On balayage and B-balayage operators

Here we consider the balayage operator in the setting of $H^p$ spaces and its Bergman space version (B-balayage) introduced by H. Wulan, J. Yang and K. Zhu \cite{WYZ}, and extend some known results on these operators.


Introduction
Let D denote the unit disk {z : C : |z| < 1} and T the unit circle.For 0 < p < ∞, the Hardy space H p consists of all functions f which are holomorphic on D and satisfy It is known that each function f ∈ H p has the radial limit f (e it ) = lim r→1 − f (re it ) a.e. on T and f (e it ) ∈ L p (T).
For a finite positive Borel measure µ on D, the function is called the balayage of µ.It follows from Fubini's theorem that S µ (e it ) ∈ L 1 (T) (see, [3, p. 229]).
If I ⊂ is an arc of T, the Carleson square S(I) is defined as 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| s , for any arc I ⊂ T.
A 1-Carleson measure is simply called a Carleson measure.In [1] Carleson proved that if µ is a positive Borel measure in D, then for 0 < p < ∞, H p ⊂ L p (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 µ belongs to BMO(T).However, the Carleson property of measure µ is not a necessary condition for S µ being a BMO(T) function ( [5]).
In the next section we obtain an extension of the above mentioned result.More exactly, we prove that if µ is an s-Carleson measure, 0 ≤ s < 1, then S µ belongs to L 1,s .
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 It has been proved in [6] that if µ is a 2-Carleson measure, then there exists a constant where β is the hyperbolic metric on D. Here, applying a similar idea to that used in the proof of this result, we prove Theorem 1. Assume that 1 < p < ∞ and µ is a positive Borel measure on D. If µ is a 2p-Carleson measure, then there exists a positive constant for all z, w ∈ D.
Here C will denote a positive constant which can vary from line to line.

Balayage operators and Campanato spaces L 1,s
We start with the following Theorem 2. If µ is an s-Carleson measure, 0 < s ≤ 1, S µ is given by (1) and 0 ≤ γ < 1, then there exists a positive constant C such that for any Proof.Without loss of generality we can assume that |I| < 1.
Let for z ∈ D and θ ∈ R be the Poisson kernel for the disk D. By the Fubini theorem, For a subarc I of T let 2 n I, n ∈ N, denote the subarc of T with the same center as I and the length 2 n |I|.
In case (i) we have So, if e iθ , e iϕ ∈ I, then Now we turn to case (ii).Then for e iψ ∈ I, Consequently, for e iθ , e iϕ ∈ I, we get Now we put Q n = S(2 n I), n = 1, 2, . . .Then by ( 4) and ( 5), The above inequality and (3) imply The next theorem shows that if µ is an s-Carleson measure, 0 < s ≤ 1, then S µ is in the Campanato space L 1,s .Theorem 3. If µ is an s-Carleson measure on D, 0 < s ≤ 1 and S µ (t) = S µ (e it ) is the balayage operator of µ given by (1), then there exists a positive constant C such that for any and the inequality follows from Theorem 2 with γ = 0.

B-balayage for weighted Bergman spaces A p α
Recall that for 0 < p < ∞, −1 < α < ∞, the weighted Bergman space A p α is the space of all holomorphic functions in L p (D, dA α ), where and dA is the normalized Lebesgue measure on D, that is D dA = 1.If f is in L p (D, dA α ), we write It is well known that for 1 < p < ∞ the Bergman projection P α given by  for all f ∈ A p α .The next corollary is an immediate consequence of Proposition.
Corollary.[6] For α > −1, σ > 0, let µ, ν be positive Borel measures on D such that Then µ is an A p α -Carleson measure if and only if ν is an A p α+σ -Carleson measure.Recall that for 1 < p < ∞, the Besov space B p is the space of all functions f analytic on D and such that where is the Möbius invariant measure on D.
We will use the fact that the Besov space B p = P α (L p , dτ ).The proof of this equality for α = 0 is given in [8, p. 90-92] and similar arguments can be used for α > −1.In particular, if f = P α g, where g ∈ L p (dτ ), then It then follows from [4, Theorem 1.9] that (6) f Bp ≤ C p,α g L p (dτ ) .
The next theorem gives a Lipschitz type estimate for functions in the analytic Besov space.
Theorem 5. [9] For any 1 < p < ∞, there exists a constant C p > 0 such that for all f ∈ B p and z, w ∈ D, where 1 p + 1 q = 1.Proof of Theorem 1.For z, w we have Since µ is a finite measure on D, the Jensen inequality yields Let q > 1 be the conjugate index for p, that is, 1 p + 1 q = 1.Then 2p = 2+ 2 q−1 and under the assumption, µ is an A r Put g = (α + 1) α+2 q f and observe that f q,α ≤ 1 if and only if g L q (dτ ) ≤ 1.Moreover, since α = 2 q−1 satisfies α+2 q = α, we get sup where the last inequality follows from Theorem 5 and inequality (6).

log 1 +
is a bounded operator from L p (D, dA α ) onto A p α .Let for z, w ∈ D, the function ϕ z (w) = z − w 1 − zw denote the automorphism of the unit disk D. The hyperbolic metric on D is given byβ(z, w) = 1 2 |ϕ z (w)| 1 − |ϕ z (w)| .For z ∈ D and r > 0 the hyperbolic disk with center z and radius r isD(z, r) = {w ∈ D : β(z, w) < r}.For s > 1 the condition for an s-Carleson measure given in Introduction is equivalent to the condition where Carleson squares are replaced by hyperbolic disks.More exactly, we have the following Proposition.[10, 2] 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 (ii) µ(D(z, r)) ≤ C(1−|z| 2 ) s for some constant C depending only on r for all hyperbolic disk D(z, r), z ∈ D.For α > −1, (α + 2) measures are characterized by the following result.