Let $(0,L)$ be an open interval of $\mathbb{R}$. We know that if $f \in H^1(0,L)$ and $f(0) = 0$ then $||f||_{L^\infty(0,L)} \leq \sqrt{2}||f||_{L^2(0,L)}^\frac12 ||f'||_{L^2(0,L)}^\frac12$ (this is a direct consequence of the fact that $f(x)^2-f(0)^2 = \int_0^x2f(y)f'(y)\,dy$).
Suppose now that the function $f$ belongs to a fractional Sobolev space, i.e. $f\in H^s(0,L)$, where, for $\frac12<s\leq1$, we define $$ [f]_{H^s(0,L)}^2 = \int_0^L\int_0^L \frac{|f(y)-f(x)|^2}{|y-x|^{1+2s}}\,dydx $$ and $H^s(0,L) = \{ f \in L^2(0,L) \,|\, [f]_{H^s(0,L)}^2 < \infty\}$. Supposing again that $f(0)=0$, can we still find a similar estimate? Something like $||f||_{L^\infty(0,L)} \leq C||f||_{L^2(0,L)}^\alpha [f]_{H^s(0,L)}^\beta$ for some suitable exponents $\alpha,\beta$?