The author studies \(Q_p\) spaces, which interpolate the well known BMOA and the Bloch space on the unit ball \(B\) of \(\mathbf C^n\). The spaces \(Q_p\) are defined by certain integral conditions involving the invariant Green’s function and the invariant gradient. In a previous work [C. H. Ouyang, W. S. Yang and R. H. Zhao, Pac. J. Math. 182, 69-99 (1998; Zbl 0893.32005)], it is shown that \(Q_p\) is trivial for \(0<p\leq\frac{n-1}{n}\) or \(p\geq \frac{n}{n-1}\). A positive Borel measure \(\mu\) on \(B\) is called a \(p\)-Carleson measure if \(\sup \mu(\beta)\delta^{-np}<\infty\) where the sup is taken over all standard Carleson balls \(\beta=\beta(\zeta,\delta)=\{z\in B: |1-<z,\zeta>|<\delta\}\), \(\zeta\in\partial B\).
The author proves a Carleson measure characterization of \(Q_p\) spaces: Let \(\frac{n-1}{n}<p<\frac{n}{n-1}\) and \(f\) be a function holomorphic on \(B\). Then, \(f\in Q_p\) if and only if \(|\widetilde\nabla f(z)|^2(1-|z|^2)^{np} d\lambda(z)\) is a \(p\)-Carleson measure where \(\widetilde\nabla f\) is the invariant gradient of \(f\) and \(\lambda\) is the invariant measure on \(B\).
This result extends the same type of characterizations for BMOA and the Bloch space.