S closed = clos S compact (cid:98) O. (12)There is an open subset U (cid:98) O such that S ⊂ U , ∂U is a C surface and that each pointof ∂U is a one-sided boundary point of U [9, 1.6]. In particular ∂U ⊂ O \ S . If we applyGreen’s identity to U \ B ( x, r ) and let r tend to 0, we obtain [9, 1.6] u ( y ) = c d (cid:90) ∂U (cid:16) K d − ( y, x ) ∂u∂(cid:126)n x ( x ) − u ( x ) ∂∂(cid:126)n x K d − ( y, x ) (cid:17) d σ ( x ) for each y ∈ S, (13)6here σ denotes surface area measure on ∂U , (cid:126)n x denotes the outer unit normal to ∂U at x ∈ ∂U and c d ∈ R + \ δ and the measure ω , we obtain, respectively,1 c d (cid:90) supp δ u ( y ) d δ ( y ) (12) = (cid:90) S (cid:90) ∂U K d − ( y, x ) ∂u∂(cid:126)n x ( x ) d σ ( x ) d δ ( y ) − (cid:90) S (cid:90) ∂U u ( x ) ∂∂(cid:126)n x K d − ( y, x ) d σ ( x ) d δ ( y ) , (14 δ )1 c d (cid:90) supp δ u ( y ) d ω ( y ) (12) = (cid:90) S (cid:90) ∂U K d − ( y, x ) ∂u∂(cid:126)n x ( x ) d σ ( x ) d ω ( y ) − (cid:90) S (cid:90) ∂U u ( x ) ∂∂(cid:126)n x K d − ( y, x ) d σ ( x ) d ω ( y ) . (14 ω )Hence, using Fubini’s theorem and differentiation under the integral sign, we have1 c d (cid:90) supp δ u ( y ) d δ ( y ) (14 δ ) = (cid:90) ∂U (cid:18)(cid:90) S K d − ( y, x ) d δ ( y ) (cid:19) ∂u∂(cid:126)n x ( x ) d σ ( x ) − (cid:90) ∂U u ( x ) ∂∂(cid:126)n x (cid:18)(cid:90) S K d − ( y, x ) d δ ( y ) (cid:19) d σ ( x ) , (15 δ )1 c d (cid:90) supp δ u ( y ) d ω ( y ) (14 ω ) = (cid:90) ∂U (cid:18)(cid:90) S K d − ( y, x ) d ω ( y ) (cid:19) ∂u∂(cid:126)n x ( x ) d σ ( x ) − (cid:90) ∂U u ( x ) ∂∂(cid:126)n x (cid:18)(cid:90) S K d − ( y, x ) d ω ( y ) (cid:19) d σ ( x ) . (15 ω )According to equality (11), for each x ∈ ∂U ⊂ O \ S , the internal integrals on the right-handsides of equalities (15 δ ) and (15 ω ) coincide, and the external integrals on the right-handsides of equalities (15 δ ) and (15 ω ) are of the same form. Therefore, the integrals onthe left-hand sides of equalities (15 δ ) and (15 ω ) also coincide for each harmonic function u ∈ har ( O ). By Definition 2, formula (4), this means that the measure ω is har ( O )-balayage of the measure δ , i.e., we have (9).It remains to consider the case when ω is a balayage of δ with respect to the class (7s).It has already been shown above that in this case we have (9), i.e., δ (cid:22) har ( O ) ω .Let u ∈ sbh ∗ ( O ) with the Riesz measure ∆ u (3) ∈ Meas + ( O ). By the Riesz DecompositionTheorem [24, Theorem 3.7.1], [14, Theorem 3.9], [1, Theorem 4.4.1], [13, Theorem 6.18],there exist an open set O (cid:48) (cid:98) O and a harmonic functions h ∈ har ( O (cid:48) ) such that S O (8) (cid:98) O (cid:48) and u ( y ) = (cid:90) clos O (cid:48) K d − ( x, y ) d ∆ u ( x ) + h ( y ) for each y ∈ S O (cid:98) O (cid:48) , (16r) S := supp δ ∪ supp ω, S O (8) = in-fill S (cid:98) O (cid:48) , (16S)7ntegrating the representation (16r) with respect to the measures δ and ω , we obtain (cid:90) S u ( y ) d δ ( y ) = (cid:90) S (cid:90) clos O (cid:48) K d − ( x, y ) d ∆ u ( x ) d δ ( y ) + (cid:90) S h ( y ) d δ ( y ) , (17 δ ) (cid:90) S u ( y ) d ω ( y ) = (cid:90) S (cid:90) clos O (cid:48) K d − ( x, y ) d ∆ u ( x ) d ω ( y ) + (cid:90) S h ( y ) d ω ( y ) . (17 ω )Hence, by Fubini’s theorem and in view of the symmetry of the kernel K d − from (6K),we can rewrite (17) in the form (cid:90) S u ( y ) d δ ( y ) (17 δ ) = (cid:90) clos O (cid:48) (cid:18)(cid:90) S K d − ( y, x ) d δ ( y ) (cid:19) d ∆ u ( x ) + (cid:90) S h ( y ) d δ ( y ) , (18 δ ) (cid:90) S u ( y ) d ω ( y ) (17 δ ) = (cid:90) clos O (cid:48) (cid:18)(cid:90) S K d − ( y, x ) d ω ( y ) (cid:19) d ∆ u ( x ) + (cid:90) S h ( y ) d ω ( y ) . (18 ω )By Proposition 3 there are harmonic functions h j ∈ j ∈ N har ( O ) such that the sequence( h j ) j ∈ N converges to this harmonic function h (16S) ∈ har ( S O ) in C ( S O ). Hence, (cid:90) S h d δ (8) = (cid:90) S O h d δ = (cid:90) S O lim j →∞ h j d δ = lim j →∞ (cid:90) S O h j d δ = lim j →∞ (cid:90) O h j d δ (9) , (5) = lim j →∞ (cid:90) O h j d ω = lim j →∞ (cid:90) S O h j d ω = (cid:90) S O lim j →∞ h j d ω = (cid:90) S O h d ω (8) = (cid:90) S h d ω. (19)By construction of class (7s), we have δ (cid:22) K ( O ) ω . Therefore, (cid:90) S K d − ( y, x ) d δ ( y ) ≤ (cid:90) S K d − ( y, x ) d ω ( y ) for each y ∈ O ⊃ clos O (cid:48) . (20)According to equality (19), the last integrals on the right-hand sides of equalities (18 δ )and (18 ω ) also coincide, and, in view of (20), we have (cid:90) clos O (cid:48) (cid:18)(cid:90) S K d − ( y, x ) d δ ( y ) (cid:19) d ∆ u ( x ) (20) ≤ (cid:90) clos O (cid:48) (cid:18)(cid:90) S K d − ( y, x ) d ω ( y ) (cid:19) d ∆ u ( x )Hence, by representations (18 δ ) and (18 ω ), we obtain (cid:90) O u ( y ) d δ ( y ) (16S) = (cid:90) S u ( y ) d δ ( y ) (18) ≤ (cid:90) S u ( y ) d ω ( y ) (16S) = (cid:90) O u ( y ) d ω ( y ) . The latter, by Definition 2, formula (4), means that the measure ω is sbh ( O )-balayage of δ . 8 Integration of measures and balayage
Let S ∈ Bor ( O ). Consider a function Θ : S → Meas +cmp ( O ) such that Θ : S → Meas +cmp ( O ) , ϑ x := Θ ( x ) , (cid:91) x ∈ S supp ϑ x (cid:98) O, sup x ∈ S ϑ x ( O ) < + ∞ , (21 ϑ ) x (cid:55)−→ x ∈ S (cid:90) O f d ϑ x is a Borel measurable function for each f ∈ C ( O ). (21B)Let ω ∈ Meas +cmp ( O ) , supp ω ⊂ S (cid:98) O. (22)Under these conditions, we can to define the integral (cid:82) Θ d ω of Θ with respect to measure ω as a Borel, or, Radon, positive measure on O [21, Introduction, § § § (cid:90) Θ d ω (21)-(22) :=: (cid:90) S ϑ x d ω ( x ) ∈ Meas +cmp ( O ) , (23I) (cid:16)(cid:90) Θ d ω (cid:17) ( B ) :=: (cid:90) S ϑ x ( B ) d ω ( x ) ∈ R for each B ∈ Bor ( O ) such that B (cid:98) O , (23B) (cid:90) Θ d ω : f (cid:55)−→ (cid:90) (cid:18)(cid:90) f d ϑ x (cid:19) d ω ( x ) ∈ R ∪ −∞ for each f ∈ usc ( O ) . (23f)Let r ∈ R + \ ϑ ∈ Meas +cmp ( r B ). For x ∈ R d , we define the shift ϑ x ∈ Meas +cmp (cid:0) B ( x, r ) (cid:1) of this measure ϑ to point x as ϑ x ( B ) := ϑ ( B − x ) for any B ∈ Bor (cid:0) B ( x, r ) (cid:1) , (24B) (cid:90) f d ϑ x := (cid:90) r B f ( x + y ) d ϑ ( y ) ∈ R ∪ −∞ for each f ∈ usc (cid:0) B ( x, r ) (cid:1) . (24f)For a measure (22), under the condition S ∪ r := (cid:91) x ∈ S B ( x, r ) (cid:98) O, (25)we can define the convolution ω ∗ ϑ ∈ Meas +cmp ( O ) of measures ω and ϑ by the the integral (cid:82) Θ d ω of Θ : S (21 ϑ ) −→ Meas +cmp ( O ) with respect to the measure ω as ω ∗ ϑ (23I) := (cid:90) Θ d ω (24) = (cid:90) S ϑ x d ω ( x ) ∈ Meas +cmp ( O ) , (26*)( ω ∗ ϑ )( B ) (24B) = (cid:90) S ϑ ( B − x ) d ω ( x ) ∈ R for each B ∈ Bor ( O ) such that B (cid:98) O ,(26B) (cid:90) f d( ω ∗ ϑ ) (24f) = (cid:90) S (cid:18)(cid:90) r B f ( x + y ) d ϑ ( y ) (cid:19) d ω ( x ) ∈ R ∪ −∞ for each f ∈ usc ( O ) . (26f)9ery special cases of the following Theorem 2 were essentially used for convolutionsin [15, Lemmata 7.1, 7.2], [2, 2.1.1, 1b)], [18, 8.1]. Theorem 2.
Let ω ∈ Meas cmp ( O ) be a measure from (22) .If ∅ (cid:54) = H ⊂ usc ( O ) and each measure ϑ x (21 ϑ ) = Θ ( x ) in (21) is H -balayage of the Diracmeasure δ x at x ∈ S , then the integral (cid:82) Θ d ω (23) ∈ Meas +cmp ( O ) is H -balayage of ω , i.e., ω (4) (cid:22) H (cid:90) Θ d ω (23) = (cid:90) S ϑ x d ω ( x ) ∈ Meas +cmp ( O ) . (27) If H = har ( O ) (respectively, H = sbh ( O ) ), r ∈ R + \ , and a measure ϑ ∈ Meas +cmp ( r B ) is an Arens – Singer (respectively, a Jensen) measure on r B at ∈ r B , then, under condi-tion (25) , the convolution ω ∗ ϑ (26) ∈ Meas +cmp ( O ) is har ( O ) (respectively, sbh ( O ) )-balayageof ω , i.e., ω (cid:22) har ( O ) (cid:16) respectively, (cid:22) sbh ( O ) (cid:17) ω ∗ ϑ ∈ Meas +cmp ( O ) . (28) Proof.
Under conditions (21)–(22), by definition (23) and by Definition 2 for H -balayage δ x (cid:22) H ϑ x , for each function h ∈ H ⊂ usc ( O ), we have (cid:90) h d ω = (cid:90) (cid:90) h d δ x d ω ≤ (cid:90) S (cid:18)(cid:90) supp ϑ x h d ϑ x (cid:19) d ω ( x ) (23f) = (cid:90) h d (cid:90) Θ d ω for each h ∈ H .(29)By Definition 2, the latter means (27). By definition (26) of convolution ω ∗ ϑ , the finalpart of Theorem 2 with formula (28) is a special case of the proved part (27) of Theorem2. Remind that a set E ⊂ R d is polar if there is u ∈ sbh ∗ ( R d ) such that E ⊂ (cid:8) x ∈ R d : u ( x ) = −∞ (cid:9) , or, in equivalent form, Cap ∗ E = 0 if we use the outer capacity Cap ∗ ( E ) := inf E ⊂ O (cid:48) open = int O (cid:48) sup C closed= clos C compact (cid:98) O ν ∈ Meas ( C ) k − d − (cid:18)(cid:90) (cid:90) K d − ( x, y ) d ν ( x ) d ν ( y ) (cid:19) . (30) Theorem 3.
If a measure ω ∈ Meas +cmp ( O ) is sbh ( O ) -balayage of a measure δ ∈ Meas +cmp ( O ) ,i.e., ω (cid:22) sbh ( O ) δ , and E ⊂ R d is polar, i.e., Cap ∗ E (30) = 0 , then ω ( O ∩ E \ supp δ ) = 0 . Remark 1.
A special case of this Theorem 3 is noted in [6, Corollary 1.8] for a Jensenmeasure ω ∈ J x ( O ) on O at x ∈ O and the Dirac measure δ := δ x . It was used in [16,Lemma 3.1]. 10 roof. There is k ∈ N such that B ( x, /k ) (cid:98) O for all x ∈ supp δ . For any k ∈ k + N there exists an finite cover of supp δ by balls B ( x j , /k ) (cid:98) O such that the open subsets O k := (cid:91) j B ( x j , /k ) (cid:98) O, supp δ (cid:98) O k ⊃ O k +1 , k ∈ k + N , supp δ = (cid:92) k ∈ k + N O k , have complements R d ∞ \ O k in R d ∞ without isolated points. Then every open set O k (cid:98) O is regular for the Dirichlet problem. It suffices to prove that the equality ω ( O k ∩ E ) = 0holds for every number k ∈ k + N . By definition of polar sets, there is a function u ∈ sbh ∗ ( O ) such that u ( E ) = {−∞} . Consider the functions U k = (cid:40) u on O \ O k , the harmonic extension of u from ∂O k into O k on O k , k ∈ k + N . (31)We have U k ∈ sbh ∗ ( O ), and U k is bounded from below in supp δ (cid:98) O k . Hence − ∞ < (cid:90) O U k d δ (4) ≤ (cid:90) O U k d ω = (cid:18)(cid:90) O \ ( O k ∩ E ) + (cid:90) O k ∩ E (cid:19) U k d ω = (cid:90) O \ ( O k ∩ E ) U k d ω + ( −∞ ) · ω ( O k ∩ E ) ≤ ω ( O ) sup supp ω U k + ( −∞ ) · ω ( O k ∩ E ) . Thus, we have ω ( O k ∩ E ) = 0.Generally speaking, Theorem 3 is not true for har ( O )-balayage. An implicit exampleis built in [23, Example]. We get in Example 5 another already constructive way to buildsuch examples. Example 3 (development of one example of T. Lyons [20, XIB2]) . Let λ be the Lebesguemeasure on R d , and let b be the volume of the unit ball B ⊂ R d . Consider O := B , < t < r < , δ := 1 b t d λ (cid:12)(cid:12) t B , ω := 1 b r d λ (cid:12)(cid:12) r B . (32)Easy to see that δ (cid:22) sbh ( B ) ω . Let E = ( e j ) j ∈ N (cid:98) r B \ t B be a polar countable set withoutlimit point in r B \ t B . Surround each point e j ∈ E with a ball B ( e j , r j ) of such a smallradius r j > r B \ t B . Consider a measure µ E := ω − b r d (cid:88) j ∈ N λ (cid:12)(cid:12) B ( e j ,r j ) + 1 b r d (cid:88) λ ( e j , r j ) δ e j (32) = 1 b r d λ (cid:12)(cid:12) r B − b r d (cid:88) j ∈ N λ (cid:12)(cid:12) B ( e j ,r j ) + 1 r d (cid:88) j ∈ N r d j δ e j . By construction, the measure µ E is har ( B )-balayage of measure δ , but µ E ( E ) = 1 r d (cid:88) j ∈ N r d j > Balayage for three measures
Proposition 7.
Let ω ∈ Meas cmp ( O ) and δ ∈ Meas cmp ( O ) .If ω is sbh ( O ) -balayage of δ , then (cid:90) u d δ ≤ (cid:90) u d ω for each u ∈ sbh ( S O ) , where S O = in-fill O S , S := supp ω ∪ supp δ, (33) i. e., if O (cid:48) ⊃ S O is an open subset in R d , then ω is sbh ( O (cid:48) ) -balayage of δ .If ω is har ( O ) -balayage of δ , then (cid:90) h d δ = (cid:90) h d ω for each h ∈ har ( S O ) , (34) i. e., if O (cid:48) ⊃ S O is an open subset in R d , then ω is har ( O (cid:48) ) -balayage of δ .Proof. If u ∈ sbh ( S O ), then, by Proposition 5, there is a function U ∈ sbh (cid:0) O ) such that u = U on S O , and, in the case δ (cid:22) sbh ( O ) ω , we have (cid:90) S O u d δ = (cid:90) S O U d δ = (cid:90) O U d δ (4) ≤ (cid:90) O U d ω = (cid:90) S O U d ω = (cid:90) S O u d ω, that gives (33). If δ (cid:22) har ( O ) ω , then we can repeat (19) using Proposition 3, and we obtain(34).Very special cases of the following Theorem 4 were essentially used in [2, Proposition3] only for special Jensen measures on the complex plane on the complex plane identifiedwith R . Theorem 4.
Suppose that measures β, δ, ω ∈ Meas +cmp ( O ) satisfy the conditions (cid:40) β (cid:22) har ( O ) δ,β (cid:22) sbh ( O ) ω, and in-fill(supp β ∪ supp δ ) ⊂ O (cid:48) , (35) where O (cid:48) (cid:98) O is an open subset such that O (cid:48) ∩ supp ω = ∅ . Then δ (cid:22) sbh ( O ) ω .Proof. It suffices to consider the case when D := O and D (cid:48) := O (cid:48) are domains. Thereexists a regular (for the Dirichlet problem) domain D (cid:48)(cid:48) such thatin-fill(supp β ∪ supp δ ) (35) ⊂ D (cid:48)(cid:48) (cid:98) D (cid:48) ⊂ D (36)since in-fill(supp β ∪ supp δ ) is compact subset in D (cid:48) by Proposition 1(i).Let u ∈ sbh ∗ ( D ). Then we can build a new subharmonic function (cid:101) u ∈ sbh ∗ ( D ) suchthat (cid:101) u (cid:12)(cid:12) D (cid:48)(cid:48) ∈ har ( D (cid:48)(cid:48) ) , (cid:101) u = u on D \ D (cid:48)(cid:48) , u ≤ (cid:101) u on D. (37)12y Proposition 7, in view of the inclusion in (36), we have (cid:90) D u d δ (35) = (cid:90) D (cid:48)(cid:48) u d δ (37) ≤ (cid:90) D (cid:48)(cid:48) (cid:101) u d δ (36) , (34) = (cid:90) D (cid:48)(cid:48) (cid:101) u d β = (cid:90) D (cid:101) u d β (35) ≤ (cid:90) D (cid:101) u d ω. (38)Since supp ω ⊂ D \ D (cid:48) , we can continue this chain of (in)equalities (38) as (cid:90) D u d δ (38) ≤ (cid:90) D (cid:101) u d ω = (cid:90) D \ D (cid:48) (cid:101) u d ω (37) = (cid:90) D \ D (cid:48) u d ω = (cid:90) D u d ω. This completes the proof.This research was supported by a grant of the Russian Science Foundation (ProjectNo. 18-11-00002).
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