If $\int_Uf(x,z)dz$ and $\int_Ug(z,y)dz$ are finite, is $\int_Uf(x,z)g(z,y)dz$ finite for all $x,y\in U$?

Question

If $\int_Uf(x,z)dz$ and $\int_Ug(z,y)dz$ are finite, is $\int_Uf(x,z)g(z,y)dz$ finite for all $x,y\in U$?

At first I tried using the schwartz's inequality, but didn't reach anywhere. Any help on this would be great. Thanks.

Answer 1

Consider \begin{align} f(x, z) = \frac{x}{\sqrt{|z|}} \ \text{ and } \ \ g(z, y) = \frac{y}{\sqrt{|z|}} \end{align} and $U=[-1, 1]$. Observe \begin{align} \int^1_{-1} f(x, z)\ dz = 4x \ \ \text{ and } \ \ \int^1_{-1} g(z, y)\ dz = 4y \end{align} but \begin{align} \int^1_{-1} \frac{xy}{|z|}\ dz = \infty. \end{align}

Answer 2

If $U=[0,1]$ an $f(x,z)=g(z,y) =\frac 1 {\sqrt z}$ for $z \neq 0$, $f(x,z)=g(z,y) =0$ for $z=0$, then $f$ and $g$ are integrable on $U$ but $fg$ is not.