If $u \in H^{\frac 12}(\Omega)$ and $c \in \mathbb{R}$ is $(u-c)^+ \in H^{\frac 12}(\Omega)$ too?

Question

If $u \in H^{\frac 12}(\Omega)$ and $c$ is a constant, is the function $$(u-c)^+ \in H^{\frac 12}(\Omega)?$$

Here $(x)^+$ is $x$ when $x > 0$ and $0$ otherwise.

If it were $H^1$ then it is a true fact. Can I cite this from somewhere? Thanks.

Answer 1

Assume $c=0$. The desired result follows easily when we consider the Gagliardo norm on $H^{\frac 12}$ and from the identity $|u^+(x)-u^+(y)| \leq |u(x)-u(y)|$ for any $u \in H^{\frac 12}$.

Am I right?