Bergman projections and duality in weighted mixed-norm spaces of analytic functions
The Bergman operator \(T_ \alpha\) on the unit ball \(U\) of \(C^ N\) is given by \[ T_ \alpha f(z)={N+\alpha \choose N} \int_ U(1-\langle z,w \rangle)^{-N-1-\alpha} f(w) c_ \alpha(1-| w |^ 2)^ \alpha dm \] where \(c_ \alpha\) is a normalising constant. The operator \(T^*_ \alpha\) is obtained by replacing \(1-\langle z,w \rangle\) with its norm. Békollé studied the continuity of these operators on weighted \(L_ p\)-spaces. In this article his results are extended to mixed norm spaces with radial symmetry, i.e. with norms of the form \[ \| f \|=\int_ 0^ 1\| f_ r \|^ q_ pw (r)dr \] where \(w\) is a weight function on (0,1) and \(\| f_ r \|_ p\) is the unweighted \(L_ p\) norm on the unit sphere. This generalised result is used to obtain some information on mixed norm spaces of holomorphic functions, the most important being the characterisation of the dual of some of these spaces.
scientific article; zbMATH DE number 63012