  If $\mathcal{H}$ denotes a Hilbert space of analytic functions on a region
$\Omega \subseteq \mathbb{C}^d$, then the weak product is defined by
$$\mathcal{H}\odot\mathcal{H}=\left\{h=\sum_{n=1}^\infty f_n g_n :
\sum_{n=1}^\infty \|f_n\|_{\mathcal{H}}\|g_n\|_{\mathcal{H}} <\infty\right\}.$$
We prove that if $\mathcal{H}$ is a first order holomorphic Besov Hilbert space
on the unit ball of $\mathbb{C}^d$, then the multiplier algebras of
$\mathcal{H}$ and of $\mathcal{H}\odot\mathcal{H}$ coincide.
