Consider the function

$$F(x):=\int_{\mathbb R} f(t+x)f(t-x) \ dt .$$

Clearly, we have by Cauchy-Schwarz

$$\vert F(x) \vert\le \Vert f \Vert^2_{L^2} $$ $$\vert F'(x)\vert\le 2\Vert f' \Vert_{L^2} \Vert f \Vert_{L^2} \le 2\Vert f \Vert_{H^1}^2$$

where $H^1$ is the $L^2$-Sobolev space of order 1.

This shows that to bound the sup norm of $F$ we require $f \in L^2$ and to bound the $C^1$ norm of $F$ it suffices to have $f \in H^1.$

I wonder now if it is true that the Hölder norm $C^{\gamma}$ with $\gamma \in (0,1)$ can be bounded by the $H^{\gamma}$ norm of $f$ and if that is not the case, I would be curious to learn what the correct interpolation space for $f$ is to bound the Hölder norm of $F$.

require$f\in L^2$ or $f\in H^1$, they merely show that such conditions aresufficient. $\endgroup$ – Yemon Choi Aug 22 at 23:25