6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Paweł Sobolewski

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

Sμ (eit)=∫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 Sμ 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

Gμ (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

|Gμ(z)-Gμ(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.

  1. Hasi Wulan, Jun Yang and Kehe Zhu: Balayage for the Bergman spaceComplex Var. Elliptic Equ., 59 (2014), no. 12, 1775 -1782.
  2. Someone: ABC’s of Operator Theory. Publishing House, 2016.
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