For random variables $A, Y, U$, they have density functions $p(a,u,y)$. Now I know $p(u|a)\in L^2(P_a)$ for $\forall u$, namely, $$ \int{\left( p\left( u|a \right) \right) ^2p\left( a \right) da}<\infty $$ My question is: Is $p(y|a)\in L^2(P_a)$ for $\forall y$ satisfied? If not, what other conditions do I need to satisfy this integrability?
My idea is to rely on the decomposition of $p(y|a)$ to prove because we have
$$ p\left( y|a \right) =\int{p\left( y|u,a \right) p\left( u|a \right) du} $$
We have some additional conditions in hand: $A\perp Y|U$.