Suppose we have the following implicit functional inequality,
$$ |h(x,t)| \leq \int_0^t \int_{\mathbb{R}}\Phi(x-y,s-t)|h(y,s)| \, dy \, ds $$
where $\Phi$ is the Gaussian kernel, $\Phi(x,t) = \dfrac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$. Is there anything we can conclude about $h$ if the only condition on $h$ is that for any $t > 0$, $h \in C^1(\mathbb{R})$?
Perhaps there's a fixed point argument we can use here? Ideally, I would like to show that the only solution here is $h \equiv 0$. Is the $C^1$ condition enough or should there be another, like a decay condition, $\lim\limits_{x\rightarrow\pm\infty}|h(x,t)| = 0$?