Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$ for all $u\in C^{\infty}_0((-1,1))$?
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$ for all $u\in C^{\infty}_0((-1,1))$?
It is not true. Start with a function $u$ which vanishes for $x<0$ and is equal to $1$ for $0 \leq x \leq \frac 12$ and then smooth from $x \geq \frac 12$. The Fourier coefficients behave like $1/n$ so that $ u \not \in H^{\frac 12}$. However it can be approximated in $L^2$ with Lipschitz functions $u_\epsilon$ having $\int x |u_\epsilon'|^2 \leq C$ joining $(0,0)$ with $(\epsilon, 1)$ by a straight line. The $H^{\frac 12}$ norms of $u_\epsilon$ blow up since $u \not \in H^{\frac 12}$.