On the Korenblum maximum principle

Abstract

We prove that Korenblum maximum principle holds in mixed norm spaces $H^{p,q,a}$ when $1 \leq p \leq q < \infty$ and does not hold when $0 <q <1$. As an immediate consequence, we obtain that Korenblum maximum principle holds in weighted Bergman spaces $A^p_{\gamma}$, if and only if $p \geq 1$. This improves the well known result which states that Korenblum maximum principle holds in Bergman spaces $A^p$ if and only if $p \geq 1$.