multiplying a $L^2(\Omega)$ function by a test-function

Question

Suppose we have $f \in L^2(\Omega)$ and $v \in D(\Omega)$ a test-function. If we multiply $f$ by $v$, the $(fv)$ function belongs to which space?

Answer 1

$fv \in L^{2}$ because $v$ is bounded; $fv \in L^{1}$ by Holder's inequality: $\int |fv| \leq \sqrt {\int |f|^{2} } \sqrt {\int |v|^{2} }$.