Mixed-norm spaces and interpolation

Let D be a bounded strictly pseudoconvex domain of ${\displaystyle {ℂ}^n}$ with smooth boundary. We consider the weighted mixed-norm spaces ${\displaystyle A_{\delta ,k}^{p,q}\left(D\right)}$ of holomorphic functions with norm ${\displaystyle \parallel f{\parallel }_{p,q,\delta ,k}={\left(\sum _{|\alpha |\le k}{\int }_0^{r_0}{\left({\int }_{\partial D_r}{\left|D^{\alpha }f\right|}^{p}d{\sigma }_r\right)}^{q/p}{r}^{\delta q/p-1}dr\right)}^{1/q}}$. We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces ${\displaystyle A_{\delta ,k}^p\left(D\right)}$ and we give results about real and complex interpolation between them. We apply these results to prove that ${\displaystyle A_{\delta ,k}^{p,q}\left(D\right)}$ is the intersection of a Besov space ${\displaystyle B_s^{p,q}\left(D\right)}$ with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm spaces.