We generalize some criteria of boundedness of $𝐋$-index in joint variables for in a bidisc analytic functions. Our propositions give an estimate the maximum modulus on a skeleton in a bidisc and an estimate of $(p+1)$th partial derivative by lower order partial derivatives (analogue of Hayman’s theorem).

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...

This paper concerns difference equations $y(x+1)=G(x,y)$ where $G$ takes values in ${\mathbf{C}}^n$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty$ in $\mathbf{C}$ and holomorphic in a neighborhood of 0 in ${\mathbf{C}}^n$. It is shown that under certain conditions on the linear part of $G$, formal power series solutions in $x^{-1/p},p\in \mathbf{N},$ are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.