  We introduce two new conditions for bounded domains, namely
$A^p$-completeness and boundary blow down type, and show that, for two bounded
domains $D_1$ and $D_2$ that are $A^p$-complete and not of boundary blow down
type, if there exists a linear isometry from $A^p(D_1)$ to $A^{p}(D_2)$ for
some real number $p>0$ with $p\neq $ even integers, then $D_1$ and $D_2$ must
be holomorphically equivalent, where for a domain $D$, $A^p(D)$ denotes the
space of $L^p$ holomorphic functions on $D$.
