I would like to know whether multiplication defines a bounded map $$H^{1/2} \otimes H^{1/2} \to H^{-1/2}$$ dimension of the domain is $3$.

I have checked two different sources and one said that it works but the other that this map is bounded as a map $$H^{1/2} \otimes H^{1/2} \to H^{s}$$ where $s < -\frac{1}{2}$. That is why I am confused.

Notation: $H^{1/2} = W^{1/2, 2} = L^2_{1/2}$

I would be satissfied with a source i can rely on.