**On balayage and B-balayage**

Let D and T denote the open unit disk and the unit circle, respectively, in the complex plane C. For a finite positive Borel measure μ on D the function

S_{μ} (e^{it})=∫_{D }^{(1-|z|2)}⁄_{|1-ze-it|2} dμ(z),

is called the balayage (or sweep) of μ. One of the most important results about the balayage states that if μ is a Carleson measure for the Hardy spaces, then S_{μ} belongs to BMO(T). Recently, an analogue of the balayage in the context of Bergman space, the so-called B-balayage, was introduced by Hasi Wulan, Jun Yang and Kehe Zhu [1]. For a finite positive Borel measure μ on D, the B-balayage of μ is given by

G_{μ} (z)=∫_{D} ^{(1-|a|2)2}⁄_{|1-āz|4} dμ(a), z ∈ D.

In [1] the authors prove that if μ is Carleson measure for the Bergman space, then

|G_{μ}(z)-G_{μ}(w)| ≤ β(z,w),

where β is the hyperbolic metric on D. In the talk we present some extensions of these results on balayage and B-balayage operators.

The talk is based on joint work with Maria Nowak.

- Hasi Wulan, Jun Yang and Kehe Zhu: Balayage for the Bergman space
*Complex Var. Elliptic Equ*., 59 (2014), no. 12, 1775 -1782. - Someone:
*ABCs of Operator Theory*. Publishing House, 2016.