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 $u(0) \in H^{k}(S^2)$?

What if we in addition know that $u' \in L^2[0,1]; H^{k'}(S^2))$ for some $k'\leq k$?

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