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. > 1 and s ≥ 1 , a finite measure σ on T is s -Carleson if and only if there exists C > 0 such that for all f ∈ L p ( ∂ T ) , ‖ P f ‖ L s p ( σ ) ≤ C ‖ f ‖ L p ( ∂ T ) , where P f denotes the Poisson integral of f . L p ( σ ) is the space of functions g defined on T such that | g | p is integrable with respect to σ and L p ( ∂ 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.