Carleson measures for the Dirichlet space on the polydisc

The Dirichlet space on the polydisc consists of analytic functions defined on the cartesian product of n-copies of a disc, having finite Sobolev norm. In the one-dimensional case (d = 1) the Carleson measures were first described by Stegenga (’80) in terms of capacity, further development was achieved in papers by Arcozzi, Rochberg, Sawyer, Wick and others. Following Arcozzi et al. we consider the equivalent problem in the discrete setting - characterization of trace measures for the Hardy operator on the polytree. For d = 2 we present a description of such measures in terms of bilogarithmic capacity (which, in turn, gives the description of Carleson measures for the Dirichlet space on the bidisc in the sense of Stegenga). We also discuss some arising combinatorial problems. This talk is based on joint work with N. Arcozzi, K.-M. Perfekt, G. Sarfatti.