# Carleson measures and interpolating sequences for Besov spaces on complex
balls

### About this Title

**N. Arcozzi**, **R. Rochberg** and **E. Sawyer**

Publication: Memoirs of the American Mathematical Society

Publication Year
2006: Volume 182, Number 859

ISBNs: 978-0-8218-3917-1 (print); 978-1-4704-0463-5 (online)

DOI: http://dx.doi.org/10.1090/memo/0859

MathSciNet review: 2229732

MSC: Primary 46E15; Secondary 30H05, 32A37, 46E22, 46E35

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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

**Chapters**

- 1. Introduction
- 2. A tree structure for the unit ball $\mathbb {B}_n$ in $\mathbb {C}^n$
- 3. Carleson measures
- 4. Pointwise multipliers
- 5. Interpolating sequences
- 6. An almost invariant holomorphic derivative
- 7. Besov spaces on trees
- 8. Holomorphic Besov spaces on Bergman trees
- 9. Completing the multiplier interpolation loop
- 10. Appendix