$L^2$ bound and interpolation of Hölder norm

Question

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 $F$.

Answer 1

Your conjecture is correct. Define the bilinear operator $T$ by $$ T(f,g)(x)=\int_{\mathbb{R}}f(t+x)g(t-x)\,dt. $$ As you have shown, this operator is bounded from $L^2\times L^2\to C^0$ and from $H^1\times H^1\to C^1$. It follows from bilinear real interpolation theory (see e.g. Zafran, A multilinear interpolation theorem) that $T$ is also bounded from $(L^2,H^1)_{\theta,p}\times (L^2,H^1)_{\theta,q}\to (C^0,C^1)_{\theta,r}$ whenever $\theta\in (0,1)$ and $p,q,r\in [1,\infty]$ satisfiy $1/r+1=1/p+1/q$.

If we take $r=\infty$ and $p=q=2$ and use $(L^2,H^1)_{\gamma,2}=H^\gamma$, $(C^0,C^1)_{\gamma,\infty}=C^\gamma$, we obtain the desired boundedness from $H^\gamma\times H^\gamma\to C^\gamma$.