Carleson measures for the Drury–Arveson Hardy space and other Besov–Sobolev spaces on complex balls

For 0⩽σ<1/20⩽σ<1/2 we characterize Carleson measures μ   for the analytic Besov–Sobolev spaces B2σ on the unit ball BnBn in CnCn by the discrete tree condition∑β⩾α[2σd(β)I*μ(β)]2⩽CI*μ(α)<∞,α∈Tn, on the associated Bergman tree TnTn. Combined with recent results about interpolating sequences this leads, for this range of σ  , to a characterization of universal interpolating sequences for B2σ and also for its multiplier algebra.However, the tree condition is not   necessary for a measure to be a Carleson measure for the Drury–Arveson Hardy space Hn2=B21/2. We show that μ   is a Carleson measure for B21/2 if and only if both the simple condition2d(α)I*μ(α)⩽C,α∈Tn, and the split tree condition∑k⩾0∑γ⩾α2d(γ)−k∑(δ,δ′)∈G(k)(γ)I*μ(δ)I*μ(δ′)⩽CI*μ(α),α∈Tn, hold. This gives a sharp estimate for Drury's generalization of von Neumann's operator inequality to the complex ball, and also provides a universal characterization of Carleson measures, up to dimensional constants, for Hilbert spaces with a complete continuous Nevanlinna–Pick kernel function.We give a detailed analysis of the split tree condition for measures supported on embedded two manifolds and we find that in some generic cases the condition simplifies. We also establish a connection between function spaces on embedded two manifolds and Hardy spaces of plane domains.