A Carleson type condition for interpolating sequences in the Hardy spaces of the ball of \(\mathbb{C}^{n}\) (Q2271358)

The following theorem concerning interpolating property in the ball \({\mathbb B}\) in \({\mathbb C}^n\) is proved. Theorem. Let \(S \subset {\mathbb B}\) be dual in \(H^p({\mathbb B})\), then \(S\) is \(H^s({\mathbb B})\) an interpolating with linear extension property, for any \(s\in [1, p[\). Here, a sequence \(S \subset {\mathbb B}\) is \(H^s({\mathbb B})\), interpolating for \(1 < p < \infty\) with interpolating constant \(C_I > 0\) if \[ \forall \lambda\in l^p,\;\exists \in H^p({\mathbb B})\text{ such that } \forall a\in S, f(a) = \lambda_a \|k_a\|_{p^{\prime}}\; {\text{and}}\; \|f\|_p \leq C_I \|\lambda\|_p, \] \(k_a = \frac{1}{(1 - \overline{a} \cdot z)^n}\); the sequence \(S\) has the linear extension property if there is a bounded linear operator \(E : l^p \rightarrow H^p({\mathbb B})\) such that \(\| E\| < \infty\) and \(\lambda\in l^p\), \( E\lambda\) interpolates the sequence \(\lambda\) in \(H^p({\mathbb B})\) on \(S\).