Carleson measures and interpolating sequences for Besov spaces on complex balls
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Bibliographic Information
Carleson measures and interpolating sequences for Besov spaces on complex balls
(Memoirs of the American Mathematical Society, no. 859)
American Mathematical Society, c2006
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Note
"Volume 182, number 859 (third of 4 numbers)"
Bibliography: p. 163
Description and Table of Contents
Description
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 Boe 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 $1\less p \less \infty$ 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
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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