Mixed-norm spaces and interpolation

Joaquín M. Ortega ; 

Studia Mathematica, Tome 109 (1994) no. 3, pp. 233-254

Résumé

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

Détail

DOI : 10.4064/sm-109-3-233-254

Keywords: analytic functions, mixed-norm spaces, real interpolation, complex interpolation, Besov spaces of holomorphic functions