I wonder if the following holds:
Let $\Omega:= \mathbb{T}^2 \times (-h,0)$, $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ and denote by $L_{1,\text{loc}}(\Omega;\mathbb{K})$ the space locally integrable functions on $\Omega$ over $\mathbb{K}$. If now $u \in L_{1,\text{loc}}(\Omega;\mathbb{K})$ and $\partial_3 = 0$, then there exists $\widetilde{u}$ in $L_1(\mathbb{T}^2)$ such that $u = \widetilde{u}$ for a.e. on $\Omega$.
Intuitively, I would say that this holds. Before I start proving this result, I wanted to know if I'm correct with my intuition.
I probably could have formulated the above result in a more general context, but since I'm still a little unsure about how to work with locally integrable functions on $\Omega= \mathbb{T}^2 \times (-h,0)$, I chose this context.
Could eventually someone help me out here?
Thank you in advance!