We characterize Carleson measures for the analytic Besov spaces \(B_{p}\) on the unit ball \(\mathbb{B}_{n}\) in \(\mathbb{C}^{n}\) in terms of a discrete tree condition on the associated Bergman tree \(\mathcal{T}_{n}\). We also characterize the pointwise multipliers on \(B_{p}\) in terms of Carleson measures. We then apply these results to characterize the interpolating sequences in \(\mathbb{B}_{n}\) for \(B_{p}\) and their multiplier spaces \(M_{B_{p}}\), generalizing a theorem of Böe in one dimension. The interpolating sequences for \(B_{p}\) and for \(M_{B_{p}}\) are precisely those sequences satisfying a separation condition and a Carleson embedding condition. These results hold for \(1 < p < \infty\) with the exceptions that for \(2+\frac{1}{n-1}\leq p < \infty\), the necessity of the tree condition for the Carleson embedding is left open, and for \(2+\frac{1}{n-1}\leq p\leq2n\), the sufficiency of the separation condition and the Carleson embedding for multiplier interpolation is left open; the separation and tree conditions are however sufficient for multiplier interpolation. Novel features of our proof of the interpolation theorem for \(M_{B_{p}}\) include the crucial use of the discrete tree condition for sufficiency, and a new notion of holomorphic Besov space on a Bergman tree, one suited to modeling spaces of holomorphic functions defined by the size of higher order derivatives, for necessity.

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Table of Contents

• Introduction
• A tree structure for the unit ball \(\mathbb{B}_n\) in \(\mathbb{C}^n\)
• Carleson measures
• Pointwise multipliers
• Interpolating sequences
• An almost invariant holomorphic derivative
• Besov spaces on trees
• Holomorphic Besov spaces on Bergman trees
• Completing the multiplier interpolation loop
• Appendix
• Bibliography