Mixed-norm spaces and interpolation

Let D be a bounded strictly pseudoconvex domain of $ℂ^n$ with smooth boundary. We consider the weighted mixed-norm spaces $A_{δ,k}^{p,q}(D)$ of holomorphic functions with norm $∥f∥_{p,q,δ,k} = (sum_{|α|≤k} \int_{0}^{r_0} (\int_{∂D_{r}} |D^{α} f|^p dσ_{r})^{q//p} r^{δq//p-1} dr)^{1//q}$. We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces $A_{δ,k}^{p}(D)$ and we give results about real and complex interpolation between them. We apply these results to prove that $A_{δ,k}^{p,q}(D)$ is the intersection of a Besov space $B_{s}^{p,q}(D)$ with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm spaces.