We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain $D$ of dimension $N$. If $\Sigma \subset bD$ is a smooth manifold of dimension $N$ and a maximum modulus set, then it admits a unique foliation by compact interpolation manifolds. There is a semiglobal converse in the real analytic case. Two functions in $A^2\left(D\right)$ with the same smooth $N$-dimensional maximum modulus set are analytically related and are polynomially related if a certain homology class in $H_1(D,\mathbf{R})$ vanishes or if $\bar{D}\subset {\mathbf{C}}^N$ is polynomially...

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