Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$ \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u \rVert_{L^2(M)}, $$ and $$ \inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2(M)} \le \lVert \nabla u \rVert_{L^2(M)}. $$ how do I obtain $$|Tu|_{H^{\frac 12}(\partial M)} \leq C\lVert \nabla u \rVert_{L^2(M)}?$$ (The $C$ here can be a different constant)
This is apparently true from an answer on Mathoverflow.. I asked the author but he hasn't responded. Does anyone know how to get it??!
Since constant functions have zero $H^{1/2}$ seminorm, it follows that $$\vert T u \vert_{H^{1/2} (\partial M)} = \vert (T u) -c \vert_{H^{1/2} (\partial M)} = \vert T (u -c) \vert_{H^{1/2} (\partial M)}\tag{1}$$ for every $c\in\mathbb{R}$. (Trace operator commutes with adding a constant, because the trace of a constant function is that constant function.)
Therefore, the first inequality you cited yields $$\vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u-c \rVert_{L^2(M)}\tag{2}$$ Take infimum over $c$ and use the Poincaré inequality (which is missing a constant in your question): $$\vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + C'\lVert \nabla u \rVert_{L^2(M)}\tag{3}$$