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Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls
N. Arcozzi, Università di Bologna, Italy, R. Rochberg, Washington University, St. Louis, MO, and E. Sawyer, McMaster University, Hamilton, ON, Canada
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Memoirs of the American Mathematical Society
2006; 163 pp; softcover
Volume: 182
ISBN-10: 0-8218-3917-9
ISBN-13: 978-0-8218-3917-1
List Price: US$65
Individual Members: US$39
Institutional Members: US$52
Order Code: MEMO/182/859
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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

  • 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
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