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

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 $1less p less infty$ 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 $2+frac{1}{n-1}leq pleq2n$, 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.