# Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls

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*N. Arcozzi; R. Rochberg; E. Sawyer*

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.

#### Table of Contents

# Table of Contents

## Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls

- Contents v6 free
- CARLESON MEASURES AND INTERPOLATING SEQUENCES 18 free
- 1. Introduction 18
- 2. A tree structure for the unit ball B[sub(n)] in C[sup(n)] 1118
- 3. Carleson measures 2128
- 4. Pointwise multipliers 3441
- 5. Interpolating sequences 3643
- 6. An almost invariant holomorphic derivative 8087
- 7. Besov spaces on trees 9097
- 8. Holomorphic Besov spaces on Bergman trees 108115
- 9. Completing the multiplier interpolation loop 155162
- 10. Appendix 160167

- Bibliography 163170