In this paper, we give a new characterization of Carleson measures for the generalized Bergman spaces. We show first that this problem is equivalent to a T(1)-type problem. Using an idea of Verdera (see [V]), we introduce a sort of curvature in the unit ball adapted to our kernel. We establish a good λ inequality which then yields the solution of this T(1)-type problem.

Two overlapping Schwarz algorithms are developed for a discontinuous Galerkin finite element approximation of second order scalar elliptic problems in both two and three dimensions. The discontinuous Galerkin formulation is based on a staggered discretization introduced by Chung and Engquist [<i>SIAM J. Numer. Anal.</i>, 47 (2009), pp. 3820--3848] for the acoustic wave equation. Two types of coarse problems are introduced for the two-level Schwarz algorithms. The first is built on a nonoverlapping subdomain partition, which allows quite general subdomain partitions, and the second on introducing an additional coarse triangulation that can also be quite independent of the fine triangulation. Condition number bounds are established and numerical results are presented.

Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a selfsimilar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009) [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.

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