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

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For 0 ⩽ σ < 1 / 2 we characterize Carleson measures μ for the analytic Besov–Sobolev spaces B 2 σ on the unit ball B n in C n by the discrete tree condition ∑ β ⩾ α [ 2 σ d ( β ) I * μ ( β ) ] 2 ⩽ C I * μ ( α ) < ∞ , α ∈ T n , on the associated Bergman tree T n . Combined with recent results about interpolating sequences this leads, for this range of σ, to a characterization of universal interpolating sequences for B 2 σ 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 H n 2 = B 2 1 / 2 . We show that μ is a Carleson measure for B 2 1 / 2 if and only if both the simple condition 2 d ( α ) I * μ ( α ) ⩽ C , α ∈ T n , and the split tree condition ∑ k ⩾ 0 ∑ γ ⩾ α 2 d ( γ ) − k ∑ ( δ , δ ′ ) ∈ G ( k ) ( γ ) I * μ ( δ ) I * μ ( δ ′ ) ⩽ C I * μ ( α ) , α ∈ T n , 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.