Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.

Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that $u(0) \in H^{k-1/2}(S^2)$? Or does it hold that $u(0) \in H^{k}(S^2)$?

If you know any references that study the trace theorem for such spaces, please send them over.

The motivation for this question is the following. Consider the ODE $$u'(r) + u(r) = F, \quad r \in [0,1]$$ $$u(0) = u_0,\quad u(1) = u_1$$ where $F \in L^2([0,1]; H^k(S^2))$, $u_0,u_1 \in H^{k-1/2}(S^2)$. Is it true that there exists a unique solution to the above system $u \in H^1((0,1);H^{k}(S^2))$? This is clearly true when $u_0, u_1 \in H^k(S^2)$, but I'm not sure if it is when $u_0,u_1 \in H^{k-1/2}(S^2)$.