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

## Arcozzi, N.; Rochberg, R.; Sawyer, E.

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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 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.

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## 1.Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls (Memoirs of the American Mathematical Society,)

Editore: American Mathematical Society (2006)
ISBN 10: 0821839179 ISBN 13: 9780821839171
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Sequitur Books
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Descrizione libro American Mathematical Society, 2006. Paperback. Condizione libro: New. Brand new. We distribute directly for the publisher. 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 $1\less p \less \infty$ 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. Codice libro della libreria 1005250203

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## 2.Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls (Memoirs of the American Mathematical Society)

Editore: American Mathematical Society (2006)
ISBN 10: 0821839179 ISBN 13: 9780821839171
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Descrizione libro American Mathematical Society, 2006. Paperback. Condizione libro: New. book. Codice libro della libreria 0821839179

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## 3.Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls (Memoirs of the American Mathem)

Editore: American Mathematical Society (2006)
ISBN 10: 0821839179 ISBN 13: 9780821839171
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Descrizione libro American Mathematical Society, 2006. Paperback. Condizione libro: New. Codice libro della libreria DADAX0821839179

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## 4.Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls (Memoirs of the American Mathematical Society,)

Editore: American Mathematical Society (2006)
ISBN 10: 0821839179 ISBN 13: 9780821839171
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Revaluation Books
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Descrizione libro American Mathematical Society, 2006. Paperback. Condizione libro: Brand New. 163 pages. 9.20x6.90x0.40 inches. In Stock. Codice libro della libreria 0821839179

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