Maximal estimate and integral operators in Bergman spaces with doubling measure

The boundedness of the maximal operator on the upper half-plane Π + \Pi ^{+} is established. Here Π + \Pi ^+ is equipped with a positive Borel measure d ω ( y ) d x d\omega (y)dx satisfying the doubling property ω ( ( 0 , 2 t ) ) ≤ C ω ( ( 0 , t ) ) \omega ((0,2t))\leq C\omega ((0,t)) . This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces A ω p ( Π + ) A_{\omega }^{p}(\Pi ^{+}) .