Let $X=M \times \{0,1\}$ with $M$ a smooth compact manifold without boundary.

Define the fractional Sobolev space $H^{\frac 12}(X) = (L^2(X), H^1(X))_{\frac 12}$, as the real interpolation space midway between $H^1$ and $L^2$.

My question is: is it true that $$\lVert u \rVert_{H^{\frac 12}(X)}^2 = \lVert u(\cdot,0) \rVert_{H^{\frac 12}(M)}^2 + \lVert u(\cdot,1) \rVert_{H^{\frac 12}(M)}^2$$ whenever $u \in H^{\frac 12}(X)$? The norm is defined through the interpolation method (say the J method).

It seems intuitively true but I don't see how to make it rigorous.??

Let $X=M \times \{0,1\}$ with $M$ a smooth compact manifold without boundary.

Define the fractional Sobolev space $H^{\frac 12}(X) = (L^2(X), H^1(X))_{\frac 12}$, as the real interpolation space midway between $H^1$ and $L^2$.

Is it true that $$\lVert u \rVert_{H^{\frac 12}(X)}^2 = \lVert u(\cdot,0) \rVert_{H^{\frac 12}(M)}^2 + \lVert u(\cdot,1) \rVert_{H^{\frac 12}(M)}^2$$ whenever $u \in H^{\frac 12}(X)$? The norm is defined through the interpolation method (say the J method).

It seems intuitively true but I don't see how to make it rigorous?