Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of $\textbf {C}^{n}$ $(n>1)$

Abstract:

Let ${A^{p,\alpha }}(B),{A^{q,\alpha }}(B)$ and ${A^{l,\alpha }}(B)$ be weighted Bergman spaces on the unit ball of ${{\text {C}}^{n}} (n > 1)$. We prove: Theorem 1. If $1/l = 1/p + 1/q$ then ${A^{p,\alpha }}(B) \cdot {A^{q,\alpha }}(B)$ is of first category in ${A^{l,\alpha }}(B)$. Theorem 2. Theorem 1 holds for Hardy spaces in place of weighted Bergman spaces. We also show that Theorems 1 and 2 hold for the polydisc ${U^n}$ in place of $B$.