Carleson measures on a homogeneous tree

We introduce the notion of s-Carleson measure (s≥1) on a homogeneous tree T and give several characterizations of such measures. In particular, we prove the following discrete version of the extension of Carleson's theorem due to Duren.For p>1and s≥1, a finite measure σon Tis s-Carleson if and only if there exists C>0such that for all f∈Lp(∂T), ‖Pf‖Lsp(σ)≤C‖f‖Lp(∂T),where Pfdenotes the Poisson integral of f.Here, Lp(σ) is the space of functions g defined on T such that |g|p is integrable with respect to σ and Lp(∂T) is the space of functions f defined on the boundary of T such that |f|p is integrable with respect to the representing measure of the harmonic function 1.