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On Balayaga and B-Balayage Operators

On Balayaga and B-Balayage Operators Here, we consider the balayage operator in the setting of H spaces and its Bergman space version (B-balayage) introduced by Wulan et al. (Complex Var Ellipt Equ 59(12):1775–1782, 2014), and extend some known results on these operators. Keywords Carleson measure · Balayage · BMO · Bergman spaces · Analytic Besov spaces Mathematics Subject Classification 30H25 · 30H35 1 Introduction Let D denote the unit disk {z : C :|z| < 1} and T the unit circle. For 0 < p < ∞,the Hardy space H consists of all functions f which are holomorphic on D and satisfy 2π it p f  = sup | f (re )| dt < ∞. 2π 0<r <1 0 p it it It is known that each function f ∈ H has the radial limit f (e ) = lim − f (re ) r →1 it p a.e. on T and f (e ) ∈ L (T). Communicated by Ilpo Laine. B Maria Nowak mt.nowak@poczta.umcs.lublin.pl Paweł Sobolewski pawel.sobolewski@umcs.eu Instytut Matematyki UMCS, Pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland 123 510 M. Nowak, P. Sobolewski For φ ∈ L (T), we say that φ ∈ BMO(T) if it φ = sup |φ(e ) − φ |dt < ∞, ∗ I |I | I ⊂T I where I denotes any arc of T, |I | is its arc length and it φ = φ(e )dt . |I | p,λ In [7], the authors have recently considered Campanato spaces L (T) defined p,λ as follows. For λ ≥ 0 and 1 ≤ p < ∞, the space L (T) consists of all functions φ ∈ L (T) for which it p sup |φ(e ) − φ | dt < ∞. |I | I ⊂T I p,1 We note that BMO(T) = L , 1 ≤ p < ∞, (see [3, pp. 222-235]). For a finite positive Borel measure μ on D, the function 1 −|z| it S (e ) = dμ(z), (1) −it 2 |1 − ze | it 1 is called the balayage of μ. It follows from Fubini’s theorem that S (e ) ∈ L (T) (see [3, p. 229]). If I is an arc of T, the Carleson square S(I ) is defined as |I | it it S(I ) = re : e ∈ I , 1 − ≤ r < 1 . 2π A positive Borel measure μ is called an s-Carleson measure, 0 < s < ∞, if there exists a positive constant C = C (μ), such that μ(S(I )) ≤ C (μ)|I | , for any arc I ⊂ T. A 1-Carleson measure is simply called a Carleson measure. In [1], Carleson proved p p that if μ is a positive Borel measure in D, then, for 0 < p < ∞, H ⊂ L (dμ) if and only if μ is a Carleson measure. It has been proved in [3, p. 229] that if μ is the Carleson measure, then S belongs to BMO(T). However, the Carleson property of measure μ is not a necessary condition for S being a BMO(T) function [5]. In the next section, we obtain an extension of the result mentioned above. More precisely, we prove that if μ is an s-Carleson measure, 0 < s ≤ 1, then S belongs to 1,s L . 123 On Balayaga and B-Balayage Operators 511 In [6], H. Wulan, J. Yang, and K. Zhu introduced the Bergman space version of the balayage operator on the unit disk that was called B-balayage. The B-balayage of a finite complex measure μ on D is given by 2 2 (1 −|w| ) G (z) = dμ(w), z ∈ D. |1 −¯ zw| It has been proved in [6] that if μ is a 2-Carleson measure, then there exists a constant C > 0, such that |G (z) − G (w)|≤ Cβ(z,w), z,w ∈ D, (2) μ μ where β is the hyperbolic metric on D. Here, applying a similar idea to that used in the proof of this result, we prove the following theorem. Theorem 1 Assume that 1 < p < ∞ and μ is a positive Borel measure on D.If μ is a 2 p-Carleson measure, then there exists a positive constant C = C ( p), such that |G (z) − G (w)|≤ C (β(z,w)) μ μ for all z,w ∈ D. Actually, this theorem is a special case of a more general theorem stated in Sect. 3. Here, C will denote a positive constant which can vary from line to line. 1,s 2 Balayage Operators and Campanato Spaces L We start with the following result. Theorem 2 If μ is an s-Carleson measure, 0 < s ≤ 1,S is given by (1) and 0 ≤ γ< 1, then there exists a positive constant C, such that for any I ⊂ T: i θ i ϕ 1 |S (e ) − S (e )| μ μ dθ dϕ ≤ C . 1+s−γ i θ i ϕ γ |I | |e − e | I I Proof Without loss of generality, we can assume that |I | < 1. Let, for z ∈ D and θ ∈ R: 2 −i θ 1 −|z| 1 + ze P (θ ) = = Re −i θ 2 −i θ |1 − ze | 1 − ze be the Poisson kernel for the disk D. By the Fubini theorem: i θ i ϕ |S (e ) − S (e )| | P (θ ) − P (ϕ)| μ μ z z dθ dϕ ≤ dμ(z) dθ dϕ i θ i ϕ γ i θ i ϕ γ |e − e | |e − e | I I I I D | P (θ ) − P (ϕ)| z z = dθ dϕ dμ(z). (3) i θ i ϕ γ |e − e | D I I 123 512 M. Nowak, P. Sobolewski For a subarc I of T,let 2 I , n ∈ N denote the subarc of T with the same center as I and the length 2 |I |. In view of the equality 2π P (θ )dθ = 2π, we have P (θ ) dϕ 1−γ dθ dϕ = P (θ ) dθ ≤ C |I | . i θ i ϕ γ i θ i ϕ γ |e − e | |e − e | I I I I Consequently | P (θ ) − P (ϕ)| z z 1−γ 1+s−γ dθ dϕ dμ(z) ≤ 2C |I | dμ(z) ≤ C |I | . i θ i ϕ γ |e − e | S(2I ) I I S(2I ) (4) Since P (θ ) ≤ 4for |z|≤ , we get | P (θ ) − P (ϕ)| dθdϕ z z dθ dϕ dμ(z) ≤ 8μ(D) i θ i ϕ γ i θ i ϕ γ |e − e | |e − e | |z|≤ I I I I 2−γ 1+s−γ ≤ C |I | ≤ C |I | . 1 i ω n+1 n Now, we assume that ≤|z| < 1 and z =|z|e ∈ S(2 I )\S(2 I ). We consider i ω n i ω n+1 n two cases: (i ) e ∈ 2 I and (ii ) e ∈ 2 I \2 I . In case (i ),wehave n n+1 2 |I | 2 |I | < 1 −|z|≤ . 2π 2π Thus (1 −|z| )2|z|| cos(θ − ω) − cos(ϕ − ω)| | P (θ ) − P (ϕ)|= z z ϕ−ω 2 θ −ω 2 2 2 (1 −|z|) + 4|z| sin (1 −|z|) + 4|z| sin 2 2 (θ −ω)+(ϕ−ω) (θ −ϕ) 8| sin || sin | 2 2 (1 −|z|) (|θ − ω|+|ϕ − ω|) |θ − ϕ| ≤ 2 . (1 −|z|) 123 On Balayaga and B-Balayage Operators 513 i θ i ϕ Therefore, if e , e ∈ I , then 1−γ | P (θ ) − P (ϕ)| (|θ − ω|+|ϕ − ω|) |θ − ϕ| z z ≤ C i θ i ϕ γ 3 |e − e | (1 −|z|) n 1−γ −1−γ 2 |I ||I | |I | ≤ C = C . (5) n 3 2n (2 |I |) 2 i ψ Now, we turn to case (ii ). Then, for e ∈ I , n−2 n 2 |I|≤|ψ − ω|≤ 2 |I |. i θ i ϕ Consequently, for e , e ∈ I , we get −i θ 2 −i ϕ 2 |(1 − ze | −|1 − ze | | P (θ ) − P (ϕ)| z z ≤ 2 i θ i ϕ γ i θ i ϕ γ −i θ −i ϕ 2 |e − e | |e − e | |1 − ze ||1 − ze | 1−γ (|θ − ω|+|ϕ − ω|) |θ − ϕ| ≤ C |θ − ω||ϕ − ω| −1−γ |I | ≤ C . (6) 2n Now, we put Q = S(2 I ), n = 1, 2,... Then, by (5) and (6), 1−γ 1+s−γ | P (θ ) − P (ϕ)| |I | |I | z z dθdϕdμ(z) ≤ C dμ(z) ≤ C . Q \Q n+1 n i θ i ϕ γ 2n n(2−s) |e − e | 2 2 I I Q n+1 |z|≥ The above inequality and (4)imply | P (θ ) − P (ϕ)| | P (θ ) − P (ϕ)| z z z z dθ dϕ dμ(z) ≤ dθdϕdμ(z) i θ i ϕ γ i θ i ϕ γ |e − e | |e − e | D I I Q I I | P (θ ) − P (ϕ)| z z + dθdϕdμ(z) i θ i ϕ γ |e − e | Q \Q I I n+1 n n=1 s+1−γ 1+s−γ ≤ C |I | = C |I | . n(2−s) n=1 The next theorem shows that if μ is an s-Carleson measure, 0 < s ≤ 1, then S is 1,s in the Campanato space L . it Theorem 3 If μ is an s-Carleson measure on D, 0 < s ≤ 1 and S (t ) = S (e ) is μ μ the balayage operator of μ given by (1), then there exists a positive constant C, such that for any I ⊂ T 123 514 M. Nowak, P. Sobolewski |S (t ) − (S ) |dt ≤ C . μ μ I |I | Proof It is enough to observe that 1 1 |S (t ) − (S ) |dt ≤ |S (t ) − S (u)|dtdu μ μ I μ μ s s+1 |I | |I | I I I and the inequality follows from Theorem 2 with γ = 0. 3 B-Balayage for Weighted Bergman Spaces A Recall that, for 0 < p < ∞, −1 <α < ∞, the weighted Bergman space A is the space of all holomorphic functions in L (D, d A ), where 2 α d A (z) = (α + 1)(1 −|z| ) d A(z) and d A is the normalized Lebesgue measure on D; that is, d A = 1. If f is in L (D, d A ), we write f  = | f (z)| d A (z). p,α α It is well known that, for 1 < p < ∞, the Bergman projection P given by f (w) P f (z) = d A (w) α α 2+α (1 − zw) ¯ is a bounded operator from L (D, d A ) onto A . α α Let for z,w ∈ D, the function z − w ϕ (w) = 1 −¯ zw denote the automorphism of the unit disk D. The hyperbolic metric on D is given by 1 1 +|ϕ (w)| β(z,w) = log . 2 1 −|ϕ (w)| For z ∈ D and r > 0, the hyperbolic disk with center z and radius r is D(z, r ) ={w ∈ D : β(z,w) < r }. For s > 1, the condition for an s-Carleson measure given in Introduction is equiva- lent to the condition where Carleson squares are replaced by hyperbolic disks. More exactly, the following result is known. 123 On Balayaga and B-Balayage Operators 515 Proposition [2,10] Let μ be a positive Borel measure on D and 1 < s < ∞. Then, the following statements are equivalent (i) μ is an s-Carleson measure, 2 s (ii) μ(D(z, r )) ≤ C (1 −|z| ) for some constant C depending only on r for all hyperbolic disk D(z, r ),z ∈ D. A positive Borel measure μ on D is called an A -Carleson measure if there exists a positive constant C, such that p p | f (z)| dμ(z) ≤ C | f (z)| d A (z) D D for all f ∈ A . It is well known that μ is an A -Carleson measure if and only if μ is (2+α)-Carleson measure (see [10, p. 133]). This means that A -Carleson measures are independent of p. The next corollary is an immediate consequence of the last proposition. Corollary [6] For α> −1,σ > 0,let μ, ν be positive Borel measures on D,such that dν(z) = (1 −|z|) dμ(z). p p Then, μ is an A -Carleson measure if and only if ν is an A -Carleson measure. α+σ Recall that, for 1 < p < ∞, the Besov space B is the space of all functions f analytic on D, such that p 2 p f  = | f (z)| (1 −|z| ) dτ(z)< ∞, where d A(z) dτ(z) = 2 2 (1 −|z| ) is the Möbius invariant measure on D. We will use the fact that the Besov space B = P (L , dτ). The proof of this p α equality is given in [9, p. 119]. Moreover, if f = P g, where g ∈ L (dτ), then g(w)w ¯ 2 2 (1 −|z| ) f (z) = (α + 2)(1 −|z| ) d A (w). 3+α (1 − zw) ¯ It then follows from [4, Thm. 1.9] that f  ≤ C g . (7) B p,α L (dτ) The next theorem gives a Lipschitz type estimate for functions in the analytic Besov space. 123 516 M. Nowak, P. Sobolewski Theorem 4 [8] For any 1 < p < ∞, there exists a constant C > 0, such that | f (z) − f (w)|≤ C  f  (β(z,w)) p B 1 1 for all f ∈ B and z,w ∈ D, where + = 1. p q In [6], the authors also consider a version of the balayage of a measure μ on D defined by 2 2+α (1 −|z| ) G (z) = dμ(w). μ,α 4+2α |1 −¯ zw| They have proved the following generalization of inequality (2). If μ is an A -Carleson measure, then the generalized balayage G satisfies the α μ,α Lipschitz condition: |G (z) − G (w)|≤ Cβ(z,w), z,w ∈ D, μ,α μ,α where C is independent of z and w. It is worth noting here that the balayage given by (1) is in a certain sense a limit case of G as α →−1. Since an A -Carleson measure is actually a (2 + α)-Carleson μ,α α measure, the last inequality gives a necessary condition for a measure μ to be an s-Carleson measure, as 1 < s < ∞. Theorem 1 is a special case of the following more general theorem. Theorem 5 Assume that 1 < p < ∞, −1 <α < ∞, and μ is a positive Borel measure on D.If μ is a p(2 + α)-Carleson measure, then there exists a positive constant C = C ( p,α), such that |G (z) − G (w)|≤ C (β(z,w)) μ,α μ,α for all z,w ∈ D. Proof For z,w,wehave 2 2+α 2 2+α (1 −|a| ) (1 −|a| ) |G (z) − G (w)|≤ − dμ(a) μ,α μ,α 4+2α 4+2α |1 − az ¯| |1 − aw ¯ | 2 2+α 2 2+α (1 −|a| ) (1 −|a| ) ≤ − dμ(a). 4+2α 4+2α (1 − az ¯) (1 − aw) ¯ Since μ is a finite measure on D, the Jensen’s inequality yields 2 2+α 2 2+α (1 −|a| ) (1 −|a| ) |G (z) − G (w)| ≤ C − dμ(a) μ μ 4+2α 4+2α (1 − az ¯) (1 − aw) ¯ 1 1 2 (2+α) p = C − (1 −|a| ) dμ(a). 4+2α 4+2α (1 − az ¯) (1 − aw) ¯ 123 On Balayaga and B-Balayage Operators 517 2 p(2+α) By the Corollary, (1 −|a| ) dμ(a) is an A -Carleson measure, because 2 p(2+α)−2 μ is an A -Carleson measure. Consequently, p(2+α)−2 1 1 2 p(2+α) − (1 −|a| ) dμ(a) 4+2α 4+2α (1 − az ¯) (1 − aw) ¯ 1 1 2 2 p(2+α)−2 ≤ C − (1 −|a| ) d A(a) 4 4 (1 − az ¯) (1 − aw) ¯ 2 2+2α 2 2+2α (1 −|a| ) (1 −|a| ) = C − d A 2 (a), 4+2α 4+2α q−1 (1 −¯ az) (1 −¯ aw) 1 1 where q is the conjugate index for p, that is, + = 1. p q Now, set β = and note that q−1 2 2+2α 2 2+2α (1 −|a| ) (1 −|a| ) − d A (a) 4+2α 4+2α (1 −¯ az) (1 −¯ aw) 2 2+α 2 2+2α (1 −|a| ) (1 −|a| ) = sup − f (a)d A (a) . 4+2α 4+2α (1 −¯ az) (1 −¯ aw) f  ≤1 D q,β β+2 q q Put g = (α+1) (1−|a| ) f and observe that  f  ≤ 1 if and only if g ≤ q,β L (dτ) β+2 1. Moreover, since β = satisfies = β, we get q−1 q 2 2+2α 2 2+2α (1 −|a| ) (1 −|a| ) sup − f (a)d A (a) 4+2α 4+2α (1 −¯ az) (1 −¯ aw) f  ≤1 D q,β 2 2+2α 2 2+2α (1 −|a| ) (1 −|a| ) = C sup − g(a)d A(a) 4+2α 4+2α (1 −¯ az) (1 −¯ aw) g q ≤1 D L (dτ) g(a) g(a) = C sup − d A (a) 2+α 4+2α 4+2α (1 −¯ az) (1 −¯ aw) g q ≤1 D L (dτ) = C sup | P g(z) − P g(w)| ≤ C (β(z,w)) , 2+α 2+α g q ≤1 L (dτ) where the last inequality follows from Theorem 4 and inequality (7). Acknowledgements The authors are grateful to the referee for suggesting Theorem 5. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 123 518 M. Nowak, P. Sobolewski References 1. Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76(2), 547–559 (1962) 2. Duren, P., Schuster, A.: Bergman Spaces. American Mathematical Society, Providence (2004) 3. Garnett, J.B.: Bounded analytic functions. Academic, New York (1981) 4. Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, New York (2000) 5. Pott, S., Volberg, A.: Carleson measure and balayage. Int. Math. Res. Not. IMRN 13, 2427–2436 (2010) 6. Wulan, H., Yang, J., Zhu, K.: Balayage for the Bergman space. Complex Var. Ellipt. Equ. 59(12), 1775–1782 (2014) 7. Xiao, J., Yuan, C.: Analytic Campanato spaces and their compositions. Indiana Univ. Math. J. 64(4), 1001–1025 (2015) 8. Zhu, K.: Analytic Besov spaces. J. Math. Anal. Appl. 157(2), 318–336 (1991) 9. Zhu, K.: Operator theory in function spaces, 2nd edn. AMS, Rhode Island (2007) 10. Zhu, K.: An integral representation for Besov and Lipschitz spaces. J. Aust. Math. Soc. 98(1), 129–144 (2015) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On Balayaga and B-Balayage Operators

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Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
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Abstract

Here, we consider the balayage operator in the setting of H spaces and its Bergman space version (B-balayage) introduced by Wulan et al. (Complex Var Ellipt Equ 59(12):1775–1782, 2014), and extend some known results on these operators. Keywords Carleson measure · Balayage · BMO · Bergman spaces · Analytic Besov spaces Mathematics Subject Classification 30H25 · 30H35 1 Introduction Let D denote the unit disk {z : C :|z| < 1} and T the unit circle. For 0 < p < ∞,the Hardy space H consists of all functions f which are holomorphic on D and satisfy 2π it p f  = sup | f (re )| dt < ∞. 2π 0<r <1 0 p it it It is known that each function f ∈ H has the radial limit f (e ) = lim − f (re ) r →1 it p a.e. on T and f (e ) ∈ L (T). Communicated by Ilpo Laine. B Maria Nowak mt.nowak@poczta.umcs.lublin.pl Paweł Sobolewski pawel.sobolewski@umcs.eu Instytut Matematyki UMCS, Pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland 123 510 M. Nowak, P. Sobolewski For φ ∈ L (T), we say that φ ∈ BMO(T) if it φ = sup |φ(e ) − φ |dt < ∞, ∗ I |I | I ⊂T I where I denotes any arc of T, |I | is its arc length and it φ = φ(e )dt . |I | p,λ In [7], the authors have recently considered Campanato spaces L (T) defined p,λ as follows. For λ ≥ 0 and 1 ≤ p < ∞, the space L (T) consists of all functions φ ∈ L (T) for which it p sup |φ(e ) − φ | dt < ∞. |I | I ⊂T I p,1 We note that BMO(T) = L , 1 ≤ p < ∞, (see [3, pp. 222-235]). For a finite positive Borel measure μ on D, the function 1 −|z| it S (e ) = dμ(z), (1) −it 2 |1 − ze | it 1 is called the balayage of μ. It follows from Fubini’s theorem that S (e ) ∈ L (T) (see [3, p. 229]). If I is an arc of T, the Carleson square S(I ) is defined as |I | it it S(I ) = re : e ∈ I , 1 − ≤ r < 1 . 2π A positive Borel measure μ is called an s-Carleson measure, 0 < s < ∞, if there exists a positive constant C = C (μ), such that μ(S(I )) ≤ C (μ)|I | , for any arc I ⊂ T. A 1-Carleson measure is simply called a Carleson measure. In [1], Carleson proved p p that if μ is a positive Borel measure in D, then, for 0 < p < ∞, H ⊂ L (dμ) if and only if μ is a Carleson measure. It has been proved in [3, p. 229] that if μ is the Carleson measure, then S belongs to BMO(T). However, the Carleson property of measure μ is not a necessary condition for S being a BMO(T) function [5]. In the next section, we obtain an extension of the result mentioned above. More precisely, we prove that if μ is an s-Carleson measure, 0 < s ≤ 1, then S belongs to 1,s L . 123 On Balayaga and B-Balayage Operators 511 In [6], H. Wulan, J. Yang, and K. Zhu introduced the Bergman space version of the balayage operator on the unit disk that was called B-balayage. The B-balayage of a finite complex measure μ on D is given by 2 2 (1 −|w| ) G (z) = dμ(w), z ∈ D. |1 −¯ zw| It has been proved in [6] that if μ is a 2-Carleson measure, then there exists a constant C > 0, such that |G (z) − G (w)|≤ Cβ(z,w), z,w ∈ D, (2) μ μ where β is the hyperbolic metric on D. Here, applying a similar idea to that used in the proof of this result, we prove the following theorem. Theorem 1 Assume that 1 < p < ∞ and μ is a positive Borel measure on D.If μ is a 2 p-Carleson measure, then there exists a positive constant C = C ( p), such that |G (z) − G (w)|≤ C (β(z,w)) μ μ for all z,w ∈ D. Actually, this theorem is a special case of a more general theorem stated in Sect. 3. Here, C will denote a positive constant which can vary from line to line. 1,s 2 Balayage Operators and Campanato Spaces L We start with the following result. Theorem 2 If μ is an s-Carleson measure, 0 < s ≤ 1,S is given by (1) and 0 ≤ γ< 1, then there exists a positive constant C, such that for any I ⊂ T: i θ i ϕ 1 |S (e ) − S (e )| μ μ dθ dϕ ≤ C . 1+s−γ i θ i ϕ γ |I | |e − e | I I Proof Without loss of generality, we can assume that |I | < 1. Let, for z ∈ D and θ ∈ R: 2 −i θ 1 −|z| 1 + ze P (θ ) = = Re −i θ 2 −i θ |1 − ze | 1 − ze be the Poisson kernel for the disk D. By the Fubini theorem: i θ i ϕ |S (e ) − S (e )| | P (θ ) − P (ϕ)| μ μ z z dθ dϕ ≤ dμ(z) dθ dϕ i θ i ϕ γ i θ i ϕ γ |e − e | |e − e | I I I I D | P (θ ) − P (ϕ)| z z = dθ dϕ dμ(z). (3) i θ i ϕ γ |e − e | D I I 123 512 M. Nowak, P. Sobolewski For a subarc I of T,let 2 I , n ∈ N denote the subarc of T with the same center as I and the length 2 |I |. In view of the equality 2π P (θ )dθ = 2π, we have P (θ ) dϕ 1−γ dθ dϕ = P (θ ) dθ ≤ C |I | . i θ i ϕ γ i θ i ϕ γ |e − e | |e − e | I I I I Consequently | P (θ ) − P (ϕ)| z z 1−γ 1+s−γ dθ dϕ dμ(z) ≤ 2C |I | dμ(z) ≤ C |I | . i θ i ϕ γ |e − e | S(2I ) I I S(2I ) (4) Since P (θ ) ≤ 4for |z|≤ , we get | P (θ ) − P (ϕ)| dθdϕ z z dθ dϕ dμ(z) ≤ 8μ(D) i θ i ϕ γ i θ i ϕ γ |e − e | |e − e | |z|≤ I I I I 2−γ 1+s−γ ≤ C |I | ≤ C |I | . 1 i ω n+1 n Now, we assume that ≤|z| < 1 and z =|z|e ∈ S(2 I )\S(2 I ). We consider i ω n i ω n+1 n two cases: (i ) e ∈ 2 I and (ii ) e ∈ 2 I \2 I . In case (i ),wehave n n+1 2 |I | 2 |I | < 1 −|z|≤ . 2π 2π Thus (1 −|z| )2|z|| cos(θ − ω) − cos(ϕ − ω)| | P (θ ) − P (ϕ)|= z z ϕ−ω 2 θ −ω 2 2 2 (1 −|z|) + 4|z| sin (1 −|z|) + 4|z| sin 2 2 (θ −ω)+(ϕ−ω) (θ −ϕ) 8| sin || sin | 2 2 (1 −|z|) (|θ − ω|+|ϕ − ω|) |θ − ϕ| ≤ 2 . (1 −|z|) 123 On Balayaga and B-Balayage Operators 513 i θ i ϕ Therefore, if e , e ∈ I , then 1−γ | P (θ ) − P (ϕ)| (|θ − ω|+|ϕ − ω|) |θ − ϕ| z z ≤ C i θ i ϕ γ 3 |e − e | (1 −|z|) n 1−γ −1−γ 2 |I ||I | |I | ≤ C = C . (5) n 3 2n (2 |I |) 2 i ψ Now, we turn to case (ii ). Then, for e ∈ I , n−2 n 2 |I|≤|ψ − ω|≤ 2 |I |. i θ i ϕ Consequently, for e , e ∈ I , we get −i θ 2 −i ϕ 2 |(1 − ze | −|1 − ze | | P (θ ) − P (ϕ)| z z ≤ 2 i θ i ϕ γ i θ i ϕ γ −i θ −i ϕ 2 |e − e | |e − e | |1 − ze ||1 − ze | 1−γ (|θ − ω|+|ϕ − ω|) |θ − ϕ| ≤ C |θ − ω||ϕ − ω| −1−γ |I | ≤ C . (6) 2n Now, we put Q = S(2 I ), n = 1, 2,... Then, by (5) and (6), 1−γ 1+s−γ | P (θ ) − P (ϕ)| |I | |I | z z dθdϕdμ(z) ≤ C dμ(z) ≤ C . Q \Q n+1 n i θ i ϕ γ 2n n(2−s) |e − e | 2 2 I I Q n+1 |z|≥ The above inequality and (4)imply | P (θ ) − P (ϕ)| | P (θ ) − P (ϕ)| z z z z dθ dϕ dμ(z) ≤ dθdϕdμ(z) i θ i ϕ γ i θ i ϕ γ |e − e | |e − e | D I I Q I I | P (θ ) − P (ϕ)| z z + dθdϕdμ(z) i θ i ϕ γ |e − e | Q \Q I I n+1 n n=1 s+1−γ 1+s−γ ≤ C |I | = C |I | . n(2−s) n=1 The next theorem shows that if μ is an s-Carleson measure, 0 < s ≤ 1, then S is 1,s in the Campanato space L . it Theorem 3 If μ is an s-Carleson measure on D, 0 < s ≤ 1 and S (t ) = S (e ) is μ μ the balayage operator of μ given by (1), then there exists a positive constant C, such that for any I ⊂ T 123 514 M. Nowak, P. Sobolewski |S (t ) − (S ) |dt ≤ C . μ μ I |I | Proof It is enough to observe that 1 1 |S (t ) − (S ) |dt ≤ |S (t ) − S (u)|dtdu μ μ I μ μ s s+1 |I | |I | I I I and the inequality follows from Theorem 2 with γ = 0. 3 B-Balayage for Weighted Bergman Spaces A Recall that, for 0 < p < ∞, −1 <α < ∞, the weighted Bergman space A is the space of all holomorphic functions in L (D, d A ), where 2 α d A (z) = (α + 1)(1 −|z| ) d A(z) and d A is the normalized Lebesgue measure on D; that is, d A = 1. If f is in L (D, d A ), we write f  = | f (z)| d A (z). p,α α It is well known that, for 1 < p < ∞, the Bergman projection P given by f (w) P f (z) = d A (w) α α 2+α (1 − zw) ¯ is a bounded operator from L (D, d A ) onto A . α α Let for z,w ∈ D, the function z − w ϕ (w) = 1 −¯ zw denote the automorphism of the unit disk D. The hyperbolic metric on D is given by 1 1 +|ϕ (w)| β(z,w) = log . 2 1 −|ϕ (w)| For z ∈ D and r > 0, the hyperbolic disk with center z and radius r is D(z, r ) ={w ∈ D : β(z,w) < r }. For s > 1, the condition for an s-Carleson measure given in Introduction is equiva- lent to the condition where Carleson squares are replaced by hyperbolic disks. More exactly, the following result is known. 123 On Balayaga and B-Balayage Operators 515 Proposition [2,10] Let μ be a positive Borel measure on D and 1 < s < ∞. Then, the following statements are equivalent (i) μ is an s-Carleson measure, 2 s (ii) μ(D(z, r )) ≤ C (1 −|z| ) for some constant C depending only on r for all hyperbolic disk D(z, r ),z ∈ D. A positive Borel measure μ on D is called an A -Carleson measure if there exists a positive constant C, such that p p | f (z)| dμ(z) ≤ C | f (z)| d A (z) D D for all f ∈ A . It is well known that μ is an A -Carleson measure if and only if μ is (2+α)-Carleson measure (see [10, p. 133]). This means that A -Carleson measures are independent of p. The next corollary is an immediate consequence of the last proposition. Corollary [6] For α> −1,σ > 0,let μ, ν be positive Borel measures on D,such that dν(z) = (1 −|z|) dμ(z). p p Then, μ is an A -Carleson measure if and only if ν is an A -Carleson measure. α+σ Recall that, for 1 < p < ∞, the Besov space B is the space of all functions f analytic on D, such that p 2 p f  = | f (z)| (1 −|z| ) dτ(z)< ∞, where d A(z) dτ(z) = 2 2 (1 −|z| ) is the Möbius invariant measure on D. We will use the fact that the Besov space B = P (L , dτ). The proof of this p α equality is given in [9, p. 119]. Moreover, if f = P g, where g ∈ L (dτ), then g(w)w ¯ 2 2 (1 −|z| ) f (z) = (α + 2)(1 −|z| ) d A (w). 3+α (1 − zw) ¯ It then follows from [4, Thm. 1.9] that f  ≤ C g . (7) B p,α L (dτ) The next theorem gives a Lipschitz type estimate for functions in the analytic Besov space. 123 516 M. Nowak, P. Sobolewski Theorem 4 [8] For any 1 < p < ∞, there exists a constant C > 0, such that | f (z) − f (w)|≤ C  f  (β(z,w)) p B 1 1 for all f ∈ B and z,w ∈ D, where + = 1. p q In [6], the authors also consider a version of the balayage of a measure μ on D defined by 2 2+α (1 −|z| ) G (z) = dμ(w). μ,α 4+2α |1 −¯ zw| They have proved the following generalization of inequality (2). If μ is an A -Carleson measure, then the generalized balayage G satisfies the α μ,α Lipschitz condition: |G (z) − G (w)|≤ Cβ(z,w), z,w ∈ D, μ,α μ,α where C is independent of z and w. It is worth noting here that the balayage given by (1) is in a certain sense a limit case of G as α →−1. Since an A -Carleson measure is actually a (2 + α)-Carleson μ,α α measure, the last inequality gives a necessary condition for a measure μ to be an s-Carleson measure, as 1 < s < ∞. Theorem 1 is a special case of the following more general theorem. Theorem 5 Assume that 1 < p < ∞, −1 <α < ∞, and μ is a positive Borel measure on D.If μ is a p(2 + α)-Carleson measure, then there exists a positive constant C = C ( p,α), such that |G (z) − G (w)|≤ C (β(z,w)) μ,α μ,α for all z,w ∈ D. Proof For z,w,wehave 2 2+α 2 2+α (1 −|a| ) (1 −|a| ) |G (z) − G (w)|≤ − dμ(a) μ,α μ,α 4+2α 4+2α |1 − az ¯| |1 − aw ¯ | 2 2+α 2 2+α (1 −|a| ) (1 −|a| ) ≤ − dμ(a). 4+2α 4+2α (1 − az ¯) (1 − aw) ¯ Since μ is a finite measure on D, the Jensen’s inequality yields 2 2+α 2 2+α (1 −|a| ) (1 −|a| ) |G (z) − G (w)| ≤ C − dμ(a) μ μ 4+2α 4+2α (1 − az ¯) (1 − aw) ¯ 1 1 2 (2+α) p = C − (1 −|a| ) dμ(a). 4+2α 4+2α (1 − az ¯) (1 − aw) ¯ 123 On Balayaga and B-Balayage Operators 517 2 p(2+α) By the Corollary, (1 −|a| ) dμ(a) is an A -Carleson measure, because 2 p(2+α)−2 μ is an A -Carleson measure. Consequently, p(2+α)−2 1 1 2 p(2+α) − (1 −|a| ) dμ(a) 4+2α 4+2α (1 − az ¯) (1 − aw) ¯ 1 1 2 2 p(2+α)−2 ≤ C − (1 −|a| ) d A(a) 4 4 (1 − az ¯) (1 − aw) ¯ 2 2+2α 2 2+2α (1 −|a| ) (1 −|a| ) = C − d A 2 (a), 4+2α 4+2α q−1 (1 −¯ az) (1 −¯ aw) 1 1 where q is the conjugate index for p, that is, + = 1. p q Now, set β = and note that q−1 2 2+2α 2 2+2α (1 −|a| ) (1 −|a| ) − d A (a) 4+2α 4+2α (1 −¯ az) (1 −¯ aw) 2 2+α 2 2+2α (1 −|a| ) (1 −|a| ) = sup − f (a)d A (a) . 4+2α 4+2α (1 −¯ az) (1 −¯ aw) f  ≤1 D q,β β+2 q q Put g = (α+1) (1−|a| ) f and observe that  f  ≤ 1 if and only if g ≤ q,β L (dτ) β+2 1. Moreover, since β = satisfies = β, we get q−1 q 2 2+2α 2 2+2α (1 −|a| ) (1 −|a| ) sup − f (a)d A (a) 4+2α 4+2α (1 −¯ az) (1 −¯ aw) f  ≤1 D q,β 2 2+2α 2 2+2α (1 −|a| ) (1 −|a| ) = C sup − g(a)d A(a) 4+2α 4+2α (1 −¯ az) (1 −¯ aw) g q ≤1 D L (dτ) g(a) g(a) = C sup − d A (a) 2+α 4+2α 4+2α (1 −¯ az) (1 −¯ aw) g q ≤1 D L (dτ) = C sup | P g(z) − P g(w)| ≤ C (β(z,w)) , 2+α 2+α g q ≤1 L (dτ) where the last inequality follows from Theorem 4 and inequality (7). Acknowledgements The authors are grateful to the referee for suggesting Theorem 5. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 123 518 M. Nowak, P. Sobolewski References 1. 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Soc. 98(1), 129–144 (2015) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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