Let $K\in L^1 (\Bbb {R}, C_0 (\Bbb {R })) $ be given.
$f\in L^1(\Bbb{R},C_0(\Bbb{R}))$ means that $\int_{\Bbb{R}}||f(x)||dx<\infty$.
I will denote by $K(x,y):=(K(x))(y)$.
I want to show that $\int_{\Bbb {R}^2} |K (x-y, y)|^2 dxdy $ is finite.
What do we know:
By definition, $\int_{\Bbb{R}}||K(x)||_{\infty}dx<\infty$.
Thus, for every fixed $y\in \Bbb{R}$, $\int_{\Bbb{R}}|K(x-y,y)|dx<\infty$.
Unfortunately, I do not see how to approach to it.
I'd just note that this is a part of another question that I have, so I'm not sure that this is even true.
Any idea (or counterexample) would help!
Thank you.
Note that $\iint|K(x-y,y)|^2dxdy=\int\left(\int|K(x-y,y)|^2dx\right)dy=\int\left(\int|K(x,y)|^2dx\right)$ so there is no need to consider $K(x-y,y)$.
The fact that $K(x,\cdot)$ is only $C_0$ should suggest that it's not going to happen. For instance, we could take
$$K(x,y):=\frac{e^{-|x|}}{\log(2+|y|)}.$$
For each $x$ we have $K(x,\cdot)\in C_0(\mathbb R)$ and $\|K(x,\cdot)\|_\infty =e^{-|x|}/\log2$ which is integrable, so $K$ satisfies the given assumptions, and yet
$$\iint_{\mathbb R^2}|K(x,y)|^2\,dxdy=\left(\int_\mathbb R e^{-2|x|}\,dx\right)\left(\int_\mathbb R\log^{-2}(2+|y|)\,dy\right)=+\infty.$$